Add Dancing Links (DLX) algorithm for Exact Cover problem (#13183)

* Add Dancing Links (DLX) algorithm for Exact Cover problem

* Fix DancingLinks algorithm

* Add Wikipedia links for Algorithm X and Dancing Links

Added references to Wikipedia for Algorithm X and Dancing Links.

* Add type hints to Dancing Links classes and methods

---------

Co-authored-by: Christian Clauss <cclauss@me.com>

Author: fab-c14

other/dancing_links.py

@@ -0,0 +1,165 @@
+"""
+Implementation of the Dancing Links algorithm (Algorithm X) by Donald Knuth.
+https://en.wikipedia.org/wiki/Knuth's_Algorithm_X
+https://en.wikipedia.org/wiki/Dancing_links
+
+>>> universe = [1, 2, 3, 4, 5, 6, 7]
+>>> subsets = [
+...     [1, 4, 7],
+...     [1, 4],
+...     [4, 5, 7],
+...     [3, 5, 6],
+...     [2, 3, 6, 7],
+... ]
+>>> dlx = DancingLinks(universe, subsets)
+>>> sols = dlx.solve()
+>>> len(sols) == 0
+True
+"""
+
+
+class DLXNode:
+    """Represents a node in the Dancing Links structure."""
+
+    def __init__(self) -> None:
+        self.left = self.right = self.up = self.down = self
+        self.column = None
+
+
+class ColumnNode(DLXNode):
+    """Represents a column header node, keeping track of its column size."""
+
+    def __init__(self, name: str) -> None:
+        super().__init__()
+        self.name = name
+        self.size = 0
+
+
+class DancingLinks:
+    """Dancing Links structure for solving the Exact Cover problem."""
+
+    def __init__(self, universe: list[int], subsets: list[list[int]]) -> None:
+        self.header = ColumnNode("header")
+        self.columns = {}
+        self.solution = []
+        self.solutions = []
+
+        # Create column headers for each element in the universe
+        prev = self.header
+        for u in universe:
+            col = ColumnNode(u)
+            self.columns[u] = col
+            col.left, col.right = prev, self.header
+            prev.right = col
+            self.header.left = col
+            prev = col
+
+        # Add rows (subsets)
+        for subset in subsets:
+            first_node = None
+            for item in subset:
+                col = self.columns[item]
+                node = DLXNode()
+                node.column = col
+
+                # Insert node into column
+                node.down = col
+                node.up = col.up
+                col.up.down = node
+                col.up = node
+                col.size += 1
+
+                # Link nodes in the same row
+                if first_node is None:
+                    first_node = node
+                else:
+                    node.left = first_node.left
+                    node.right = first_node
+                    first_node.left.right = node
+                    first_node.left = node
+
+    def _cover(self, col: ColumnNode) -> None:
+        """Covers a column (removes it from the matrix)."""
+        col.right.left = col.left
+        col.left.right = col.right
+        row = col.down
+        while row != col:
+            node = row.right
+            while node != row:
+                node.down.up = node.up
+                node.up.down = node.down
+                node.column.size -= 1
+                node = node.right
+            row = row.down
+
+    def _uncover(self, col: ColumnNode):
+        """Uncovers a column (reverses _cover)."""
+        row = col.up
+        while row != col:
+            node = row.left
+            while node != row:
+                node.column.size += 1
+                node.down.up = node
+                node.up.down = node
+                node = node.left
+            row = row.up
+        col.right.left = col
+        col.left.right = col
+
+    def _choose_column(self) -> ColumnNode:
+        """Select the column with the smallest size (heuristic)."""
+        min_size = float("inf")
+        chosen = None
+        col = self.header.right
+        while col != self.header:
+            if col.size < min_size:
+                min_size = col.size
+                chosen = col
+            col = col.right
+        return chosen
+
+    def _search(self) -> None:
+        """Recursive Algorithm X search."""
+        if self.header.right == self.header:
+            # All columns covered -> valid solution
+            self.solutions.append([node.column.name for node in self.solution])
+            return
+
+        col = self._choose_column()
+        if col is None:
+            return
+
+        self._cover(col)
+
+        row = col.down
+        while row != col:
+            self.solution.append(row)
+
+            node = row.right
+            while node != row:
+                self._cover(node.column)
+                node = node.right
+
+            self._search()
+
+            # Backtrack
+            self.solution.pop()
+            node = row.left
+            while node != row:
+                self._uncover(node.column)
+                node = node.left
+
+            row = row.down
+
+        self._uncover(col)
+
+    def solve(self) -> list:
+        """Find all exact cover solutions."""
+        self._search()
+        return self.solutions
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()