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+"""
+Library Sort (Gapped Insertion) — Enhanced and Documented
+
+Concept:
+- Library Sort is a gapped version of insertion sort that keeps extra empty
+  slots between placed elements, so most insertions do not require shifting
+  a long suffix of the array. The algorithm occasionally "rebalances" the
+  layout to redistribute elements and restore evenly spaced gaps. This yields
+  an expected average complexity of O(n log n) under typical (non-adversarial)
+  input distributions.
+
+API:
+- library_sort(data, key=None, reverse=False, epsilon=1.0) -> List
+  - data: any iterable of items
+  - key: optional key extractor like built-in sorted()/list.sort()
+  - reverse: set True for descending order
+  - epsilon: extra-space factor controlling how many gaps are left in the
+             backing array (larger epsilon = more gaps = fewer collisions,
+             more memory)
+
+Complexity (informal):
+- Average (with high probability): O(n log n) thanks to binary search for the
+  logical order plus amortized O(1) gap insertions.
+- Worst case: O(n^2) if inputs are adversarial and repeatedly force dense
+  regions, even after rebalancing.
+- Space: O(n * (1 + epsilon)) for the gapped array.
+
+Notes for learners:
+- We maintain two parallel arrays:
+  - A_keys: holds keys in a sparse (gapped) physical layout.
+  - A_vals: holds the original values at the same indices as A_keys.
+- We also maintain a 'pos' list that records the indices of the FILLED
+  positions in sorted, logical order. This lets us:
+  (1) binary-search the correct logical position for a new key
+  (2) ask for a "desired physical slot" roughly midway between neighbors
+      so we’re likely to find (or make) a gap near that logical position.
+- If there is no gap at the desired slot, we "rebalance" by redistributing
+  the existing items farther apart, restoring even gaps before retrying.
+
+This annotated implementation focuses on clarity and pedagogy rather than
+micro-optimizations, so readers can trace each step of the algorithm.
+"""
+
+from typing import Callable, Iterable, List, Optional, Tuple
+import bisect
+
+
+def library_sort(
+    data: Iterable,
+    key: Optional[Callable] = None,
+    reverse: bool = False,
+    epsilon: float = 1.0,
+) -> List:
+    """
+    Sort 'data' using Library Sort (gapped insertion) and return a new list.
+
+    Parameters
+    ----------
+    data : Iterable
+        Items to sort.
+    key : Callable | None
+        Optional key extractor (like built-in sorted()).
+    reverse : bool
+        If True, return results in descending order.
+    epsilon : float
+        Extra-space factor controlling gap density; larger values create
+        more gaps, which reduces collisions but uses more memory.
+
+    Returns
+    -------
+    List
+        A new list containing the sorted items.
+
+    Teaching tip:
+    - Think of 'pos' as the logical, in-order view (where elements "should"
+      be if there were no gaps), and 'A_keys/A_vals' as the physical shelves
+      that include empty spots to make insertions cheap.
+    """
+    # Materialize input and handle trivial sizes.
+    items = list(data)
+    n = len(items)
+    if n < 2:
+        return items.copy()
+
+    # Normalize to a key/value representation so we can sort arbitrary objects.
+    key_fn = key if key is not None else (lambda x: x)
+    keyed: List[Tuple] = [(key_fn(x), x) for x in items]
+
+    # Capacity with slack: leave about 'epsilon' * n empty slots as gaps.
+    # The +1 ensures at least one spare slot even for small inputs.
+    cap = max(3, int((1.0 + epsilon) * n) + 1)
+
+    # Sparse physical storage for keys/values; None marks a gap (empty slot).
+    A_keys: List[Optional[Tuple]] = [None] * cap
+    A_vals: List[Optional[object]] = [None] * cap
+
+    # 'pos' tracks indices of FILLED slots in sorted order of keys.
+    # This lets us binary-search by logical rank, independent of gaps.
+    pos: List[int] = []
+
+    # Seed the structure by placing the first element near the middle so
+    # we can grow to both sides without immediate rebalancing.
+    mid = cap // 2
+    A_keys[mid] = keyed[0][0]
+    A_vals[mid] = keyed[0][1]
+    pos.append(mid)
+
+    def rebalance(target_count: int) -> None:
+        """
+        Redistribute elements with fresh gaps.
+
+        Given we currently have 'target_count' filled items, rebuild 'pos'
+        into a new array of size ≈ (1 + epsilon) * target_count, spacing
+        items roughly evenly so subsequent insertions are likely to find
+        nearby gaps.
+        """
+        nonlocal A_keys, A_vals, pos, cap
+
+        # Grow capacity if needed to preserve slack proportional to item count.
+        cap = max(cap, int((1.0 + epsilon) * target_count) + 3)
+
+        # Compute a stride so that (target_count) items are spaced out with gaps.
+        step = max(1, cap // (target_count + 1))
+        start = step // 2  # small offset so ends aren't packed
+
+        new_keys: List[Optional[Tuple]] = [None] * cap
+        new_vals: List[Optional[object]] = [None] * cap
+        new_pos: List[int] = []
+
+        # Copy each existing filled slot to its new, spaced-out location.
+        for i, old_idx in enumerate(pos):
+            new_index = start + i * step
+            new_keys[new_index] = A_keys[old_idx]
+            new_vals[new_index] = A_vals[old_idx]
+            new_pos.append(new_index)
+
+        A_keys, A_vals, pos = new_keys, new_vals, new_pos
+
+    def desired_slot(rank: int) -> int:
+        """
+        Given the logical insertion rank (the index where the new key would go
+        in sorted order), return a physical index that lies between its neighbors.
+
+        This heuristic aims for the midpoint between adjacent filled indices
+        to maximize the chance we land on, or near, a gap.
+        """
+        if rank == 0:
+            return pos[0] - 1  # just before first filled slot
+        if rank == len(pos):
+            return pos[-1] + 1  # just after last filled slot
+        return (pos[rank - 1] + pos[rank]) // 2  # midpoint between neighbors
+
+    # Insert remaining items one by one.
+    for k, v in keyed[1:]:
+        # Binary-search the logical order of existing keys using 'pos'.
+        logical_keys = [A_keys[i] for i in pos]
+        ins_rank = bisect.bisect_left(logical_keys, k)
+
+        tries = 0
+        while True:
+            # Ask for a good physical slot near the desired rank.
+            idx = desired_slot(ins_rank)
+
+            # If it's a valid gap, claim it and record the new filled position.
+            if 0 <= idx < cap and A_keys[idx] is None:
+                A_keys[idx] = k
+                A_vals[idx] = v
+                pos.insert(ins_rank, idx)
+                break
+
+            # Otherwise, things are too dense around there; rebalance to
+            # re-open gaps and try again.
+            tries += 1
+            rebalance(len(pos) + 1)
+
+            # Safety valve: if local density remains high after a few passes,
+            # gradually increase epsilon (more gaps) and rebalance again.
+            if tries > 3:
+                epsilon *= 1.25
+                rebalance(len(pos) + 1)
+                tries = 0
+
+    # Stitch the final, in-order values back together using 'pos'.
+    out = [A_vals[i] for i in pos]  # type: ignore
+    if reverse:
+        out.reverse()
+    return out
+
+
+if __name__ == "__main__":
+    # Minimal demo to visualize behavior.
+    data = [34, 7, 23, 32, 5, 62, 14, 19, 45, 38]
+    print("Before:", data)
+    print("After: ", library_sort(data))