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      <title>From Oberon to Go: A Short Family Tree</title>
      <link>/post/oberon-to-golang/</link>
      <pubDate>Sun, 02 Feb 2025 12:00:00 +0000</pubDate>
      <guid>/post/oberon-to-golang/</guid>
      <description>&lt;p&gt;A compact overview of the Oberon family of languages and how their ideas lead to Go. The lineage runs from Pascal and Modula-2 through Oberon and its variants to Go, with shared emphasis on simplicity, strong typing, and clear semantics.&lt;/p&gt;
&lt;h2 id=&#34;oberon-1986--niklaus-wirth--jurg-gutknecht&#34;&gt;Oberon (1986) — Niklaus Wirth &amp;amp; Jurg Gutknecht&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;Evolved from Pascal and Modula-2; designed for simplicity and efficiency.&lt;/li&gt;
&lt;li&gt;Strong typing, modular programming, integrated development environment.&lt;/li&gt;
&lt;li&gt;Used in software and hardware (e.g. Ceres workstation); part of Project Oberon at ETH Zurich.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&#34;oberon-2-1991--hanspeter-mössenböck&#34;&gt;Oberon-2 (1991) — Hanspeter Mössenböck&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;Extension of Oberon with object-oriented features.&lt;/li&gt;
&lt;li&gt;Type-bound procedures, read-only export of variables and methods.&lt;/li&gt;
&lt;li&gt;Improved type safety and reusability.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&#34;modula-3-1988--luca-cardelli-james-donahue-greg-nelson-paul-rovner-andrew-birrell&#34;&gt;Modula-3 (1988) — Luca Cardelli, James Donahue, Greg Nelson, Paul Rovner, Andrew Birrell&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;Evolution of Modula-2: simplicity, safety, systems programming.&lt;/li&gt;
&lt;li&gt;Garbage collection, exception handling, strong type system with modules.&lt;/li&gt;
&lt;li&gt;Concurrency and generic programming.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&#34;active-oberon-1997--jurg-gutknecht&#34;&gt;Active Oberon (1997) — Jurg Gutknecht&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;Extends Oberon with active objects and concurrency.&lt;/li&gt;
&lt;li&gt;Combines object-oriented and system-programming features.&lt;/li&gt;
&lt;li&gt;Runtime and language support for concurrent processes.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&#34;zonnon-2005--jurg-gutknecht&#34;&gt;Zonnon (2005) — Jurg Gutknecht&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;Draws on Oberon, Active Oberon, and Modula-2.&lt;/li&gt;
&lt;li&gt;Component-oriented programming, concurrency and parallelism.&lt;/li&gt;
&lt;li&gt;Refined syntax and semantics.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&#34;oberon-07-2007--niklaus-wirth&#34;&gt;Oberon-07 (2007) — Niklaus Wirth&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;Simplified, modernized Oberon.&lt;/li&gt;
&lt;li&gt;Removed deprecated features; clearer syntax and safer semantics.&lt;/li&gt;
&lt;li&gt;Streamlined compiler and language report.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&#34;component-pascal-1997&#34;&gt;Component Pascal (1997)&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Clemens Szyperski:&lt;/strong&gt; Component Pascal at Oberon Microsystems (1991–1997); component-based software; author of &lt;em&gt;Component Software: Beyond Object-Oriented Programming&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;K. John Gough:&lt;/strong&gt; Co-developed Gardens Point Component Pascal for .NET.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Derived from Oberon-2 with stronger object-oriented and component-oriented features; designed for component-based software engineering and .NET integration (Gardens Point).&lt;/p&gt;
&lt;h2 id=&#34;go--robert-griesemer-rob-pike-ken-thompson&#34;&gt;Go — Robert Griesemer, Rob Pike, Ken Thompson&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;Designed at Google (from 2007); open-sourced 2009, first stable release 2012.&lt;/li&gt;
&lt;li&gt;Influenced by the simplicity and efficiency of Oberon and the Wirth tradition.&lt;/li&gt;
&lt;li&gt;Targets multicore and networked systems: goroutines, channels, fast compilation.&lt;/li&gt;
&lt;li&gt;Garbage-collected, statically typed, simple syntax.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&#34;how-it-connects&#34;&gt;How it connects&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Simplicity and efficiency:&lt;/strong&gt; Go inherits Oberon’s focus on clear, maintainable code.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Concurrency:&lt;/strong&gt; Builds on ideas from Modula-3 and Active Oberon.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Type safety:&lt;/strong&gt; Strong static typing throughout the family.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Components:&lt;/strong&gt; Component Pascal’s component-oriented design is part of the same design culture.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Implementation and runtimes:&lt;/strong&gt; Work on JIT and dynamic optimization (e.g. Michael Franz) influenced modern compilers and runtimes; Go prioritizes fast compilation and efficient execution in that spirit.&lt;/li&gt;
&lt;/ul&gt;
</description>
    </item>
    
    <item>
      <title>From Oberon to Go in Faces</title>
      <link>/post/oberon-in-faces/</link>
      <pubDate>Sat, 01 Feb 2025 12:00:00 +0000</pubDate>
      <guid>/post/oberon-in-faces/</guid>
      <description>&lt;p&gt;Short portraits of people in the Oberon and Modula lineage—language designers, compiler writers, and researchers—whose work leads, in spirit or in person, toward Go.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Niklaus Wirth&lt;/strong&gt;&lt;br&gt;
Inventor of Pascal (1970), Modula-2 (1978), and Oberon (1986). He also designed Oberon-07 (2007). His work stressed simplicity and efficiency and influenced many modern languages.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Hanspeter Mössenböck&lt;/strong&gt;&lt;br&gt;
Developed Oberon-2 (1991) and the compiler generator Coco/R. His work on compiler construction and development tools has had a lasting impact.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Jurg Gutknecht&lt;/strong&gt;&lt;br&gt;
Co-developer of Oberon. Designed Active Oberon (1997) and Zonnon (2005), extending the Oberon family and exploring concurrency and new language paradigms.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Michael Franz&lt;/strong&gt;&lt;br&gt;
His 1994 PhD thesis was on code generation on the fly. He pioneered techniques in just-in-time (JIT) compilation and dynamic optimization that influenced modern runtimes and language implementations.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Clemens Szyperski&lt;/strong&gt;&lt;br&gt;
Worked on Component Pascal at Oberon Microsystems (1991–1997). His PhD focused on object-orientation in operating systems. He is known for component-based software engineering and the book &lt;em&gt;Component Software: Beyond Object-Oriented Programming&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Cuno Pfister&lt;/strong&gt;&lt;br&gt;
Known for embedded systems and the Internet of Things (IoT). He has authored books and developed tools and frameworks in these areas.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Josef Templ&lt;/strong&gt;&lt;br&gt;
His PhD thesis focused on metaprogramming in Oberon, contributing to the use of metaprogramming techniques in practice.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Ralph Sommerer&lt;/strong&gt;&lt;br&gt;
His PhD addressed integrating online documents. He developed Oberon Script—a lightweight compiler and runtime that compiles Oberon to JavaScript for interactive web clients (e.g. JMLC 2006).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;K. John Gough&lt;/strong&gt;&lt;br&gt;
Co-developed Gardens Point Component Pascal for .NET, enabling Component Pascal on the .NET framework.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;J. Stanley Warford&lt;/strong&gt;&lt;br&gt;
Known for computer science education, especially his textbooks on computer architecture and assembly language.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Michael Spivey&lt;/strong&gt;&lt;br&gt;
Known for the Oxford Oberon-2 compiler, contributing to the development and optimization of Oberon-2.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Luca Cardelli, James Donahue, Greg Nelson, Paul Rovner, Andrew Birrell&lt;/strong&gt;&lt;br&gt;
Designers of Modula-3 (1988), an evolution of Modula-2 emphasizing simplicity, safety, systems programming, and generics.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Robert Griesemer&lt;/strong&gt;&lt;br&gt;
Co-designer of Go at Google. His PhD focused on programming languages for vector computers. Go is influenced by Oberon’s simplicity and efficiency and is widely used for its performance and concurrency support, continuing the Wirth tradition.&lt;/p&gt;
</description>
    </item>
    
    <item>
      <title>Solving a Three-Variable Recursion via Generating Functions</title>
      <link>/post/three-var-recursive/</link>
      <pubDate>Tue, 10 Apr 2018 03:14:00 +0000</pubDate>
      <guid>/post/three-var-recursive/</guid>
      <description>&lt;p&gt;This post extends the generating-function technique from the &lt;a href=&#34;/post/two-var-recursive-func/&#34;&gt;two-variable recursion&lt;/a&gt; to a three-variable case. I originally wrote this as an answer to a &lt;a href=&#34;https://math.stackexchange.com/questions/1093271/how-to-solve-this-multivariable-recursion/2730331#2730331&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Math Stack Exchange question&lt;/a&gt;; here it is adapted for the blog with clearer exposition and code.&lt;/p&gt;
&lt;h2 id=&#34;the-problem&#34;&gt;The Problem&lt;/h2&gt;
&lt;p&gt;We want to solve the recurrence&lt;/p&gt;
&lt;p&gt;$a_{n,m,k} = 2a_{n-1,m-1,k-1} + a_{n-1,m-1,k}$
$ + a_{n,m-1,k-1} + a_{n-1,m,k-1}$&lt;/p&gt;
&lt;p&gt;where $m$, $n$, $k$ are nonnegative integers, with boundary conditions:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;$a_{0,0,0} = a_{0,1,0} = a_{1,0,0} = a_{0,0,1} = 1$&lt;/li&gt;
&lt;li&gt;$a_{n,0,0} = a_{0,n,0} = a_{0,0,n} = 0$ for any $n &amp;gt; 1$&lt;/li&gt;
&lt;li&gt;$a_{n,m,k}$ is symmetric in $n$, $m$, $k$&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;A subtlety: $a_{1,0,1}$ is not defined by the recurrence alone, since it would require values like $a_{0,-1,0}$. We take $a_{n,m,k} = 0$ whenever any subscript is negative.&lt;/p&gt;
&lt;h2 id=&#34;the-generating-function&#34;&gt;The Generating Function&lt;/h2&gt;
&lt;p&gt;Define&lt;/p&gt;
&lt;p&gt;$$\Phi(x,y,z) = \sum_{n,m,k \geq 0} a_{n,m,k} \cdot x^n y^m z^k$$&lt;/p&gt;
&lt;p&gt;Using the initial values above, we can write the recurrence including boundary terms as follows:&lt;/p&gt;
&lt;p&gt;$a_{n,m,k} = 2a_{n-1,m-1,k-1} + a_{n-1,m-1,k}$
$ + a_{n,m-1,k-1} + a_{n-1,m,k-1}$
$ + [n=m=k=0] + [n=m=0 \wedge k=1]$
$ + [n=k=0 \wedge m=1] + [m=k=0 \wedge n=1]$&lt;/p&gt;
&lt;p&gt;Substituting the recurrence into the generating function and collecting terms:&lt;/p&gt;
&lt;p&gt;$\Phi(x,y,z) = \sum_{n,m,k} a_{n,m,k} \cdot x^n y^m z^k $
$ = 2\sum_{n,m,k} a_{n,m,k} \cdot x^{n+1} y^{m+1} z^{k+1} $
$   + \sum_{n,m,k} a_{n,m,k} \cdot x^{n+1} y^{m+1} z^k $
$   + \sum_{n,m,k} a_{n,m,k} \cdot x^{n+1} y^m z^{k+1} $
$   + \sum_{n,m,k} a_{n,m,k} \cdot x^n y^{m+1} z^{k+1} $
$   + 1 + x + y + z $
$ = 2 \Phi \cdot x y z + \Phi \cdot x y + \Phi \cdot x z + \Phi \cdot y z + 1 + x + y + z$
$ = \Phi \cdot ( 2 x y z + x y + x z + y z ) + 1 + x + y + z$&lt;/p&gt;
&lt;p&gt;where the boundary terms $1 + x + y + z$ come from $a_{0,0,0}$, $a_{1,0,0}$, $a_{0,1,0}$, $a_{0,0,1}$.&lt;/p&gt;
&lt;p&gt;Solving for $\Phi$:&lt;/p&gt;
&lt;p&gt;$$\Phi(x,y,z) = \frac{1 + x + y + z}{1 - 2xyz - xy - xz - yz}$$&lt;/p&gt;
&lt;h2 id=&#34;from-generating-function-to-closed-form&#34;&gt;From Generating Function to Closed Form&lt;/h2&gt;
&lt;p&gt;Using&lt;/p&gt;
&lt;p&gt;$$\frac{1}{1-\rho} = \sum_{i \geq 0} \rho^i$$&lt;/p&gt;
&lt;p&gt;and the multinomial expansion&lt;/p&gt;
&lt;p&gt;$$(x_1+x_2+x_3+x_4)^N = \sum_{r_1+r_2+r_3+r_4=N} \binom{N}{r_1,r_2,r_3,r_4} x_1^{r_1} x_2^{r_2} x_3^{r_3} x_4^{r_4}$$&lt;/p&gt;
&lt;p&gt;with&lt;/p&gt;
&lt;p&gt;$$\binom{N}{r_1,r_2,r_3,r_4} = \frac{N!}{r_1!\cdot r_2!\cdot r_3!\cdot r_4!}$$&lt;/p&gt;
&lt;p&gt;we expand the denominator of $\Phi$. Let $\rho = 2xyz + xy + xz + yz$. Then&lt;/p&gt;
&lt;p&gt;$$\Phi = \frac{1+x+y+z}{1-\rho} = (1+x+y+z) \sum_{N \geq 0} \rho^N$$&lt;/p&gt;
&lt;p&gt;Expanding $\rho^N$ with the multinomial theorem (and writing $r_4 = N - r_1 - r_2 - r_3$):&lt;/p&gt;
&lt;p&gt;$\sum_{N \geq 0} \rho^N = \sum_{N}(2 x y z + x y + x z + y z)^N $
$ = \sum_{r_1+r_2+r_3+r_4=N} \binom{N} {r_1,r_2,r_3,r_4} (2 x y z)^{r_1} \cdot (x y)^{r_2} \cdot (x z)^{r_3} \cdot (y z)^{r_4}$
$ = \sum_{r_1+r_2+r_3+r_4=N} \binom{N} {r_1,r_2,r_3,r_4} 2^{r_1} x^{r_1+r_2+r_3} y^{r_1+r_2+r_4} z^{r_1+r_3+r_4}$
$ = \sum_{r_1+r_2+r_3 \leq N} \binom{N} {r_1,r_2,r_3, N-r_1-r_2-r_3} 2^{r_1} x^{r_1+r_2+r_3} y^{N-r_3} z^{N-r_2}$&lt;/p&gt;
&lt;p&gt;So we have&lt;/p&gt;
&lt;p&gt;$$ \Phi = (1 + x + y + z) \sum_{r_1+r_2+r_3 \leq N} \binom{N} {r_1,r_2,r_3, N-r_1-r_2-r_3} 2^{r_1} x^{r_1+r_2+r_3} y^{N-r_3} z^{N-r_2}$$&lt;/p&gt;
&lt;p&gt;Extracting the coefficient of $x^n y^m z^k$ gives the closed form. The full expression has four sums (from the numerator $1+x+y+z$):&lt;/p&gt;
&lt;p&gt;$$a_{n,m,k} = \sum_{N=\max(n,m,k)}^{ (n+m+k)/2 } \binom{N}{n+m+k-2N, N-n, N-m, N-k} 2^{n+m+k-2N}$$
$$ + \sum_{N=\max(n,m-1,k)}^{ (n+m+k-1)/2 } \binom{N}{n+m+k-2N-1, N-n, N-m+1, N-k} 2^{n+m+k-2N-1}$$
$$ + \sum_{N=\max(n-1,m,k)}^{ (n+m+k-1)/2 } \binom{N}{n+m+k-2N-1, N-n+1, N-m, N-k} 2^{n+m+k-2N-1}$$
$$ + \sum_{N=\max(n,m,k-1)}^{ (n+m+k-1)/2 } \binom{N}{n+m+k-2N-1, N-n, N-m, N-k+1} 2^{n+m+k-2N-1}$$&lt;/p&gt;
&lt;p&gt;There may be room to simplify this further; the symmetry in $n,m,k$ could help.&lt;/p&gt;
&lt;h2 id=&#34;complexity&#34;&gt;Complexity&lt;/h2&gt;
&lt;p&gt;&lt;strong&gt;Recursion with memoization (DP):&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;time and space: $\Theta(n \cdot m \cdot k)$.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Closed form:&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Precompute factorials, then loop over $N$;&lt;/li&gt;
&lt;li&gt;time and space: $\Theta(n+m+k)$,&lt;/li&gt;
&lt;li&gt;we ignore the cost of arithmetic on large integers.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&#34;implementation&#34;&gt;Implementation&lt;/h2&gt;
&lt;p&gt;Both the recursive and closed-form versions in Python:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;import&lt;/span&gt; functools
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;import&lt;/span&gt; math
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#a6e22e&#34;&gt;@functools.lru_cache&lt;/span&gt;(maxsize&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;None&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;a_rec&lt;/span&gt;(n: int, m: int, k: int) &lt;span style=&#34;color:#f92672&#34;&gt;-&amp;gt;&lt;/span&gt; int:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;not&lt;/span&gt; (n &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; k):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        n,m,k &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; sorted([n,m,k])
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; a_rec(n,m,k)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;==&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;or&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;==&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;or&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;==&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;==&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; int(m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; k)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; (
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; a_rec(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, m &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; a_rec(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, m &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, k)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; a_rec(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, m, k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; a_rec(n, m &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    )
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#a6e22e&#34;&gt;@functools.lru_cache&lt;/span&gt;(maxsize&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;None&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;binom4&lt;/span&gt;(N: int, r1: int, r2: int, r3: int) &lt;span style=&#34;color:#f92672&#34;&gt;-&amp;gt;&lt;/span&gt; int:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    r4 &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; r1 &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; r2 &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; r3
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; math&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;factorial(N) &lt;span style=&#34;color:#f92672&#34;&gt;//&lt;/span&gt; (
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        math&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;factorial(r1) &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; math&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;factorial(r2)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; math&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;factorial(r3) &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; math&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;factorial(r4)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    )
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;a_closed&lt;/span&gt;(n: int, m: int, k: int) &lt;span style=&#34;color:#f92672&#34;&gt;-&amp;gt;&lt;/span&gt; int:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; min(n, m, k) &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    s &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; N &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; range(max(n, m, k), (n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k) &lt;span style=&#34;color:#f92672&#34;&gt;//&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        s &lt;span style=&#34;color:#f92672&#34;&gt;+=&lt;/span&gt; binom4(N, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; n, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; m, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; k) &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;**&lt;/span&gt; (n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; N)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; N &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; range(max(n, m &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, k), (n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;) &lt;span style=&#34;color:#f92672&#34;&gt;//&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        s &lt;span style=&#34;color:#f92672&#34;&gt;+=&lt;/span&gt; binom4(N, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; n, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; k) &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;**&lt;/span&gt; (n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; N &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; range(max(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, m, k), (n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;) &lt;span style=&#34;color:#f92672&#34;&gt;//&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        s &lt;span style=&#34;color:#f92672&#34;&gt;+=&lt;/span&gt; binom4(N, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; m, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; k) &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;**&lt;/span&gt; (n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; N &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; range(max(n, m, k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;), (n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;) &lt;span style=&#34;color:#f92672&#34;&gt;//&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        s &lt;span style=&#34;color:#f92672&#34;&gt;+=&lt;/span&gt; binom4(N, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; n, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; m, N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;) &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;**&lt;/span&gt; (n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; k &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; N &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; s
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;import&lt;/span&gt; timeit
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;setup_code &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#34;from __main__ import a_rec, a_closed&amp;#34;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;execution_time &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; timeit&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;timeit(&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#39;a_rec(200, 300, 400)&amp;#39;&lt;/span&gt;, setup&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;setup_code, number&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;print(&lt;span style=&#34;color:#e6db74&#34;&gt;f&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#34;Execution time for recursive:   &lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{&lt;/span&gt;execution_time&lt;span style=&#34;color:#e6db74&#34;&gt;}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt; seconds&amp;#34;&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;execution_time &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; timeit&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;timeit(&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#39;a_closed(200, 300, 400)&amp;#39;&lt;/span&gt;, setup&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;setup_code, number&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;print(&lt;span style=&#34;color:#e6db74&#34;&gt;f&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#34;Execution time for closed form: &lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{&lt;/span&gt;execution_time&lt;span style=&#34;color:#e6db74&#34;&gt;}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt; seconds&amp;#34;&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-text&#34; data-lang=&#34;text&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;Execution time for recursive:   8.308788761030883 seconds
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;Execution time for closed form: 0.003819321980699897 seconds
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#75715e&#34;&gt;# Sanity check&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;res0 &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; a_rec(&lt;span style=&#34;color:#ae81ff&#34;&gt;100&lt;/span&gt;, &lt;span style=&#34;color:#ae81ff&#34;&gt;200&lt;/span&gt;, &lt;span style=&#34;color:#ae81ff&#34;&gt;210&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;res1 &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; a_closed(&lt;span style=&#34;color:#ae81ff&#34;&gt;100&lt;/span&gt;, &lt;span style=&#34;color:#ae81ff&#34;&gt;200&lt;/span&gt;, &lt;span style=&#34;color:#ae81ff&#34;&gt;210&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;print(&lt;span style=&#34;color:#e6db74&#34;&gt;f&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#34;Recursive: &lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{&lt;/span&gt;res0&lt;span style=&#34;color:#e6db74&#34;&gt;}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#34;&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;print(&lt;span style=&#34;color:#e6db74&#34;&gt;f&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#34;Closed:    &lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{&lt;/span&gt;res1&lt;span style=&#34;color:#e6db74&#34;&gt;}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#34;&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;print(&lt;span style=&#34;color:#e6db74&#34;&gt;f&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#34;Match: &lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{&lt;/span&gt;res0 &lt;span style=&#34;color:#f92672&#34;&gt;==&lt;/span&gt; res1&lt;span style=&#34;color:#e6db74&#34;&gt;}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#34;&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We did not exploit the symmetry $a_{n,m,k} = a_{\sigma(n,m,k)}$ for permutations $\sigma$; it could speed up computation but does not obviously simplify the closed expression.&lt;/p&gt;
&lt;hr&gt;
&lt;p&gt;&lt;em&gt;Originally answered on &lt;a href=&#34;https://math.stackexchange.com/questions/1093271/how-to-solve-this-multivariable-recursion/2730331#2730331&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Math Stack Exchange&lt;/a&gt;.&lt;/em&gt;&lt;/p&gt;
</description>
    </item>
    
    <item>
      <title>Multiplicative Hashing Functions -- Notes on Primes, Golden Ratio, and Evil</title>
      <link>/post/multiplicative-hash/</link>
      <pubDate>Wed, 02 Aug 2017 07:29:43 +0000</pubDate>
      <guid>/post/multiplicative-hash/</guid>
      <description>&lt;h2 id=&#34;introduction&#34;&gt;Introduction&lt;/h2&gt;
&lt;p&gt;Mapping partitions (or keys) to slices (or buckets) in a distributed or sharded system has a large impact on performance.
Different hash-based solutions for this problem exist; each has drawbacks.
The goal is to choose a hash function that is simple to implement and gives acceptable performance with as few collisions as possible.&lt;/p&gt;
&lt;p&gt;The problem is defined as follows. Given a logical partition number $P$, compute the corresponding slice number $S = s(P)$.
Where $0 ≤ P &amp;lt; 2^{32}$, $0 ≤ S &amp;lt; M$, and $M$ denotes the table size (e.g. $M = 2^{14}$).
In our &amp;ldquo;binary&amp;rdquo; world, assumptions that input data (partition numbers) have uniform distribution are not always correct.
Therefore, the hash function $s: P \to S$ must be designed very carefully.
In addition to providing uniform distribution of hash values (slice numbers), it has to add some randomness.
Luckily, this field is well studied: well-known textbooks of Corman&amp;rsquo;s and Knuth&amp;rsquo;s have a good introduction to this field.
The last one has more detail explanation; therefore, without hesitation, we make use of Knuth&amp;rsquo;s book (Section 6.4.):
usually explaining in two paragraphs what the book is saying just in one sentence.&lt;/p&gt;
&lt;h2 id=&#34;basic-ideas&#34;&gt;Basic Ideas&lt;/h2&gt;
&lt;p&gt;A first approach is the division-remainder method with $M$ a power of 2:&lt;/p&gt;
&lt;p&gt;$s(P) = P \bmod M$,&lt;/p&gt;
&lt;p&gt;where $M = 2^{14}$ is the table size (number of slices). A variant is to take $M$ prime instead:&lt;/p&gt;
&lt;p&gt;$s(P) = P \bmod M$,&lt;/p&gt;
&lt;p&gt;where $M = 16411$ is a prime $&amp;gt; 2^{14}$. Both forms are called &amp;ldquo;Division Remainder Method&amp;rdquo;. These notes focus on another method, the &amp;ldquo;Multiplicative Method&amp;rdquo;:&lt;/p&gt;
&lt;p&gt;$$f(P) = A \cdot P \bmod W$$&lt;/p&gt;
&lt;p&gt;$$s(P) = f(P) \div (W \div M)$$&lt;/p&gt;
&lt;p&gt;Where $W = 2^{32}$ &amp;ndash; the machine word, $M = 2^{14}$ &amp;ndash; the table size, $\div$ denotes an integer division, $W \div M = 2^{32 - 14} = 2^{18}$, $A = 2654435761$ &amp;ndash; some &amp;ldquo;magic&amp;rdquo; integer.
It will be explained later, for now we assume that due to this magic, the value of $f(P)$ is almost random.
It is easy to see that $s(P)$ is in valid interval of slices, between 0 and $M = 2^{14}$.&lt;/p&gt;
&lt;h3 id=&#34;example-of-hash-codes&#34;&gt;Example of Hash Codes&lt;/h3&gt;
&lt;table&gt;
  &lt;thead&gt;
      &lt;tr&gt;
          &lt;th&gt;partition $P$&lt;/th&gt;
          &lt;th&gt;dec $P$&lt;/th&gt;
          &lt;th&gt;hex $P$&lt;/th&gt;
          &lt;th&gt;binary $f(P)$&lt;/th&gt;
          &lt;th&gt;binary $S$&lt;/th&gt;
          &lt;th&gt;hex $S$&lt;/th&gt;
          &lt;th&gt;dec $S$&lt;/th&gt;
      &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
      &lt;tr&gt;
          &lt;td&gt;$0$&lt;/td&gt;
          &lt;td&gt;0&lt;/td&gt;
          &lt;td&gt;00000000&lt;/td&gt;
          &lt;td&gt;00000000000000000000000000000000&lt;/td&gt;
          &lt;td&gt;00000000000000&lt;/td&gt;
          &lt;td&gt;0000&lt;/td&gt;
          &lt;td&gt;0&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$1$&lt;/td&gt;
          &lt;td&gt;1&lt;/td&gt;
          &lt;td&gt;00000001&lt;/td&gt;
          &lt;td&gt;10011110001101110111100110110001&lt;/td&gt;
          &lt;td&gt;10011110001101&lt;/td&gt;
          &lt;td&gt;278D&lt;/td&gt;
          &lt;td&gt;10125&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2$&lt;/td&gt;
          &lt;td&gt;2&lt;/td&gt;
          &lt;td&gt;00000002&lt;/td&gt;
          &lt;td&gt;00111100011011101111001101100010&lt;/td&gt;
          &lt;td&gt;00111100011011&lt;/td&gt;
          &lt;td&gt;0F1B&lt;/td&gt;
          &lt;td&gt;3867&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$3$&lt;/td&gt;
          &lt;td&gt;3&lt;/td&gt;
          &lt;td&gt;00000003&lt;/td&gt;
          &lt;td&gt;11011010101001100110110100010011&lt;/td&gt;
          &lt;td&gt;11011010101001&lt;/td&gt;
          &lt;td&gt;36A9&lt;/td&gt;
          &lt;td&gt;13993&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{14}-1$&lt;/td&gt;
          &lt;td&gt;16383&lt;/td&gt;
          &lt;td&gt;00003FFF&lt;/td&gt;
          &lt;td&gt;01000000001101001100011001001111&lt;/td&gt;
          &lt;td&gt;01000000001101&lt;/td&gt;
          &lt;td&gt;100D&lt;/td&gt;
          &lt;td&gt;4109&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{14}$&lt;/td&gt;
          &lt;td&gt;16384&lt;/td&gt;
          &lt;td&gt;00004000&lt;/td&gt;
          &lt;td&gt;11011110011011000100000000000000&lt;/td&gt;
          &lt;td&gt;11011110011011&lt;/td&gt;
          &lt;td&gt;379B&lt;/td&gt;
          &lt;td&gt;14235&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{14}+1$&lt;/td&gt;
          &lt;td&gt;16385&lt;/td&gt;
          &lt;td&gt;00004001&lt;/td&gt;
          &lt;td&gt;01111100101000111011100110110001&lt;/td&gt;
          &lt;td&gt;01111100101000&lt;/td&gt;
          &lt;td&gt;1F28&lt;/td&gt;
          &lt;td&gt;7976&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{14}+2$&lt;/td&gt;
          &lt;td&gt;16386&lt;/td&gt;
          &lt;td&gt;00004002&lt;/td&gt;
          &lt;td&gt;00011010110110110011001101100010&lt;/td&gt;
          &lt;td&gt;00011010110110&lt;/td&gt;
          &lt;td&gt;06B6&lt;/td&gt;
          &lt;td&gt;1718&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{15}-1$&lt;/td&gt;
          &lt;td&gt;32767&lt;/td&gt;
          &lt;td&gt;00007FFF&lt;/td&gt;
          &lt;td&gt;00011110101000010000011001001111&lt;/td&gt;
          &lt;td&gt;00011110101000&lt;/td&gt;
          &lt;td&gt;07A8&lt;/td&gt;
          &lt;td&gt;1960&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{15}$&lt;/td&gt;
          &lt;td&gt;32768&lt;/td&gt;
          &lt;td&gt;00008000&lt;/td&gt;
          &lt;td&gt;10111100110110001000000000000000&lt;/td&gt;
          &lt;td&gt;10111100110110&lt;/td&gt;
          &lt;td&gt;2F36&lt;/td&gt;
          &lt;td&gt;12086&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{15}+1$&lt;/td&gt;
          &lt;td&gt;32769&lt;/td&gt;
          &lt;td&gt;00008001&lt;/td&gt;
          &lt;td&gt;01011011000011111111100110110001&lt;/td&gt;
          &lt;td&gt;01011011000011&lt;/td&gt;
          &lt;td&gt;16C3&lt;/td&gt;
          &lt;td&gt;5827&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{15}+2$&lt;/td&gt;
          &lt;td&gt;32770&lt;/td&gt;
          &lt;td&gt;00008002&lt;/td&gt;
          &lt;td&gt;11111001010001110111001101100010&lt;/td&gt;
          &lt;td&gt;11111001010001&lt;/td&gt;
          &lt;td&gt;3E51&lt;/td&gt;
          &lt;td&gt;15953&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{30}-1$&lt;/td&gt;
          &lt;td&gt;1073741823&lt;/td&gt;
          &lt;td&gt;3FFFFFFF&lt;/td&gt;
          &lt;td&gt;10100001110010001000011001001111&lt;/td&gt;
          &lt;td&gt;10100001110010&lt;/td&gt;
          &lt;td&gt;2872&lt;/td&gt;
          &lt;td&gt;10354&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{30}$&lt;/td&gt;
          &lt;td&gt;1073741824&lt;/td&gt;
          &lt;td&gt;40000000&lt;/td&gt;
          &lt;td&gt;01000000000000000000000000000000&lt;/td&gt;
          &lt;td&gt;01000000000000&lt;/td&gt;
          &lt;td&gt;1000&lt;/td&gt;
          &lt;td&gt;4096&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{30}+1$&lt;/td&gt;
          &lt;td&gt;1073741825&lt;/td&gt;
          &lt;td&gt;40000001&lt;/td&gt;
          &lt;td&gt;11011110001101110111100110110001&lt;/td&gt;
          &lt;td&gt;11011110001101&lt;/td&gt;
          &lt;td&gt;378D&lt;/td&gt;
          &lt;td&gt;14221&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{30}+2$&lt;/td&gt;
          &lt;td&gt;1073741826&lt;/td&gt;
          &lt;td&gt;40000002&lt;/td&gt;
          &lt;td&gt;01111100011011101111001101100010&lt;/td&gt;
          &lt;td&gt;01111100011011&lt;/td&gt;
          &lt;td&gt;1F1B&lt;/td&gt;
          &lt;td&gt;7963&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{31}-1$&lt;/td&gt;
          &lt;td&gt;2147483647&lt;/td&gt;
          &lt;td&gt;7FFFFFFF&lt;/td&gt;
          &lt;td&gt;11100001110010001000011001001111&lt;/td&gt;
          &lt;td&gt;11100001110010&lt;/td&gt;
          &lt;td&gt;3872&lt;/td&gt;
          &lt;td&gt;14450&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{31}$&lt;/td&gt;
          &lt;td&gt;2147483648&lt;/td&gt;
          &lt;td&gt;80000000&lt;/td&gt;
          &lt;td&gt;10000000000000000000000000000000&lt;/td&gt;
          &lt;td&gt;10000000000000&lt;/td&gt;
          &lt;td&gt;2000&lt;/td&gt;
          &lt;td&gt;8192&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{31}+1$&lt;/td&gt;
          &lt;td&gt;2147483649&lt;/td&gt;
          &lt;td&gt;80000001&lt;/td&gt;
          &lt;td&gt;00011110001101110111100110110001&lt;/td&gt;
          &lt;td&gt;00011110001101&lt;/td&gt;
          &lt;td&gt;078D&lt;/td&gt;
          &lt;td&gt;1933&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{31}+2$&lt;/td&gt;
          &lt;td&gt;2147483650&lt;/td&gt;
          &lt;td&gt;80000002&lt;/td&gt;
          &lt;td&gt;10111100011011101111001101100010&lt;/td&gt;
          &lt;td&gt;10111100011011&lt;/td&gt;
          &lt;td&gt;2F1B&lt;/td&gt;
          &lt;td&gt;12059&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;$2^{32}-1$&lt;/td&gt;
          &lt;td&gt;4294967295&lt;/td&gt;
          &lt;td&gt;FFFFFFFF&lt;/td&gt;
          &lt;td&gt;01100001110010001000011001001111&lt;/td&gt;
          &lt;td&gt;01100001110010&lt;/td&gt;
          &lt;td&gt;1872&lt;/td&gt;
          &lt;td&gt;6258&lt;/td&gt;
      &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;h3 id=&#34;implementation-in-c&#34;&gt;Implementation in C&lt;/h3&gt;
&lt;p&gt;The new hashing function looks at bit complicated, so let&amp;rsquo;s consider how it (and others) could be implemented in C code. The type uint32_t denotes 32-bit unsigned integer. We start from examples of Division Remainder hashing. For $M$ being a power of 2:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-c&#34; data-lang=&#34;c&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;const&lt;/span&gt; &lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; M &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;14&lt;/span&gt;;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;slice&lt;/span&gt;(&lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; P) {
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; P &lt;span style=&#34;color:#f92672&#34;&gt;&amp;amp;&lt;/span&gt; (M &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;);
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;}
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;For prime $M$:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-c&#34; data-lang=&#34;c&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;const&lt;/span&gt; &lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; M &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;16411&lt;/span&gt;;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;slice&lt;/span&gt;(&lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; P) {
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; P &lt;span style=&#34;color:#f92672&#34;&gt;%&lt;/span&gt; M;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;}
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The next function implements Multiplicative hashing:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-c&#34; data-lang=&#34;c&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;const&lt;/span&gt; &lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; M &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;14&lt;/span&gt;;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;const&lt;/span&gt; &lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; A &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2654435761&lt;/span&gt;;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;slice&lt;/span&gt;(&lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; P) {
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; (P &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; A) &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;&amp;gt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;18&lt;/span&gt;;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;}
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;It works well for arbitrary input data and allows using the same number of slices $M = 2^{14}$. As the reader can see, the multiplicative version uses only a multiplication and a logical shift; on some architectures it can be faster than computing a modulo.&lt;/p&gt;
&lt;p&gt;Details for Math Fans&lt;/p&gt;
&lt;p&gt;In this section we describe properties of every approach. On analysis of the simple functions, we demonstrate the informal tools that we later apply to more complex functions. We hope that even non great fan of mathematics will be able to catch the main point.&lt;/p&gt;
&lt;h2 id=&#34;the-power-of-2-is-evil&#34;&gt;The Power of 2 is Evil&lt;/h2&gt;
&lt;p&gt;The first function we will consider is $s(P) = P \bmod 2^{14}$.
Let&amp;rsquo;s recall some properties of this function.&lt;/p&gt;
&lt;p&gt;Flipping just one bit in the given partition not always leads to a change in the corresponding slice (in hash-value).
More formal: for some bit $i$, &amp;ldquo;occasionally&amp;rdquo; $s(P) = s(P + 2^i)$ holds; here we assume for simplicity that $P$ has $i$ bit cleared.
In our &amp;ldquo;binary&amp;rdquo; world, partitions differing in only one bit may have a strong correlation.
Therefore, we are expecting that a &amp;ldquo;good&amp;rdquo; hashing will produce different hash-code for them (i.e., it will place the partitions in different slices).
But for this function, the property fails on any partition $P ≥ 2^{14}$.
There are simple templates of partitions having the same hash-code.
For example, for any $i$ and $S$, partitions $P = S + 2^{14} \cdot i$ will be placed by the function to the same slice $S$.
From a &amp;ldquo;good&amp;rdquo; hashing, we are expecting not to have very common patterns, especially, of binary form.
But again, this function fails on this condition.&lt;/p&gt;
&lt;h2 id=&#34;the-art-of-primes&#34;&gt;The Art of Primes&lt;/h2&gt;
&lt;p&gt;The second function is $s(P) = P \bmod M$.
Where $M = 16411$.
For this function, flipping any bit in a partition $P$ changes the hash code, because $M$ is chosen to be a prime.
Actually, this condition on $M$ is too strong.
For satisfying this property, it is sufficient that $M ≠ 2^i$ will hold.
For example, $M = 15 · 12 · 97 = 17460$ (15 modules, 12 disks, 97 is a prime) is also &amp;ldquo;good&amp;rdquo;.&lt;/p&gt;
&lt;p&gt;Since $M$ is prime, it seems that the following pattern is not common: $P = S + M · i$. But actually, primes that are close to a power of 2 are also not good. Knuth recommends to choose such $M$ that the following condition will not hold for any small integers $a$ and $j$: $r^j ≡ ± a \pmod M$. Where $r$ denotes the base of computation. From explanation of Knuth, the meaning of $r$ for our case is not too clear: whether $r=2$, $r=16$, or $r=256$? It seems the answer very depends on the type of input data.&lt;/p&gt;
&lt;p&gt;By Knuth, if $r=2$, the chosen $M$ is not so good, since $M = 16411 = 2^{14} + 27$, and hence $2^{14} ≡ -27 \pmod M$. For $r=16$, we get that $16^4 ≡ -108 \pmod M$. For $r=256$, $256^2 ≡ -108 \pmod M$.&lt;/p&gt;
&lt;p&gt;Knuth explains that such $M$ may produce a hash code that is a simple composition of key digits (in $r$ base system). Instead of trying to understand this explanation, we will give some intuition. Working with numbers, a programmer usually chooses powers of 2 for sizes of structures and buffers (e.g., $2^{10}$ bytes). Then he defines the format of such data and introduces headers (e.g. the header size = 20 bytes). Hence, the size of data without the header becomes very close to the power of 2 (in our example, $2^{10} - 20 = 1004$). On the other hand, embedding this structure to an outer packet (assume the size of this outer header is 30 bytes) leads to the total size being also close to the power of 2 ($2^{10} + 30 = 1054$). As result, most of numbers in our &amp;ldquo;binary&amp;rdquo; world are either powers of 2 or close to them. Therefore, such choice of $M$ increases collisions. In other words, not only powers of 2 are &lt;em&gt;evil&lt;/em&gt;, but primes closing to them are &lt;em&gt;evil&lt;/em&gt; too.&lt;/p&gt;
&lt;p&gt;As an example of a &amp;ldquo;good&amp;rdquo; prime, let&amp;rsquo;s consider $M = 24571$. It is a bit smaller then the middle of $2^{14}$ and $2^{15}$.&lt;/p&gt;
&lt;p&gt;Introducing a little change in this hashing function, we get better hashing even for $M = 2^{14}$: $s(P) = P \bmod N \bmod M$. In this case $N$ must be some &amp;ldquo;big&amp;rdquo; and &amp;ldquo;far&amp;rdquo; from a power of 2 prime number. The discussion of this method is out of scope of this Notes.&lt;/p&gt;
&lt;h2 id=&#34;the-magic-of-golden-ratio&#34;&gt;The Magic of Golden Ratio&lt;/h2&gt;
&lt;p&gt;We now present in details the multiplicative hashing. First, we discuss the properties of the function $f$: $f(P) = A · P \bmod W$.&lt;/p&gt;
&lt;p&gt;The magic number $A$ is chosen to have the following nice properties: $W / A ≈ 1.6$ &amp;ndash; the golden ratio and $GCD(A, W) = 1$ ($A$ and $W$ are relatively prime or coprime). Therefore, the function $f$ is 1-1 on 32-bit integers. In other word, for any $P_1~ ≠ P_2$, $f(P_1) ≠ f(P_2)$ holds. This function has a role of perturbation or randomization of the input data (partitions).&lt;/p&gt;
&lt;p&gt;Assume given a partition $P$. Let&amp;rsquo;s set the 0 bit in $P$. Then we get $f(P + 1) = (f(P) + A) \bmod W$. So the bit 0 dirties those bits of $f(P)$ that are set in $A$. This implies that $A$ must be chosen from the following range: $2^{31} &amp;lt; A &amp;lt; 2^{32}$, and in addition it must have a half of bits being set. By Knuth, the best choice is to define it to be the golden value of $W$: $W / A = 1.6 \ldots$ . In this case setting $i$-bit in $P$ dirties in $f(P)$ all the bits from $i$ to 31.&lt;/p&gt;
&lt;p&gt;Using the properties of $f()$, we show the intuition why the function $s(P) = f(P) \div 2^{18}$ is a &amp;ldquo;good&amp;rdquo; hashing function.&lt;/p&gt;
&lt;p&gt;The function $s$ remains 14 most significant bits of $f(P)$. Therefore, from the properties of $f()$, flipping any bit in $P$ will yield flipping at least in one bit of $s(P)$.
The function $s$ puts into one slice $S$ all the partitions of the form: $P = (S · 2^{18} + i) · B \bmod 2^{32}$, where $B = 244002641$ (see for details the section Invertibility) and $0 ≤ i &amp;lt; 2^{18}$. Assume that $P_0 = S · B · 2^{18} \bmod 2^{32}$, then $P_i = (P_0 + i · B) \bmod 2^{32}$. The function $g(i) = i · B$ produces almost random 32-bit integers from 18-bit integers. Therefore, it is almost impossible to derive a &amp;ldquo;nice&amp;rdquo; sub-sequence from the values of set ${g(i) | 0 ≤ i &amp;lt; 2^{18}}$.&lt;/p&gt;
&lt;h2 id=&#34;invertibility&#34;&gt;Invertibility&lt;/h2&gt;
&lt;p&gt;Given a 14-bit slice $S$ and some 18-bit identifier $Id$, we are interested to compute the partition $P$ that is placed by the function $s$ at slice $S$. Note that given a partition $P$, the computation of $Id$ is also depends on the chosen method of hashing. We present for each hashing how the identifier and the inverse of $s()$ can be computed.&lt;/p&gt;
&lt;p&gt;For the simplest function $s(P) = P \bmod 2^{14}$, the identifier is computed as follows: $Id = P \div 2^{14}$.
Then its inverse is $p(S, Id) = S + Id · 2^{14}$.
For prime $M$, the identifier is $Id = P \div M$ and the inverse function is $p(S, Id) = S + Id · M$.
For multiplicative hashing, we have $Id = A · P \bmod 2^{14}$. Then $p(S, Id) = (S · 2^{18} + Id) · B \bmod 2^{32}$.
This implies from the fact that $A · B ≡ 1 \pmod 2^{32}$ and the inverse of $f()$ is $f^{-1}(x) = x · B \bmod 2^{32}$.&lt;/p&gt;
&lt;p&gt;We show the implementation of $p()$ in C code for the multiplicative hashing only:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-c&#34; data-lang=&#34;c&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;const&lt;/span&gt; &lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; M &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;14&lt;/span&gt;;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;const&lt;/span&gt; &lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; B &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;244002641&lt;/span&gt;;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;partition&lt;/span&gt;(&lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; S, &lt;span style=&#34;color:#66d9ef&#34;&gt;uint32_t&lt;/span&gt; Id) {
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; (S &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;18&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; Id) &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; B;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;}
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h2 id=&#34;big-numbers&#34;&gt;Big Numbers&lt;/h2&gt;
&lt;p&gt;In this section we discuss the behavior of multiplicative hashing on big partition numbers. Recall that flipping just one $i$-bit in $P$ has impact on $(32 - i)$ bits (from $i$ to 31) in $f(P)$. By the definition, $s(P)$ derives 14 most significant bits of $f(P)$. As result, for $i &amp;lt; 18$, the flipping $i$-bit dirties most of 14 bits of $s(P)$, while for $i ≥ 18$, it change less then 14 bits of $s(P)$: $32 - i ≤ 14$. For example, flipping the most significant 31-bit of the partition $P$ flips the only one 14-bit of the corresponding slice $S = s(P)$. This is not a real problem for our case, the partitions are still putted into different slides.&lt;/p&gt;
&lt;p&gt;Another problem with flipping of most significant bits is following. For a set of partitions differing in most significant bits only, the number of collisions is greater then for the case that partitions are different in less significant bits. But due to the nice choice of $A$ (being golden ratio of $W$), this is still sufficiently good distribution. Intuitively, if the partitions differ in x most significant bits, then slices for them will also differ in x bits. So most of these partitions will be placed to different slices.&lt;/p&gt;
&lt;p&gt;Note that the number of impacted bits in $s(P)$ decreases only for partitions with more then 18 bits. Even then, as described above, it does not mean that we will have a huge degradation of the hashing performance. But if we want to prevent any potentially dangerous case, we may extend the machine word to 64-bits, especially if the used architecture is also 64-bit. In this case, we just compute a new pair of integers $A$ and $B$ ($A$ is the golden ration of and is coprime with $W=2^{64}$, $B$ is coprime with $A$) and correct logical shifts. This is an example of implementation in C:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-c&#34; data-lang=&#34;c&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;const&lt;/span&gt; &lt;span style=&#34;color:#66d9ef&#34;&gt;uint64_t&lt;/span&gt; M &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;14&lt;/span&gt;;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;const&lt;/span&gt; &lt;span style=&#34;color:#66d9ef&#34;&gt;uint64_t&lt;/span&gt; A &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;11400714819323198485&lt;/span&gt;;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;uint64_t&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;slice&lt;/span&gt;(&lt;span style=&#34;color:#66d9ef&#34;&gt;uint64_t&lt;/span&gt; P) {
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; (P &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; A) &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;&amp;gt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;50&lt;/span&gt;;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;}
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h3 id=&#34;example-of-big-partitions&#34;&gt;Example of Big Partitions&lt;/h3&gt;
&lt;p&gt;The table contains computation of slices for partitions that differ only in three most significant bits. As we can see, corresponding slices also differ only in three bits. This property may be considered as drawback. But note that this case has no collisions.&lt;/p&gt;
&lt;table&gt;
  &lt;thead&gt;
      &lt;tr&gt;
          &lt;th&gt;binary $P$&lt;/th&gt;
          &lt;th&gt;hex $P$&lt;/th&gt;
          &lt;th&gt;binary $f(P)$&lt;/th&gt;
          &lt;th&gt;binary $S$&lt;/th&gt;
          &lt;th&gt;hex $S$&lt;/th&gt;
          &lt;th&gt;dec $S$&lt;/th&gt;
      &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;000&lt;/strong&gt;10101010111010100100101011001&lt;/td&gt;
          &lt;td&gt;155D4959&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;100&lt;/strong&gt;01101010010011100011110001001&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;100&lt;/strong&gt;01101010010&lt;/td&gt;
          &lt;td&gt;$2352$&lt;/td&gt;
          &lt;td&gt;$9042$&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;001&lt;/strong&gt;10101010111010100100101011001&lt;/td&gt;
          &lt;td&gt;355D4959&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;101&lt;/strong&gt;01101010010011100011110001001&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;101&lt;/strong&gt;01101010010&lt;/td&gt;
          &lt;td&gt;$2B52$&lt;/td&gt;
          &lt;td&gt;$11090$&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;010&lt;/strong&gt;10101010111010100100101011001&lt;/td&gt;
          &lt;td&gt;555D4959&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;110&lt;/strong&gt;01101010010011100011110001001&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;110&lt;/strong&gt;01101010010&lt;/td&gt;
          &lt;td&gt;$3352$&lt;/td&gt;
          &lt;td&gt;$13138$&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;011&lt;/strong&gt;10101010111010100100101011001&lt;/td&gt;
          &lt;td&gt;755D4959&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;111&lt;/strong&gt;01101010010011100011110001001&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;111&lt;/strong&gt;01101010010&lt;/td&gt;
          &lt;td&gt;$3B52$&lt;/td&gt;
          &lt;td&gt;$15186$&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;100&lt;/strong&gt;10101010111010100100101011001&lt;/td&gt;
          &lt;td&gt;955D4959&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;000&lt;/strong&gt;01101010010011100011110001001&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;000&lt;/strong&gt;01101010010&lt;/td&gt;
          &lt;td&gt;$0352$&lt;/td&gt;
          &lt;td&gt;$850$&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;101&lt;/strong&gt;10101010111010100100101011001&lt;/td&gt;
          &lt;td&gt;B55D4959&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;001&lt;/strong&gt;01101010010011100011110001001&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;001&lt;/strong&gt;01101010010&lt;/td&gt;
          &lt;td&gt;$0B52$&lt;/td&gt;
          &lt;td&gt;$2898$&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;110&lt;/strong&gt;10101010111010100100101011001&lt;/td&gt;
          &lt;td&gt;D55D4959&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;010&lt;/strong&gt;01101010010011100011110001001&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;010&lt;/strong&gt;01101010010&lt;/td&gt;
          &lt;td&gt;$1352$&lt;/td&gt;
          &lt;td&gt;$4946$&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;&lt;strong&gt;111&lt;/strong&gt;10101010111010100100101011001&lt;/td&gt;
          &lt;td&gt;F55D4959&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;011&lt;/strong&gt;01101010010011100011110001001&lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;011&lt;/strong&gt;01101010010&lt;/td&gt;
          &lt;td&gt;$1B52$&lt;/td&gt;
          &lt;td&gt;$6994$&lt;/td&gt;
      &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;h2 id=&#34;conclusion&#34;&gt;Conclusion&lt;/h2&gt;
&lt;p&gt;TODO&lt;/p&gt;
&lt;h3 id=&#34;links-and-references&#34;&gt;Links and References&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;T Corman, Ch Leiserson, R Rivest, &amp;ldquo;Introduction to Algorithms&amp;rdquo;, 1999&lt;/li&gt;
&lt;li&gt;D Knuth, &amp;ldquo;Sorting and Searching&amp;rdquo;, volume 3 of &amp;ldquo;The Art of Computer Programming&amp;rdquo;, 1973. Section 12.3&lt;/li&gt;
&lt;li&gt;Integer Hash Function, Thomas Wang&lt;/li&gt;
&lt;li&gt;Selecting a Hash Function, Scott Gasch&lt;/li&gt;
&lt;li&gt;Hash functions, Paul Hsieh&lt;/li&gt;
&lt;li&gt;Wikipedia: Hash Table &amp;ndash; sites Thomas Wang that the multiplicative hash has &amp;ldquo;particularly poor clustering behavior&amp;rdquo;.&lt;/li&gt;
&lt;li&gt;Prime Double Hash Table, Thomas Wang &amp;ndash; the wrong proof.&lt;/li&gt;
&lt;li&gt;&lt;a href=&#34;http://web.archive.org/web/19990903133921/http://www.concentric.net/~Ttwang/tech/primehash.htm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;http://web.archive.org/web/19990903133921/http://www.concentric.net/~Ttwang/tech/primehash.htm&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;Wikipedia: Talking Hash Table &amp;ndash; note that the proof is wrong, see &amp;ldquo;Info about multiplicative hashing is wrong&amp;rdquo; section.&lt;/li&gt;
&lt;/ul&gt;
</description>
    </item>
    
    <item>
      <title>Perfect Distribution: GCD in Disguise</title>
      <link>/post/perfect-distribution/</link>
      <pubDate>Tue, 01 Aug 2017 07:29:43 +0000</pubDate>
      <guid>/post/perfect-distribution/</guid>
      <description>&lt;h2 id=&#34;perfect-distribution-gcd-in-disguise&#34;&gt;Perfect Distribution: GCD in Disguise&lt;/h2&gt;
&lt;p&gt;We discuss an algorithm that distributes $a$ ones among $n$ positions so that the gaps between consecutive ones differ by at most one—a &lt;strong&gt;perfect distribution&lt;/strong&gt;. I developed it while working on profiling, stress, and negative testing of a system that needed exactly this kind of uniform spread. I am not aware of prior art; if you know of related work, I would be interested to hear.&lt;/p&gt;
&lt;p&gt;The examples below are software engineering use cases for this algorithm.&lt;/p&gt;
&lt;h3 id=&#34;use-case-0-text-spacing&#34;&gt;Use Case 0: Text spacing&lt;/h3&gt;
&lt;p&gt;Given a string of characters $s$ and a number $n$ greater than the length of $s$,
extend the string $s$ to length $n$ by inserting spaces between the words.
The key requirement is that the distances between two consecutive words be uniform.&lt;/p&gt;
&lt;h3 id=&#34;use-case-1-stress-testing&#34;&gt;Use Case 1: Stress testing&lt;/h3&gt;
&lt;p&gt;Assume we want to stress-test a server.
Our test contains positive and negative requests.
Suppose we want to send $1000$ requests, $300$ of which are negative.
Our test tool might send requests simultaneously or sequentially,
but we require that the negative requests arrive at the server with uniform distribution.
I.e., we want to avoid sending 300 negative followed by 700 positive requests.
Using the algorithm, we simply distribute the 300 negative requests by calling ${\cal PD}(300, 1000)$:
cell $i$ is $1$ if and only if request $i$ is negative.
The test tool then uses the array to send requests (possibly sequentially).&lt;/p&gt;
&lt;h3 id=&#34;use-case-2-rate-limiting--profiling&#34;&gt;Use Case 2: Rate limiting / profiling&lt;/h3&gt;
&lt;p&gt;Assume the test tool can run performance tests—creating stress and measuring throughput (requests per second).
We want to extend the test tool with a profiling feature: specify the number of requests sent per second.
E.g., a performance test shows the server handles $1000$ requests per second.
We want the test tool to send requests at a specified rate, say $500$, $200$, $20$, or even $0.2$ requests per second (one request every 5 seconds).
Such a profiling tool allows exploring the server state under various load conditions: CPU, memory, threads depending on the number of requests received.&lt;/p&gt;
&lt;p&gt;The test tool may divide 1 second into $100$ parts, each $20$ milliseconds.
Suppose we specify $230$ requests per second: then during every part, $2.3$ requests must be sent on average.
In other words, the tool must send 2 requests in most parts and 3 in some parts.
We need to choose which $30$ parts get the extra request; ${\cal PD}(30, 100)$ yields that array (a $1$ in cell $i$ means part $i$ gets 3 requests).&lt;/p&gt;
&lt;h3 id=&#34;why-naive-approaches-fail&#34;&gt;Why naive approaches fail&lt;/h3&gt;
&lt;p&gt;A natural idea is to place ones at multiples of $n/a$, e.g. &lt;code&gt;[1 if (i * a) % n &amp;lt; a else 0 for i in range(n)]&lt;/code&gt;, or to use &lt;code&gt;round(i * a / n)&lt;/code&gt; to decide. Such formulas can produce wrong counts (off-by-one), clusters at the boundaries, or gaps that differ by more than one. Another approach—repeatedly choosing the “emptiest” gap—avoids clustering but is harder to implement correctly and lacks the clean recurrence we want. ${\cal PD}$ gives the exact, unique fairest distribution in a few lines.&lt;/p&gt;
&lt;p&gt;We present the general solution ${\cal PD}$ to the problem—for any specified input in the above examples, ${\cal PD}$ produces the exact solution.
Any probabilistic or approximation solutions do not achieve the required exactness and are usually more complicated.
The algorithm ${\cal PD}$ is simple, has a clear correctness proof, and admits $O(n)$ time and space complexity.
Moreover, it has a direct relation to Euclid&amp;rsquo;s algorithm for computing the Greatest Common Divisor, denoted here by ${\cal GCD}$.&lt;/p&gt;
&lt;h3 id=&#34;notation&#34;&gt;Notation&lt;/h3&gt;
&lt;p&gt;&lt;em&gt;POST&lt;/em&gt; denotes a condition that the procedure guarantees will hold after the call.&lt;/p&gt;
&lt;h3 id=&#34;euclids-algorithm-cal-gcd&#34;&gt;Euclid&amp;rsquo;s Algorithm ${\cal GCD}$&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;POST: $res = {\cal GCD}(a, n)$.&lt;/li&gt;
&lt;/ul&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;gcd&lt;/span&gt;(a: int, n: int) &lt;span style=&#34;color:#f92672&#34;&gt;-&amp;gt;&lt;/span&gt; int:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; a &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; n
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    res &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; n
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; a &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        res &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; gcd(n &lt;span style=&#34;color:#f92672&#34;&gt;%&lt;/span&gt; a, a)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; res
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h3 id=&#34;deterministic-algorithm-of-perfect-distribution-cal-pd&#34;&gt;Deterministic algorithm of perfect distribution ${\cal PD}$&lt;/h3&gt;
&lt;p&gt;The recurrence mirrors Euclid&amp;rsquo;s algorithm: ${\cal PD}(a, n)$ calls ${\cal PD}(n \bmod a, a)$, so the recursion depth is $O(\log \min(a, n))$ and total time is $O(n)$.&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;POST: size of $res$ is $n$.&lt;/li&gt;
&lt;li&gt;POST: $\forall i,, res[i] \in {0, 1}$.&lt;/li&gt;
&lt;li&gt;POST: $\sum_{i=0}^{n-1} res[i] = a$ (exactly $a$ ones).&lt;/li&gt;
&lt;li&gt;POST: $res$ is perfectly distributed (gaps between consecutive ones differ by at most 1).&lt;/li&gt;
&lt;/ul&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;pd&lt;/span&gt;(a: int, n: int) &lt;span style=&#34;color:#f92672&#34;&gt;-&amp;gt;&lt;/span&gt; list[int]:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; a &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; n
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    res &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; [&lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;] &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; n
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; a &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        steps &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; pd(n &lt;span style=&#34;color:#f92672&#34;&gt;%&lt;/span&gt; a, a)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        step &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;//&lt;/span&gt; a
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        ofs &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; i &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; range(a):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;            res[ofs] &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;            ofs &lt;span style=&#34;color:#f92672&#34;&gt;+=&lt;/span&gt; step &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; steps[i]
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; res
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;&lt;strong&gt;Why it works.&lt;/strong&gt; The algorithm treats $n$ positions as $a$ “blocks” of size $\lfloor n/a \rfloor$ or $\lceil n/a \rceil$. The recursive call ${\cal PD}(n \bmod a, a)$ decides which blocks get the extra slot (the remainder). Each block is filled by either repeating a perfect distribution of ones in a sub-interval (when $a \le n/2$) or by inverting zeros when $a &amp;gt; n/2$. The recurrence ensures that at every level, the pattern stays uniform—no block gets more than one extra slot, so gaps never differ by more than one.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Optimality.&lt;/strong&gt; The perfect distribution (gaps differ by at most 1) is unique up to the choice of which gaps are $\lfloor (n-a)/a \rfloor$ and which are $\lceil (n-a)/a \rceil$. The algorithm constructs this by induction: the base case $a=0$ is trivial; for $a&amp;gt;0$, the recursive call yields a perfect distribution for the “remainder” subproblem, and the interleaving step preserves the gap invariant.&lt;/p&gt;
&lt;h3 id=&#34;example-cal-pd2-10-a--2-n--10-so-n---a--8-zeros&#34;&gt;Example. ${\cal PD}(2, 10)$: $a = 2$, $n = 10$, so $n - a = 8$ zeros&lt;/h3&gt;
&lt;p&gt;We want to mix $2$ ones and $8$ zeros.
The recursive call ${\cal PD}(0, 2)$ returns $[0, 0] = 0^2$, which is assigned to $arr$.
Then the array $res$ is filled as follows: $[(1, 0, 0, 0, 0), (1, 0, 0, 0, 0)] = (1, 0^4)^2$, which is returned by ${\cal PD}(2, 10)$.&lt;/p&gt;
&lt;h3 id=&#34;example-cal-pd8-10-a--8-n--10-so-n---a--2-zeros&#34;&gt;Example. ${\cal PD}(8, 10)$: $a = 8$, $n = 10$, so $n - a = 2$ zeros&lt;/h3&gt;
&lt;p&gt;We want to mix $8$ ones and $2$ zeros.
The recursive call ${\cal PD}(2, 8)$ returns $[1, 0, 0, 0, 1, 0, 0, 0] = (1, 0^3)^2$, which is assigned to $arr$.
Then the array $res$ is filled as follows: $[(1, 0, 1, 1, 1), (1, 0, 1, 1, 1)] = (1, 0, 1^3)^2$, which is returned by ${\cal PD}(8, 10)$.&lt;/p&gt;
&lt;h3 id=&#34;examples&#34;&gt;Examples.&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;
&lt;p&gt;${\cal PD}(1, 10) : [1, 0, \ldots, 0] = 1, 0^9$&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;${\cal PD}(2, 10) : [1, 0, 0, 0, 0, 1, 0, 0, 0, 0] = 1, 0^4 , 1, 0^4$&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;${\cal PD}(3, 10) : [1, 0, 0, 1, 0, 0, 1, 0, 0, 0] = 1, 0^2 , 1, 0^2 , 1, 0^3$ — visual: &lt;code&gt;1 _ _ 1 _ _ 1 _ _ _&lt;/code&gt; (gaps between 1s: 2, 2, 3)&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;${\cal PD}(4, 10) : [1, 0, 0, 1, 0, 1, 0, 0, 1, 0] = 1, 0^2 , 1, 0, 1, 0^2 , 1, 0$ $= (1, 0^2 , 1, 0)^2$&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;${\cal PD}(5, 10) : [1, 0, 1, 0, 1, 0, 1, 0, 1, 0] = (1, 0)^5$&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;${\cal PD}(6, 10) : [1, 0, 1, 1, 0, 1, 0, 1, 1, 0] = 1, 0, 1^2 , 0, 1, 0, 1^2 , 0$ $ = (1, 0, 1^2 , 0)^2$&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;${\cal PD}(0, 2) : [0, 0] = 0^2$&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;${\cal PD}(2, 6) : [1, 0, 0, 1, 0, 0] = 1, 0^2 , 1, 0^2 = (1, 0^2)^2$&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;${\cal PD}(6, 8) : [1, 0, 1, 1, 1, 0, 1, 1] = 1, 0, 1^3, 0, 1^2 = (1, 0, 1^2)^2$&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;${\cal PD}(8, 14) : [1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0] = (1, 0, 1, 1, 0, 1, 0)^2$ $ = ((1, 0, 1)^2 , 0)^2$&lt;/p&gt;
&lt;/li&gt;
&lt;/ul&gt;
&lt;h3 id=&#34;alternative-formulation-subtraction-based-slow-versions&#34;&gt;Alternative formulation: subtraction-based (slow) versions&lt;/h3&gt;
&lt;p&gt;The same recurrence can be expressed using subtraction instead of modulo, which makes the symmetry with Euclid&amp;rsquo;s algorithm more explicit. These versions have the same recursion structure but do more work per step.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Slow GCD (subtraction-based):&lt;/strong&gt;&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;gcd&lt;/span&gt;(a: int, n: int) &lt;span style=&#34;color:#f92672&#34;&gt;-&amp;gt;&lt;/span&gt; int:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; a &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; n
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    res &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; n
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; a &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        b &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; a
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; b &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;&lt;/span&gt; a:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;            res &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; gcd(a, b)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;else&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;            res &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; gcd(b, a)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; res
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;&lt;strong&gt;Slow PD (subtraction-based):&lt;/strong&gt;&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;pd&lt;/span&gt;(a: int, n: int) &lt;span style=&#34;color:#f92672&#34;&gt;-&amp;gt;&lt;/span&gt; list[int]:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; a &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; n
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    res &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; [&lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;] &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; n
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; a &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        ofs &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        b &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; a
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; b &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;&lt;/span&gt; a:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;            arr &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; pd(a, b)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;            &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; i &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; range(b):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;                res[ofs] &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; arr[i]
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;                ofs &lt;span style=&#34;color:#f92672&#34;&gt;+=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; arr[i]
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;else&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;            arr &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; pd(b, a)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;            &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; i &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; range(a):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;                res[ofs] &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; arr[i]
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;                ofs &lt;span style=&#34;color:#f92672&#34;&gt;+=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; arr[i]
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; res
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;</description>
    </item>
    
    <item>
      <title>Binomial Coefficients Modulo a Prime: Fermat&#39;s Theorem and the Non-Adjacent Selection Problem</title>
      <link>/post/binomial-modulo-prime/</link>
      <pubDate>Sat, 08 Jul 2017 07:29:43 +0000</pubDate>
      <guid>/post/binomial-modulo-prime/</guid>
      <description>&lt;p&gt;In the &lt;a href=&#34;/post/efficient-implementation-non-adjacent-selection/&#34;&gt;previous post&lt;/a&gt;, we implemented the closed form $F_{n,m} = \binom{n-m+1}{m}$ using Python&amp;rsquo;s &lt;code&gt;math.factorial&lt;/code&gt;, and with &lt;code&gt;scipy&lt;/code&gt; and &lt;code&gt;sympy&lt;/code&gt;. Here we cover the common competitive-programming case: computing the answer &lt;strong&gt;modulo a large prime&lt;/strong&gt; $M$ (e.g. $M = 10^9+7$).&lt;/p&gt;
&lt;h2 id=&#34;why-modulo&#34;&gt;Why modulo?&lt;/h2&gt;
&lt;p&gt;In counting problems, the result can be huge even for moderate input. Often the problem asks for the answer modulo a big prime so that it fits in a standard integer type. We could compute the full number and then take the remainder, but that forces expensive long-integer arithmetic. Computing &lt;strong&gt;everything&lt;/strong&gt; modulo $M$ from the start is much faster.&lt;/p&gt;
&lt;h2 id=&#34;from-binomials-to-modular-inverses&#34;&gt;From binomials to modular inverses&lt;/h2&gt;
&lt;p&gt;We have $F_{n,m} = \binom{n-m+1}{m} = \frac{(n-m+1)!}{m!,(n-2m+1)!}$. To compute this mod $M$, we need factorials mod $M$ and division mod $M$. Division mod $M$ is multiplication by the &lt;strong&gt;modular inverse&lt;/strong&gt;: for prime $M$ and $0 &amp;lt; x &amp;lt; M$, the inverse of $x$ is $x^{M-2} \bmod M$ by &lt;a href=&#34;https://en.wikipedia.org/wiki/Fermat%27s_little_theorem&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fermat&amp;rsquo;s little theorem&lt;/a&gt;. In Python we can use &lt;code&gt;pow(x, M - 2, M)&lt;/code&gt;.&lt;/p&gt;
&lt;h2 id=&#34;implementation&#34;&gt;Implementation&lt;/h2&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;import&lt;/span&gt; functools
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;M &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;10&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;**&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;9&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;7&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;f_binom_mod&lt;/span&gt;(n, m):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;and&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt;m:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; binom_mod(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, m)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;binom_mod&lt;/span&gt;(n, m):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; n
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; ((fact_mod(n) &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; inv_mod(fact_mod(m))) &lt;span style=&#34;color:#f92672&#34;&gt;%&lt;/span&gt; M &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; inv_mod(fact_mod(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; m))) &lt;span style=&#34;color:#f92672&#34;&gt;%&lt;/span&gt; M
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#a6e22e&#34;&gt;@functools.lru_cache&lt;/span&gt;(maxsize&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;None&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;fact_mod&lt;/span&gt;(m):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; (m &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; fact_mod(m &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)) &lt;span style=&#34;color:#f92672&#34;&gt;%&lt;/span&gt; M
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;inv_mod&lt;/span&gt;(x):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; pow(x, M &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;, M)
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;All operations stay in the ring of integers mod $M$. The only non-obvious part is modular division: we replace division by $d$ with multiplication by &lt;code&gt;inv_mod(d)&lt;/code&gt; using Fermat&amp;rsquo;s little theorem.&lt;/p&gt;
&lt;h2 id=&#34;benchmarks&#34;&gt;Benchmarks&lt;/h2&gt;
&lt;p&gt;Compared to computing the full binomial and then taking the remainder, the modular version avoids long arithmetic and is much faster:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;fact_mod(&lt;span style=&#34;color:#ae81ff&#34;&gt;10000&lt;/span&gt;)  &lt;span style=&#34;color:#75715e&#34;&gt;# for caching factorials&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;funcs &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; [f_binom_mod, f_binom, f_sci, f_sym]
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;test(&lt;span style=&#34;color:#ae81ff&#34;&gt;10000&lt;/span&gt;, &lt;span style=&#34;color:#ae81ff&#34;&gt;1000&lt;/span&gt;, funcs)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;test(&lt;span style=&#34;color:#ae81ff&#34;&gt;10000&lt;/span&gt;, &lt;span style=&#34;color:#ae81ff&#34;&gt;2000&lt;/span&gt;, funcs)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;test(&lt;span style=&#34;color:#ae81ff&#34;&gt;10000&lt;/span&gt;, &lt;span style=&#34;color:#ae81ff&#34;&gt;3000&lt;/span&gt;, funcs)
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Example output:&lt;/p&gt;
&lt;pre tabindex=&#34;0&#34;&gt;&lt;code&gt;f(10000,1000): 450169549
  f_binom_mod:   0.0000 sec, x 1.00
      f_binom:   0.0073 sec, x 337.60
        f_sci:   0.0011 sec, x 49.33
        f_sym:   0.0076 sec, x 353.22

f(10000,2000): 75198348
  f_binom_mod:   0.0000 sec, x 1.00
      f_binom:   0.0063 sec, x 368.94
        f_sci:   0.0026 sec, x 153.33
        f_sym:   0.0053 sec, x 308.93

f(10000,3000): 679286557
  f_binom_mod:   0.0000 sec, x 1.00
      f_binom:   0.0060 sec, x 361.12
        f_sci:   0.0056 sec, x 338.13
        f_sym:   0.0053 sec, x 319.02
&lt;/code&gt;&lt;/pre&gt;&lt;p&gt;The same pattern—factorials mod $M$ plus Fermat-based inverses—works for any combinatorial formula that can be written in terms of factorials and binomials modulo a prime.&lt;/p&gt;
</description>
    </item>
    
    <item>
      <title>Efficient Implementation of the Non-Adjacent Selection Formula</title>
      <link>/post/efficient-implementation-non-adjacent-selection/</link>
      <pubDate>Fri, 07 Jul 2017 07:29:43 +0000</pubDate>
      <guid>/post/efficient-implementation-non-adjacent-selection/</guid>
      <description>&lt;p&gt;In the &lt;a href=&#34;/post/two-var-recursive-func/&#34;&gt;previous post&lt;/a&gt;, we derived the closed form for the non-adjacent selection problem:&lt;/p&gt;
&lt;p&gt;$$ F_{n, m} = {n - m + 1 \choose m} $$&lt;/p&gt;
&lt;p&gt;Now we discuss how to implement this efficiently in Python—from a simple factorial-based solution to library implementations. For the common case of computing the answer &lt;strong&gt;modulo a large prime&lt;/strong&gt; (e.g. in competitive programming), see the &lt;a href=&#34;/post/binomial-modulo-prime/&#34;&gt;next post&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id=&#34;fast-solutions-based-on-binomials&#34;&gt;Fast Solutions Based on Binomials&lt;/h2&gt;
&lt;p&gt;We can reflect the closed form in very trivial Python code:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-Python&#34; data-lang=&#34;Python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;import&lt;/span&gt; math
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;f_binom&lt;/span&gt;(n, m):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;and&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt;m:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; binom(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, m)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;binom&lt;/span&gt;(n, m):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; n
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; math&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;factorial(n) &lt;span style=&#34;color:#f92672&#34;&gt;//&lt;/span&gt; math&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;factorial(m) &lt;span style=&#34;color:#f92672&#34;&gt;//&lt;/span&gt; math&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;factorial(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; m)
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;This implementation overperforms significantly the initial DP and memoization solutions from &lt;a href=&#34;/post/intro-to-dp/&#34;&gt;Introduction to Dynamic Programming and Memoization&lt;/a&gt;.
A naive implementation of &lt;code&gt;math.factorial()&lt;/code&gt; might make $n$ multiplications.
This could still be faster than doing $\Theta(n)$ additions in DP approach.&lt;/p&gt;
&lt;p&gt;The actual implementation of &lt;code&gt;math.factorial()&lt;/code&gt; is written in C
and probably has precomputed results for some range of $n$
and might even cache the results for bigger $n$.&lt;/p&gt;
&lt;h2 id=&#34;implementations-based-on-scipy-and-sympy-libraries&#34;&gt;Implementations Based on &lt;code&gt;scipy&lt;/code&gt; and &lt;code&gt;sympy&lt;/code&gt; Libraries&lt;/h2&gt;
&lt;p&gt;Several third party libraries provide a functionality to compute binomial coefficients.
Let&amp;rsquo;s take a look at &lt;code&gt;scipy&lt;/code&gt; and &lt;code&gt;sympy&lt;/code&gt;.&lt;/p&gt;
&lt;p&gt;We can install both of them using &lt;code&gt;pip&lt;/code&gt;-package manager:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-bash&#34; data-lang=&#34;bash&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;pip install scipy sympy
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can easily write two implementations of $F_{n,m}$ which will call &lt;code&gt;scipy.special.comb()&lt;/code&gt; or &lt;code&gt;sympy.binomial()&lt;/code&gt; functions:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-Python&#34; data-lang=&#34;Python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;import&lt;/span&gt; scipy.special
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;f_sci&lt;/span&gt;(n, m):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;and&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt;m:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; scipy&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;special&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;comb(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, m, exact&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;True&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The second one is very similar to the first one:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-Python&#34; data-lang=&#34;Python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;import&lt;/span&gt; sympy
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;f_sym&lt;/span&gt;(n, m):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;and&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt;m:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; sympy&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;binomial(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, m)
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can use the same &lt;code&gt;test()&lt;/code&gt; helper function that we defined in &lt;a href=&#34;/post/intro-to-dp/&#34;&gt;Introduction to Dynamic Programming and Memoization&lt;/a&gt;.
Let&amp;rsquo;s run it on all the 5 implementations:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;funcs &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; [f_mem, f_dp, f_binom, f_sci, f_sym]
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;test(&lt;span style=&#34;color:#ae81ff&#34;&gt;6000&lt;/span&gt;, &lt;span style=&#34;color:#ae81ff&#34;&gt;2000&lt;/span&gt;, funcs)
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;It will print something similar to following output:&lt;/p&gt;
&lt;pre tabindex=&#34;0&#34;&gt;&lt;code&gt;f(6000,2000): 192496093
       f_mem:   6.7195 sec, x 4195.10
        f_dp:   5.3249 sec, x 3324.43
     f_binom:   0.0016 sec, x 1.00
       f_sci:   0.0021 sec, x 1.32
       f_sym:   0.0043 sec, x 2.69
&lt;/code&gt;&lt;/pre&gt;&lt;p&gt;The first two methods, which are based on memoization and DP, are much slower than the last three,
which are based on the binomial coefficients.&lt;/p&gt;
&lt;h2 id=&#34;the-intuition-for-the-time-complexity-analysis&#34;&gt;The Intuition for the Time Complexity Analysis&lt;/h2&gt;
&lt;p&gt;DP and memoization makes $O(n^2)$ of &amp;ldquo;addition&amp;rdquo; operations over long integers.
The long integers are bounded by $F_{n,m}$ value, which could be bounded by $2^n$.
One &amp;ldquo;addition&amp;rdquo; operation takes $O(N)$ time for $N$-digit integer input.
For our case $N$ could be bounded by $ O(\log 2^n)$ $ = O(n)$.
So the total time complexity is bounded by
$ O(n^2 \cdot N) $
$ = O(n^2 \cdot \log 2^n) $
$ = O(n^2 \cdot n) $
$ = O(n^3)$.&lt;/p&gt;
&lt;p&gt;The binomial based solutions make $O(n)$ &amp;ldquo;multiplication&amp;rdquo; operations over long integers
The long integers could be bounded by $O(n!)$ $ = O(n^n)$.
The &amp;ldquo;multiplication&amp;rdquo; operation could be implemented in a naive way which runs $O(N^2)$ in time and is used for a small input.
But it also has a more advanced implementation, which takes $O(N^{\log_2 3})$ $ = O(N^{1.59})$ and is used on big integers.
Note that here $N$ denotes the number of digits in the input long integers for multiplication.
In this case, $N$ is bounded by
$ O(\log n!) $
$ = O(\log (n^n)) $
$ = O(n \log n) $.
The total time complexity of the binomial based implementations is bounded by
$ O\left(n \cdot N^{\log_2 3}\right) $
$ = O\left(n \cdot (\log n!)^{\log_2 3}\right)$
$ = O\left(n \cdot (n \log n)^{1.59}\right)$
$ = O\left(n^{2.59} \cdot (\log n)^{1.59}\right)$.&lt;/p&gt;
&lt;p&gt;This is not really a formal proof, but it gives some intuition why the last approach overperforms the former one.
In practice, the results of factorial computation are cached,
therefore we observe even bigger gap in performance (yeahh again not formal claim, just an intuition).&lt;/p&gt;
&lt;p&gt;You can play with running tests on different $n$ and $m$.
What I saw that actually there is no clear winner between the last 3 implementations.
Probably, the most of the time is spent on the long arithmetic computation.&lt;/p&gt;
&lt;p&gt;When the problem asks for the answer &lt;strong&gt;modulo a large prime&lt;/strong&gt; (e.g. $10^9+7$), computing everything mod $M$ from the start is much faster than using long integers. We cover that in a &lt;a href=&#34;/post/binomial-modulo-prime/&#34;&gt;separate post&lt;/a&gt;: binomial coefficients modulo a prime using Fermat&amp;rsquo;s little theorem.&lt;/p&gt;
</description>
    </item>
    
    <item>
      <title>Cracking Multivariate Recursive Equations Using Generating Functions</title>
      <link>/post/two-var-recursive-func/</link>
      <pubDate>Thu, 06 Jul 2017 07:29:43 +0000</pubDate>
      <guid>/post/two-var-recursive-func/</guid>
      <description>&lt;p&gt;In this post, we return back to the combinatorial problem discussed in &lt;a href=&#34;/post/intro-to-dp/&#34;&gt;Introduction to Dynamic Programming and Memoization&lt;/a&gt; post.
We will show that generating functions may work great not only for single variable case (see &lt;a href=&#34;/post/gen-func-art/&#34;&gt;The Art of Generating Functions&lt;/a&gt;),
but also could be very useful for hacking two-variable relations (and of course, in general for multivariate case too).&lt;/p&gt;
&lt;p&gt;For making the post self-contained, we repeat the problem definition here.&lt;/p&gt;
&lt;h2 id=&#34;the-problem&#34;&gt;The Problem&lt;/h2&gt;
&lt;blockquote&gt;
&lt;p&gt;Compute the number of ways to choose $m$ elements from $n$ elements such that selected elements in one combination are not adjacent.&lt;/p&gt;&lt;/blockquote&gt;
&lt;p&gt;For example, for $n=4$ and $m=2$, the answer is $3$, since from the $4$-element set: $\lbrace 1,2,3,4 \rbrace$,
there are three feasible $2$-element combinations: $\lbrace 1,4 \rbrace$, $\lbrace 2,4 \rbrace$, $\lbrace 1,3 \rbrace$.&lt;/p&gt;
&lt;p&gt;Another example: for $n=5$ and $m=3$, there is only one $3$-element combination: $\lbrace 1,3,5 \rbrace$.&lt;/p&gt;
&lt;p&gt;As we discussed in the first post, there is a nice recursive relation for this problem:&lt;/p&gt;
&lt;p&gt;$$ F_{n, m} = F_{n - 1, m} + F_{n - 2, m - 1} + [n=m=0] + [n=m=1] $$&lt;/p&gt;
&lt;p&gt;We assume that for any $n &amp;lt; 0$ or $m &amp;lt; 0$, $F_{n,m} = 0$.
The indicator $[P]$ gives $1$ if the predicate $P$ is true.&lt;/p&gt;
&lt;h2 id=&#34;the-generating-function-for-f_nm&#34;&gt;The Generating Function for $F_{n,m}$&lt;/h2&gt;
&lt;p&gt;Let&amp;rsquo;s introduce the generating function $\Phi(x,y)$ of two (floating point) variables $x$ and $y$:&lt;/p&gt;
&lt;p&gt;$$\Phi(x,y) = \sum_{n,m} F_{n,m} x^n y^m$$&lt;/p&gt;
&lt;p&gt;Substituting the definition of $F_{n,m}$ and simplifying the sums, we will get:&lt;/p&gt;
&lt;p&gt;$ \Phi(x,y) $
$ = \sum_{n,m} F_{n,m} x^n y^m $
$ = \sum_{n,m} \left(F_{n - 1, m} + F_{n - 2, m - 1} + [n=m=1] + [n=m=0] \right) x^n y^m$
$ = \sum_{n,m} F_{n - 1, m} x^n y^m + \sum_{n,m} F_{n - 2, m - 1} x^n y^m + x \cdot y + 1 $
$ = x \sum_{n,m} F_{n - 1, m} x^{n-1} y^m + x^2 y \sum_{n,m} F_{n - 2, m - 1} x^{n-2} y^{m-1} + x \cdot y + 1 $
$ = x \Phi(x,y) + x^2 y \Phi(x,y) + x \cdot y + 1 $&lt;/p&gt;
&lt;p&gt;We have just found the simple representation for the generating function:&lt;/p&gt;
&lt;p&gt;$$\Phi(x, y) = \frac{1 + x \cdot y}{1 - x - x^2 y}$$&lt;/p&gt;
&lt;h2 id=&#34;the-closed-form-for-f_nm&#34;&gt;The Closed Form for $F_{n,m}$&lt;/h2&gt;
&lt;p&gt;Using the infinite series: $ \frac{1}{1-z} = \sum_{k\geq 0} z^k $,
we can make the following transforms:&lt;/p&gt;
&lt;p&gt;$\Phi(x, y) $
$ = \frac{1 + x \cdot y}{1 - x - x^2 y}$
$ = (1 + x \cdot y) \sum_{k\geq 0} (x + x^2 y)^k $
$ = (1 + x \cdot y) \sum_{0 \leq i \leq k} {k \choose i} x^{k+i} y^i $
$ = \sum_{0 \leq i \leq k} {k \choose i} x^{k+i} y^i + \sum_{0 \leq i \leq k} {k \choose i} x^{k+i+1} y^{i+1}$&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;
&lt;p&gt;Introducing the new variables: $n=k+i, m=i$, we can transform the first expression as follows:
$ \sum_{0 \leq i \leq k} {k \choose i} x^{k+i} y^i $
$ = \sum_{m \geq 0, n \geq 2m} {n-m \choose m} x^{n} y^m $.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;And introducing the new variables: $n=k+i+1, m=i+1$, we can transform the second expression similarly:
$ \sum_{0 \leq i \leq k} {k \choose i} x^{k+i+1} y^{i+1} $
$ = \sum_{m \geq 1, n \geq 2m-1} {n-m \choose m-1} x^n y^m $.&lt;/p&gt;
&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;The last two transformations give us the closed form:&lt;/p&gt;
&lt;p&gt;$F_{n, m} $
$ = {n - m \choose m} + {n - m \choose m - 1}$.&lt;/p&gt;
&lt;p&gt;Which actually equals to&lt;/p&gt;
&lt;p&gt;$$ F_{n, m} = {n - m + 1 \choose m} $$&lt;/p&gt;
&lt;p&gt;In the &lt;a href=&#34;/post/efficient-implementation-non-adjacent-selection/&#34;&gt;next post&lt;/a&gt;, we implement this closed form in Python (factorial-based and with scipy/sympy). For computing the answer modulo a large prime, see &lt;a href=&#34;/post/binomial-modulo-prime/&#34;&gt;Binomial Coefficients Modulo a Prime&lt;/a&gt;.&lt;/p&gt;
</description>
    </item>
    
    <item>
      <title>The Art of Generating Functions</title>
      <link>/post/gen-func-art/</link>
      <pubDate>Wed, 05 Jul 2017 07:29:43 +0000</pubDate>
      <guid>/post/gen-func-art/</guid>
      <description>&lt;p&gt;The notion of generating functions and its application to solving recursive equations are very well-known.
For reader who did not have a chance to get familiar with the topics,
I recommend to take a look at very good book:
&lt;a href=&#34;https://en.wikipedia.org/wiki/Concrete_Mathematics&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Concrete Mathematics: A Foundation for Computer Science, by Ronald L. Graham, Donald E. Knuth, Oren Patashnik&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;Generating functions are usually applied to single variable recursive equations.
But actually, the technique may be extended to multivariate recursive equations, or even to a system of recursive equations.
Readers who are familiar with one-variable case, may jump directly to the next post:
&lt;a href=&#34;/post/two-var-recursive-func/&#34;&gt;Cracking Multivariate Recursive Equations Using Generating Functions&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id=&#34;the-generating-function-for-fibonacci-sequence&#34;&gt;The Generating Function for Fibonacci Sequence&lt;/h2&gt;
&lt;p&gt;Let&amp;rsquo;s consider the generating function&amp;rsquo;s application to Fibonacci numbers.
The Fibonacci numbers could defined by the recursive expression:&lt;/p&gt;
&lt;p&gt;$$f_n = f_{n-1} + f_{n-2} + [n=0]$$&lt;/p&gt;
&lt;p&gt;We assume that for any $n &amp;lt; 0$, $f_n = 0$.
The indicator $[n=0]$ equals to $1$ only if $n=0$.
This definition produces the following sequence of the Fibonacci numbers: $1$, $1$, $2$, $3$, $5$, $8$, $13$, $\dots$.
Note that sometime, the Fibonacci sequence is defined to start from 0, but it is really not important for the perpose of our discussion.&lt;/p&gt;
&lt;p&gt;The generating function $\Phi(x)$ on a floating point variable $x$ is defined as the infinite sum:&lt;/p&gt;
&lt;p&gt;$$\Phi(x) = \sum_{n\geq 0} f_n x^n$$&lt;/p&gt;
&lt;p&gt;Usually, it is hard to develop an intuition why it is defined that way and why it could be useful.
So let&amp;rsquo;s just focuse on what we can do with this,
and let&amp;rsquo;s start from substituting the defintion of Fibonacci recursion into the formular of generating function:&lt;/p&gt;
&lt;p&gt;$ \Phi(x) $
$ = \sum_{n\geq 0} f_n x^n $
$ = \sum_n f_n x^n $
$ = \sum_n (f_{n-1} + f_{n-2} + [n=0]) x^n $
$ = \sum_n f_{n-1} x^n + \sum_n f_{n-2} x^n + \sum_n [n=0] x^n $
$ = x \sum_n f_{n-1} x^{n-1} + x^2 \sum_n f_{n-2} x^{n-2} + 1 x^0 $
$ = x \sum_n f_n x^n + x^2 \sum_n f_n x^n + 1 $
$ = x \Phi(x) + x^2 \Phi(x) + 1 $&lt;/p&gt;
&lt;p&gt;And we can obtain the generating function for $f_n$:&lt;/p&gt;
&lt;p&gt;$$\Phi(x) = \frac{1}{1 - x - x^2}$$&lt;/p&gt;
&lt;h2 id=&#34;two-closed-forms-for-fibonacci-relation&#34;&gt;Two Closed Forms for Fibonacci Relation&lt;/h2&gt;
&lt;p&gt;Nice expression, but what can we do with this &amp;ldquo;magic&amp;rdquo; formula?
We can hack it in two different ways, and depending on our next step, we can get different closed forms for $f_n$.&lt;/p&gt;
&lt;h3 id=&#34;the-1st-closed-form&#34;&gt;The 1st closed form&lt;/h3&gt;
&lt;p&gt;One of the standard ways is to split the fraction into two simple ones of the form: $\frac{A}{x+B}$.
Note that the roots of the quadratic equation: $1 - x - x^2 = 0$ are $x_0=-\phi$ and $x_1=\phi^{-1}$,
where $\phi$ is the &lt;em&gt;Golden Ratio&lt;/em&gt;:&lt;/p&gt;
&lt;p&gt;$ \phi = \frac{1 + \sqrt{5}}{2} = 1.618\dots $.&lt;/p&gt;
&lt;p&gt;Let&amp;rsquo;s do a quick test:&lt;/p&gt;
&lt;p&gt;$ -(x-x_0) (x-x_1) $
$ = -(x+\phi) \left(x-\phi^{-1}\right) $
$ =-x^2 - x\cdot\left(\phi - \phi^{-1}\right) + 1 $
$ = -x^2 - x + 1 $.&lt;/p&gt;
&lt;p&gt;Now we can transform the generating function $\Phi(x)$ as follows:&lt;/p&gt;
&lt;p&gt;$\Phi(x)$
$ = \frac{1}{1-x-x^2}$
$ = -\frac{1}{(x+\phi) \left(x-\phi^{-1}\right)}$
$ = \frac{A}{x+\phi} - \frac{A}{x-\phi^{-1}}$
$ = A\cdot\phi^{-1} \frac{1}{1+x\cdot\phi^{-1}} + A\cdot\phi \frac{1}{1 - x\cdot\phi}$,&lt;/p&gt;
&lt;p&gt;where $A=\frac{1}{\phi+\phi^{-1}}$.&lt;/p&gt;
&lt;p&gt;The next trick is to apply the infinite series $\frac{1}{1-z} = \sum_n z^n$ to each of two expressions and to simplify the sums:&lt;/p&gt;
&lt;p&gt;$\Phi(x) $
$ = A\cdot\phi^{-1} \sum_n \left(-x\cdot \phi^{-1}\right)^n + A\cdot\phi \sum_n (x\cdot\phi)^n $
$ = A\cdot\phi^{-1} \sum_n (-\phi)^{-n} x^n + A\cdot\phi \sum_n \phi^n x^n $
$ = \sum_n A\cdot\left(\phi^{-1} (-\phi)^{-n} + \phi\cdot \phi^n \right)  x^n $
$ = \sum_n A\cdot\left(\phi^{n+1} - (-\phi)^{-n-1}\right) x^n $
$ = \sum_n \frac{\phi^{n+1} - (-\phi)^{-n-1}}{\phi+\phi^{-1}} x^n $.&lt;/p&gt;
&lt;p&gt;The term before $x^n$ is nothing but $f_n$, which gives the first closed form for the Fibonacci relation:&lt;/p&gt;
&lt;p&gt;$$f_n=\frac{\phi^{n+1} - (-\phi)^{-n-1}}{\phi+\phi^{-1}}$$&lt;/p&gt;
&lt;h3 id=&#34;the-2nd-closed-form&#34;&gt;The 2nd closed form&lt;/h3&gt;
&lt;p&gt;Another way to crack the generating function is to apply the infinite series $\frac{1}{1-z} $ $= \sum_n z^n$ directly to the generating function:&lt;/p&gt;
&lt;p&gt;$ \Phi(x) $
$ = \frac{1}{1 - x - x^2} $
$ = \frac{1}{1 - (x+x^2)}$
$ = \sum_n (x+x^2)^n $
$ = \sum_n (1+x)^n x^n $&lt;/p&gt;
&lt;p&gt;Now we can use the Binomial formula: $(a + b)^n = \sum_{0\leq k\leq n} {n \choose k}  a^{n-k} b^k$ in order to expand $(1+x)^n$:&lt;/p&gt;
&lt;p&gt;$ = \sum_{0\leq k \leq n} {n \choose k} x^{n+k} $.&lt;/p&gt;
&lt;p&gt;By introducing the new variable $t=n+k$, we obtain&lt;/p&gt;
&lt;p&gt;$ = \sum_{t/2 \leq n \leq t} {n \choose t-n} x^t $
$ = \sum_t \sum_{n=t/2}^t {n \choose t-n} x^t $&lt;/p&gt;
&lt;p&gt;Which gives us another closed form for the Fibonacci numbers:&lt;/p&gt;
&lt;p&gt;$$f_t = \sum_{n=t/2}^t {n \choose t-n}$$&lt;/p&gt;
&lt;h2 id=&#34;conclusion&#34;&gt;Conclusion&lt;/h2&gt;
&lt;p&gt;Believe it or not, but the two forms define the same Fibonacci sequence.&lt;/p&gt;
&lt;p&gt;$$ f_n = \frac{\phi^{n+1} - (-\phi)^{-n-1}}{\phi+\phi^{-1}} $$&lt;/p&gt;
&lt;p&gt;$$ f_n = \sum_{i=n/2}^n {i \choose n-i} $$&lt;/p&gt;
&lt;p&gt;Just notice how powerful the generating functions are!
An interested reader can try to prove by induction the correctness of the relations above.
If you liked the topic, you are wellcome to take a look at the next post where we extend the tools to mutivariate recursions.&lt;/p&gt;
</description>
    </item>
    
    <item>
      <title>Introduction to Dynamic Programming and Memoization</title>
      <link>/post/intro-to-dp/</link>
      <pubDate>Tue, 04 Jul 2017 07:29:43 +0000</pubDate>
      <guid>/post/intro-to-dp/</guid>
      <description>&lt;p&gt;In the post, we discuss the basics of &lt;em&gt;Recursion&lt;/em&gt;, &lt;em&gt;Dynamic Programming&lt;/em&gt; (DP), and &lt;em&gt;Memoization&lt;/em&gt;.
As an example, we take a combinatorial problem, which has very short and clear description.
This allows us to focus on DP and memoization.
Note that the topics are very popular in coding interviews.
Hopefully, this article will help to somebody to prepare for such types of questions.&lt;/p&gt;
&lt;p&gt;In the next posts,
we consider more advanced topics, like
&lt;a href=&#34;/post/gen-func-art/&#34;&gt;The Art of Generating Functions&lt;/a&gt; and
&lt;a href=&#34;/post/two-var-recursive-func/&#34;&gt;Cracking Multivariate Recursive Equations Using Generating Functions&lt;/a&gt;.
The methods can be applied to the same combinatorial question.
Let&amp;rsquo;s start from presenting the problem.&lt;/p&gt;
&lt;h2 id=&#34;the-problem&#34;&gt;The Problem&lt;/h2&gt;
&lt;blockquote&gt;
&lt;p&gt;Compute the number of ways to choose $m$ elements from $n$ elements such that selected elements in one combination are not adjacent.&lt;/p&gt;&lt;/blockquote&gt;
&lt;p&gt;For example, for $n=4$ and $m=2$, the answer is $3$, since from the $4$-element set: $\lbrace 1,2,3,4 \rbrace$,
there are three feasible $2$-element combinations: $\lbrace 1,4 \rbrace$, $\lbrace 2,4 \rbrace$, $\lbrace 1,3 \rbrace$.&lt;/p&gt;
&lt;p&gt;Another example: for $n=5$ and $m=3$, there is only one $3$-element combination: $\lbrace 1,3,5 \rbrace$.&lt;/p&gt;
&lt;p&gt;If you are asked such question during coding interview, interviewer is, probably, expecting to cover with you the following topics:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;em&gt;Brute Force&lt;/em&gt; approach that generates and counts all the feasible combinations. It will work slow even for small input.&lt;/li&gt;
&lt;li&gt;Recursive solution which counts the combination without generating them. It will work faster but still has exponential time complexity.&lt;/li&gt;
&lt;li&gt;Use DP or memoization techniques. In both cases, the time complexity becomes linear in $n$ and $m$.&lt;/li&gt;
&lt;li&gt;Corner cases, recursion termination, call stack, testing.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;Let&amp;rsquo;s talk about most important topics: building a recursive solution and optimize it using DP or memoization.&lt;/p&gt;
&lt;h2 id=&#34;recursive-relation&#34;&gt;Recursive Relation&lt;/h2&gt;
&lt;p&gt;Let&amp;rsquo;s define $F_{n, m}$ to be the function that computes the answer for given $n$ and $m$.
Let&amp;rsquo;s look at the $n, m$-task. We have two non-overlapping sub-tasks (or cases):&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Skip the $n$-th element, then $ F_{n, m} = F_{n-1, m} $.&lt;/li&gt;
&lt;li&gt;Pick the $n$-th element, then $ F_{n, m} = F_{n-2, m-1} $.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;From the above, we can define the solution in the recursive form:&lt;/p&gt;
&lt;p&gt;$ F_{n, m} $ $ = F_{n - 1, m} + F_{n - 2, m - 1} $,&lt;/p&gt;
&lt;p&gt;let&amp;rsquo;s not forget to write down the corner cases:&lt;/p&gt;
&lt;p&gt;$F_{0, 0}$ $= F_{1, 1}$ $= 1$.&lt;/p&gt;
&lt;p&gt;Basically, we can combine the general and the corner cases into one expression:&lt;/p&gt;
&lt;p&gt;$$ F_{n, m} = F_{n - 1, m} + F_{n - 2, m - 1} + [n=m=0] + [n=m=1] $$&lt;/p&gt;
&lt;p&gt;It is very common to define $F_{n,m} = 0$ for any negative $n$ or $m$.
The indicator $[P]$ gives $1$ if the predicate $P$ is true.
But what about special cases, like $n=1$ and $m=0$?
Actually, the relation covers this:&lt;/p&gt;
&lt;p&gt;$F_{1,0}$ $= F_{0,0} + F_{-1,-1} + [1=0=1] + [1=m=1]$ $=F_{0,0} + 0 + 0 + 0$ $=[0=0=0]$ $=1$.&lt;/p&gt;
&lt;p&gt;It is non-intuitive, but there is no need to add extra corner cases!&lt;/p&gt;
&lt;h2 id=&#34;memoization&#34;&gt;Memoization&lt;/h2&gt;
&lt;p&gt;The function $F_{n,m}$ has a straighforward recursive implementation in any programming language.
But the naive recursive solution will have exponential in $n$ time complexity and will be very slow.
Let&amp;rsquo;s look at the following &lt;code&gt;Python 3&lt;/code&gt; code snippet that uses memoization technique:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;import&lt;/span&gt; functools
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;import&lt;/span&gt; sys
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;sys&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;setrecursionlimit(&lt;span style=&#34;color:#ae81ff&#34;&gt;100000&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#a6e22e&#34;&gt;@functools.lru_cache&lt;/span&gt;(maxsize&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;None&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;f_mem&lt;/span&gt;(n: int, m: int) &lt;span style=&#34;color:#f92672&#34;&gt;-&amp;gt;&lt;/span&gt; int:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;or&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt;m:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;==&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;==&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;==&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;==&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; f_mem(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, m) &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; f_mem(n &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;, m &lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Nice &lt;a href=&#34;https://docs.python.org/3/library/functools.html#functools.lru_cache&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;@functools.lru_cache&lt;/a&gt;
notation creates a wrapper on the &lt;code&gt;f_mem&lt;/code&gt; function and internally caches the results of all calls.
Such caching (or memoization) significantly improves the speed of the recursion,
and basically reduces the number of calls to something like $O(n \cdot m)$.
The recursion still may fail on the stack overflow even on relatively small values of $n$.
That is why we increase the stack size for the Python interpreter by calling
&lt;a href=&#34;https://docs.python.org/3/library/sys.html#sys.setrecursionlimit&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;sys.setrecursionlimit()-method&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id=&#34;dynamic-programming-dp&#34;&gt;Dynamic Programming (DP)&lt;/h2&gt;
&lt;p&gt;The next iterative implementation uses DP.
Basically, it fills out the $n \times m$ table starting from the low values of $n$ and $m$.&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;f_dp&lt;/span&gt;(n: int, m: int) &lt;span style=&#34;color:#f92672&#34;&gt;-&amp;gt;&lt;/span&gt; int:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;and&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt;m:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    table &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; [[&lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;] &lt;span style=&#34;color:#f92672&#34;&gt;*&lt;/span&gt; (m &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;) &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; _ &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; range(n &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;)]
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    table[&lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;][&lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;] &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; n &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        table[&lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;][&lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;] &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; m &lt;span style=&#34;color:#f92672&#34;&gt;&amp;gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;            table[&lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;][&lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;] &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; i &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; range(&lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;, n&lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        table[i][&lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;] &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; j &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; range(&lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, min(m, (i &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;) &lt;span style=&#34;color:#f92672&#34;&gt;//&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;) &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;            table[i][j] &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; table[i&lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;][j] &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; table[i&lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;2&lt;/span&gt;][j&lt;span style=&#34;color:#f92672&#34;&gt;-&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;]
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;return&lt;/span&gt; table[n][m]
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The time and space complexities are $O(n\cdot m) $.
More accurately, time complexity might be bounded by $\Theta(n\cdot \min(n,m))$.
It is not hard to notice that the space consumption could be reduced to $O(\min(n, m))$.&lt;/p&gt;
&lt;h2 id=&#34;testing&#34;&gt;Testing&lt;/h2&gt;
&lt;p&gt;In order to measure the time and to check the correctness, let&amp;rsquo;s write the following helper function:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;import&lt;/span&gt; timeit
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;M &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1000&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;**&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;3&lt;/span&gt; &lt;span style=&#34;color:#f92672&#34;&gt;+&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;7&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;&lt;span style=&#34;color:#66d9ef&#34;&gt;def&lt;/span&gt; &lt;span style=&#34;color:#a6e22e&#34;&gt;test&lt;/span&gt;(n, m, funcs, number&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;, module&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;__name__):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    f_mem&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;cache_clear()
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    results &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; []
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    func_times &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; []
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; func &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; funcs:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        stmt&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;(&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;,&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;)&amp;#39;&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;format(func&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;__name__, n, m)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        setup&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#39;from &lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt; import &lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;format(module, func&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;__name__)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        t &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; timeit&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;timeit(stmt&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;stmt, setup&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;setup, number&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;number)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        func_times&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;append(t)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        results&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;append(func(n, m) &lt;span style=&#34;color:#f92672&#34;&gt;%&lt;/span&gt; M)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;assert&lt;/span&gt; len(set(results)) &lt;span style=&#34;color:#f92672&#34;&gt;&amp;lt;=&lt;/span&gt; &lt;span style=&#34;color:#ae81ff&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;if&lt;/span&gt; results:
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        print(&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#39;f(&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;,&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;): &lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;format(n, m, results[&lt;span style=&#34;color:#ae81ff&#34;&gt;0&lt;/span&gt;]))
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    best_time &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; min(func_times)
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;    &lt;span style=&#34;color:#66d9ef&#34;&gt;for&lt;/span&gt; i, func &lt;span style=&#34;color:#f92672&#34;&gt;in&lt;/span&gt; enumerate(funcs):
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        func_time &lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt; func_times[i]
&lt;/span&gt;&lt;/span&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;        print(&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{:&amp;gt;13}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;: &lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{:8.4f}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt; sec, x &lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;{:.2f}&lt;/span&gt;&lt;span style=&#34;color:#e6db74&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;format(func&lt;span style=&#34;color:#f92672&#34;&gt;.&lt;/span&gt;__name__, func_time, func_time&lt;span style=&#34;color:#f92672&#34;&gt;/&lt;/span&gt;best_time))
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can run it as following:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; style=&#34;color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span style=&#34;display:flex;&#34;&gt;&lt;span&gt;test(n&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;6000&lt;/span&gt;, m&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;&lt;span style=&#34;color:#ae81ff&#34;&gt;2000&lt;/span&gt;, funcs&lt;span style=&#34;color:#f92672&#34;&gt;=&lt;/span&gt;[f_mem, f_dp])
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Which prints output similar to this one:&lt;/p&gt;
&lt;pre tabindex=&#34;0&#34;&gt;&lt;code&gt;f(6000,2000): 192496093
        f_mem:   6.5852 sec, x 1.28
         f_dp:   5.1507 sec, x 1.00
&lt;/code&gt;&lt;/pre&gt;&lt;p&gt;The function &lt;code&gt;test()&lt;/code&gt; computes $F_{6000, 2000}$ using two different ways.
It validates that all the solutions are consistent and produce the same result.
It also measures the time it takes to execute each method.
For each run, it also prints the relative factor computed based on the fastest solution.
The function&amp;rsquo;s result is printed modulo $M=1000^3+7$.&lt;/p&gt;
&lt;p&gt;Memoization and DP techniques have $\Theta(n \cdot m)$ time complexity.
Note that we ignore here a lot of interesting details, for example,
Python has built-in long arithmetics for integers, which is used here and definetely not cheap.&lt;/p&gt;
&lt;p&gt;Testing the implementations on different parameters,
we may notice that there is no clear winner between the two algorithms.
Probably, most of the time is spent on the long arithmetic computation.&lt;/p&gt;
&lt;p&gt;Can we do even better?
Definitely, we can!
The table (or cache) could be preprocessed.
Then computing the function $F_{n,m}$ will require just one query from the table (or from the cache, for memoization solution).
As we mentioned before, such approach requires $\Theta(n\cdot m)$ space, which could be huge for $n=10000$.
Actually, there is a way, how we can remain in very low memory consuption and still get very fast solution.
We will discuss this in the next two posts.&lt;/p&gt;
</description>
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