Add Dancing Links (DLX) algorithm for Exact Cover problem

Author: fab-c14

other/dancing_links.py

@@ -0,0 +1,82 @@
+"""
+Dancing Links (DLX) Algorithm for Exact Cover Problem
+
+Author: fab-c14
+Reference: Donald Knuth, "Dancing Links" (Algorithm X)
+Wikipedia: https://en.wikipedia.org/wiki/Dancing_Links
+
+DLX is an efficient algorithm for solving the Exact Cover problem, such as
+tiling, polyomino puzzles, or Sudoku.
+
+This implementation demonstrates DLX for a small exact cover problem.
+
+Usage Example:
+>>> universe = [1, 2, 3, 4, 5, 6, 7]
+>>> subsets = [
+...     [1, 4, 7],
+...     [1, 4],
+...     [4, 5, 7],
+...     [3, 5, 6],
+...     [2, 3, 6, 7],
+...     [2, 7]
+... ]
+>>> for solution in dlx(universe, subsets):
+...     print(solution)
+[0, 3, 4]
+[1, 2, 5]
+"""
+
+from collections.abc import Iterator
+
+
+def dlx(universe: list[int], subsets: list[list[int]]) -> Iterator[list[int]]:
+    """Yields solutions to the Exact Cover problem using Algorithm X (Dancing Links)."""
+    cover: dict[int, set[int]] = {u: set() for u in universe}
+    for idx, subset in enumerate(subsets):
+        for elem in subset:
+            cover[elem].add(idx)
+    partial: list[int] = []
+
+    def search() -> Iterator[list[int]]:
+        if not cover:
+            yield list(partial)
+            return
+        # Choose column with fewest rows (heuristic)
+        c = min(cover, key=lambda col: len(cover[col]))
+        for r in list(cover[c]):
+            partial.append(r)
+            removed: dict[int, set[int]] = {}
+            for j in subsets[r]:
+                for i in cover[j].copy():
+                    for k in subsets[i]:
+                        if k == j:
+                            continue
+                        if k in cover:
+                            cover[k].discard(i)
+                removed[j] = cover.pop(j)
+            yield from search()
+            # Backtrack
+            for j, s in removed.items():
+                cover[j] = s
+                for i in cover[j]:
+                    for k in subsets[i]:
+                        if k != j and k in cover:
+                            cover[k].add(i)
+            partial.pop()
+
+    yield from search()
+
+
+if __name__ == "__main__":
+    # Example: Solve the cover problem from Knuth's original paper
+    universe = [1, 2, 3, 4, 5, 6, 7]
+    subsets = [
+        [1, 4, 7],
+        [1, 4],
+        [4, 5, 7],
+        [3, 5, 6],
+        [2, 3, 6, 7],
+        [2, 7],
+    ]
+    for solution in dlx(universe, subsets):
+        print("Solution:", solution)