Add Hopcroft-Karp algorithm for maximum bipartite matching

graphs/hopcroft_karp.py

@@ -0,0 +1,325 @@
+"""Hopcroft-Karp algorithm for finding maximum cardinality matching in bipartite graphs.
+
+Reference:
+    https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm
+
+The Hopcroft-Karp algorithm finds a maximum cardinality matching in an unweighted
+bipartite graph in O(|E| * sqrt(|V|)) time.
+
+Key Concepts and Conditions:
+1. Bipartite Condition:
+   A graph G = (U union V, E) is bipartite if its vertices can be partitioned into
+   two disjoint sets U (left partition) and V (right partition) such that every
+   edge connects a vertex in U to a vertex in V. No edges may exist between two
+   vertices within the same partition (U intersect V = empty set). Vertices cannot
+   be None.
+
+2. Matching Condition:
+   A matching M is a subset of edges such that no two edges share a common vertex.
+   A vertex is 'free' (unmatched) if it is not incident to any edge in M.
+
+3. Alternating and Augmenting Paths:
+   - Alternating path: A path whose edges alternate between unmatched edges
+     (not in M) and matched edges (in M).
+   - Augmenting path: An alternating path that starts and ends at distinct free
+     vertices.
+   - Berge's Lemma: A matching is of maximum cardinality if and only if no
+     augmenting paths exist.
+
+4. Hopcroft-Karp Layering and Augmentation Conditions:
+   Instead of searching for augmenting paths one-by-one (O(|V| * |E|)), Hopcroft-Karp
+   operates in phases:
+   - BFS Phase (Layering): Simultaneously searches from all free vertices in U to
+     find the length of the shortest augmenting paths. It builds a layered DAG of
+     alternating levels. If no free vertex in V is reachable, the algorithm terminates.
+   - DFS Phase (Augmentation): Discovers a maximal set of vertex-disjoint augmenting
+     paths of the shortest length found by BFS. It only traverses edges satisfying:
+     distance_map[matched_left] == distance_map[curr_left] + 1.
+   - Symmetric Difference: Matching edges along each augmenting path are flipped
+     (unmatched becomes matched, matched becomes unmatched).
+   - Iterative DFS: The DFS phase is implemented iteratively using an explicit stack
+     to prevent RecursionError on graphs with large alternating path diameters.
+
+Complexity:
+    Time Complexity:  O(|E| * sqrt(|V|))
+    Space Complexity: O(|V| + |E|)
+"""
+
+from __future__ import annotations
+
+import math
+from collections import deque
+
+_NIL = object()
+
+
+class HopcroftKarp[T]:
+    """Class implementing the Hopcroft-Karp maximum bipartite matching algorithm.
+
+    >>> hk = HopcroftKarp({"u1": ["v1", "v2"], "u2": ["v1"], "u3": ["v2", "v3"]})
+    >>> hk.maximum_matching()
+    {'u1': 'v2', 'u2': 'v1', 'u3': 'v3'}
+    """
+
+    def __init__(self, graph: dict[T, list[T]]) -> None:
+        """Initialize bipartite partitions and match pairing dictionaries.
+
+        Raises:
+            ValueError: If partitions overlap or if any vertex is None.
+
+        >>> hk = HopcroftKarp({"u1": ["v1"]})
+        >>> hk.left_vertices
+        ['u1']
+        >>> hk.right_vertices
+        ['v1']
+        >>> HopcroftKarp({"A": ["A"]})
+        Traceback (most recent call last):
+            ...
+        ValueError: Partitions must be disjoint: found vertices in both sets: ['A']
+        >>> HopcroftKarp({"u1": [None]})
+        Traceback (most recent call last):
+            ...
+        ValueError: Vertices cannot be None
+        """
+        self.graph = graph
+        self.left_vertices = list(graph.keys())
+        self.right_vertices = sorted(
+            {
+                right_vertex
+                for neighbors in graph.values()
+                for right_vertex in neighbors
+            },
+            key=repr,
+        )
+
+        if any(vertex is None for vertex in self.left_vertices) or any(
+            vertex is None for vertex in self.right_vertices
+        ):
+            msg = "Vertices cannot be None"
+            raise ValueError(msg)
+
+        overlap = set(self.left_vertices) & set(self.right_vertices)
+        if overlap:
+            msg = (
+                f"Partitions must be disjoint: found vertices in both sets: "
+                f"{sorted(overlap, key=repr)}"
+            )
+            raise ValueError(msg)
+
+        # pair_left[u] stores matched vertex in V for u in U (or _NIL if free)
+        self.pair_left: dict[T, T | object] = dict.fromkeys(self.left_vertices, _NIL)
+        # pair_right[v] stores matched vertex in U for v in V (or _NIL if free)
+        self.pair_right: dict[T, T | object] = dict.fromkeys(self.right_vertices, _NIL)
+        # distance_map stores the BFS level from free vertices in U
+        self.distance_map: dict[T | object, float] = {}
+
+    def breadth_first_search(self) -> bool:
+        """BFS Phase: Layer the graph and find shortest augmenting path length.
+
+        Returns:
+            True if at least one augmenting path to a free vertex in V exists,
+            False otherwise (termination condition).
+
+        >>> hk = HopcroftKarp({"u1": ["v1"]})
+        >>> hk.breadth_first_search()
+        True
+        >>> hk.pair_left["u1"] = "v1"
+        >>> hk.pair_right["v1"] = "u1"
+        >>> hk.breadth_first_search()
+        False
+        """
+        queue: deque[T] = deque()
+
+        # Enqueue all free vertices in the left partition at level 0
+        for left_vertex in self.left_vertices:
+            if self.pair_left[left_vertex] is _NIL:
+                self.distance_map[left_vertex] = 0.0
+                queue.append(left_vertex)
+            else:
+                self.distance_map[left_vertex] = math.inf
+
+        # distance_map[_NIL] represents distance to a free vertex in right partition
+        self.distance_map[_NIL] = math.inf
+
+        while queue:
+            left_vertex = queue.popleft()
+            if self.distance_map[left_vertex] < self.distance_map[_NIL]:
+                for right_vertex in self.graph[left_vertex]:
+                    matched_left = self.pair_right[right_vertex]
+                    if self.distance_map.get(matched_left, math.inf) == math.inf:
+                        self.distance_map[matched_left] = (
+                            self.distance_map[left_vertex] + 1.0
+                        )
+                        if matched_left is not _NIL:
+                            queue.append(matched_left)  # type: ignore[arg-type]
+
+        return self.distance_map[_NIL] != math.inf
+
+    def depth_first_search(self, start_left: T) -> bool:
+        """DFS Phase: Find and augment along shortest augmenting paths iteratively.
+
+        Implemented iteratively with an explicit stack to prevent RecursionError
+        on graphs with deep alternating paths (diameter > 1000).
+
+        Parameters:
+            start_left: The free vertex in the left partition to start the search from.
+
+        Returns:
+            True if an augmenting path was found and augmented, False otherwise.
+
+        >>> hk = HopcroftKarp({"u1": ["v1"]})
+        >>> _ = hk.breadth_first_search()
+        >>> hk.depth_first_search("u1")
+        True
+        >>> hk.pair_left["u1"]
+        'v1'
+        >>> hk.depth_first_search("u1")
+        False
+        """
+        stack: list[T] = [start_left]
+        neighbor_indices: list[int] = [0]
+        path: list[tuple[T, T]] = []
+
+        while stack:
+            curr_left = stack[-1]
+            curr_index = neighbor_indices[-1]
+            neighbors = self.graph[curr_left]
+
+            found_next = False
+            for idx in range(curr_index, len(neighbors)):
+                right_vertex = neighbors[idx]
+                matched_left = self.pair_right[right_vertex]
+
+                # Augmentation Condition: Only step along shortest layer paths
+                if (
+                    self.distance_map.get(matched_left, math.inf)
+                    == self.distance_map[curr_left] + 1.0
+                ):
+                    neighbor_indices[-1] = idx + 1
+                    path.append((curr_left, right_vertex))
+
+                    if matched_left is _NIL:
+                        # Reached a free right vertex: augment matching along path
+                        for path_left, path_right in path:
+                            self.pair_right[path_right] = path_left
+                            self.pair_left[path_left] = path_right
+                        return True
+
+                    stack.append(matched_left)  # type: ignore[arg-type]
+                    neighbor_indices.append(0)
+                    found_next = True
+                    break
+
+            if not found_next:
+                # Dead end: prune curr_left from this phase
+                self.distance_map[curr_left] = math.inf
+                stack.pop()
+                neighbor_indices.pop()
+                if path:
+                    path.pop()
+
+        return False
+
+    def maximum_matching(self) -> dict[T, T]:
+        """Compute and return the maximum cardinality matching.
+
+        >>> hk = HopcroftKarp({"u1": ["v1"], "u2": ["v1"]})
+        >>> hk.maximum_matching()
+        {'u1': 'v1'}
+        """
+        while self.breadth_first_search():
+            for left_vertex in self.left_vertices:
+                if self.pair_left[left_vertex] is _NIL:
+                    self.depth_first_search(left_vertex)
+
+        return {
+            left_vertex: matched_right  # type: ignore[misc]
+            for left_vertex, matched_right in self.pair_left.items()
+            if matched_right is not _NIL
+        }
+
+
+def hopcroft_karp[T](graph: dict[T, list[T]]) -> dict[T, T]:
+    """Find a maximum cardinality matching in a bipartite graph using Hopcroft-Karp.
+
+    Parameters:
+        graph: An adjacency list mapping each vertex in the left partition (U) to
+            a list of adjacent vertices in the right partition (V). The two
+            partitions must be disjoint, and vertices cannot be None.
+
+    Returns:
+        A dictionary representing the matching, mapping each matched vertex in
+        the left partition to its matched partner in the right partition.
+
+    Raises:
+        ValueError: If any vertex appears in both partitions or if any vertex is None.
+
+    Examples:
+        >>> # Standard bipartite matching
+        >>> graph = {"u1": ["v1", "v2"], "u2": ["v1"], "u3": ["v2", "v3"]}
+        >>> hopcroft_karp(graph)
+        {'u1': 'v2', 'u2': 'v1', 'u3': 'v3'}
+
+        >>> # Empty graph condition
+        >>> hopcroft_karp({})
+        {}
+
+        >>> # Isolated vertices (no incident edges)
+        >>> hopcroft_karp({"u1": []})
+        {}
+
+        >>> # Competing vertices (more left vertices than right vertices)
+        >>> hopcroft_karp({"u1": ["v1"], "u2": ["v1"]})
+        {'u1': 'v1'}
+
+        >>> # Bipartite cycle (6 vertices)
+        >>> cycle_graph = {
+        ...     "u1": ["v1", "v2"],
+        ...     "u2": ["v2", "v3"],
+        ...     "u3": ["v3", "v1"],
+        ... }
+        >>> hopcroft_karp(cycle_graph)
+        {'u1': 'v1', 'u2': 'v2', 'u3': 'v3'}
+
+        >>> # Error condition: Overlapping partitions (not a valid bipartite graph)
+        >>> hopcroft_karp({"A": ["A"]})
+        Traceback (most recent call last):
+            ...
+        ValueError: Partitions must be disjoint: found vertices in both sets: ['A']
+
+        >>> # Error condition: None vertex
+        >>> hopcroft_karp({"u": [None]})
+        Traceback (most recent call last):
+            ...
+        ValueError: Vertices cannot be None
+    """
+    return HopcroftKarp(graph).maximum_matching()
+
+
+def test_hopcroft_karp() -> None:
+    """Pytest test function to verify maximum bipartite matching functionality.
+
+    >>> test_hopcroft_karp()
+    """
+    assert hopcroft_karp({"u1": ["v1", "v2"], "u2": ["v1"], "u3": ["v2", "v3"]}) == {
+        "u1": "v2",
+        "u2": "v1",
+        "u3": "v3",
+    }
+    assert hopcroft_karp({}) == {}
+    assert hopcroft_karp({"u1": []}) == {}
+    assert hopcroft_karp({"u1": ["v1"], "u2": ["v1"]}) == {"u1": "v1"}
+    assert hopcroft_karp(
+        {"u1": ["v1", "v2"], "u2": ["v2", "v3"], "u3": ["v3", "v1"]}
+    ) == {"u1": "v1", "u2": "v2", "u3": "v3"}
+
+    # Test deep alternating path to ensure no RecursionError occurs
+    chain_length = 1500
+    chain_graph = {f"u{i}": [f"v{i}", f"v{i + 1}"] for i in range(chain_length)}
+    assert len(hopcroft_karp(chain_graph)) == chain_length
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()