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+# Sorting Algorithms
+
+## Overview
+Sorting algorithms reorder elements of a collection according to a defined order
+(e.g. ascending or descending). They are fundamental building blocks used in
+searching, optimization, and data processing.
+
+Different sorting algorithms make different trade-offs in terms of:
+- time complexity
+- space usage
+- stability
+- adaptability to input characteristics
+
+---
+
+## Classification
+
+### Comparison-based
+- Merge Sort
+- Quick Sort
+- Heap Sort
+
+### Non-comparison-based
+- Counting Sort
+- Radix Sort
+- Bucket Sort
+
+---
+
+## When to use which
+
+| Algorithm      | Time (avg) | Space | Stable | Notes |
+|---------------|-----------|-------|--------|------|
+| Merge Sort    | O(n log n) | O(n)  | Yes    | Predictable, good for linked lists |
+| Quick Sort    | O(n log n) | O(log n) | No | Fast in practice, bad worst-case |
+| Heap Sort     | O(n log n) | O(1)  | No     | In-place, not stable |
+| Counting Sort | O(n + k)   | O(k)  | Yes    | Only for small integer ranges |
+
+---
+
+## Related techniques
+- Divide and Conquer
+- Heap data structure
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+# Merge Sort
+
+## Problem
+Given a collection of elements, sort them in non-decreasing order.
+
+Merge Sort is a **comparison-based, divide-and-conquer sorting algorithm**.
+
+---
+
+## High-level idea
+The algorithm recursively divides the collection into two halves until subarrays
+of size one are reached. These subarrays are then merged back together in sorted
+order.
+
+The key insight is that **merging two already sorted arrays can be done efficiently**.
+
+---
+
+## Algorithm steps
+1. Divide the array into two halves.
+2. Recursively apply Merge Sort to each half.
+3. Merge the two sorted halves into a single sorted array.
+
+---
+
+## Correctness argument
+Merge Sort is correct by induction on the size of the array.
+
+**Base case:**  
+An array of size 0 or 1 is trivially sorted.
+
+**Inductive step:**  
+Assume Merge Sort correctly sorts arrays of size less than `n`.  
+For an array of size `n`, the algorithm:
+- divides it into two smaller arrays,
+- recursively sorts each one (by the inductive hypothesis),
+- merges the two sorted arrays into a fully sorted array.
+
+Since the merge step preserves order, the final array is sorted.
+
+---
+
+## Complexity
+- **Time complexity:** `O(n log n)` in all cases
+- **Space complexity:** `O(n)` due to auxiliary arrays
+
+---
+
+## Properties
+- Stable: ✅
+- In-place: ❌
+- Comparison-based: ✅
+
+---
+
+## When to use
+- When predictable performance is required
+- When stability matters
+- When working with linked lists or external sorting
+
+---
+
+## When NOT to use
+- When memory usage must be minimal
+- When in-place sorting is required
+
+---
+
+## Related concepts
+- Divide and Conquer
+- Recursion
+- External sorting
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