Package edu.princeton.cs.algs4

Class HopcroftKarp

  • public class HopcroftKarp
extends Object
    The HopcroftKarp class represents a data type for computing a maximum (cardinality) matching and a minimum (cardinality) vertex cover in a bipartite graph. A bipartite graph in a graph whose vertices can be partitioned into two disjoint sets such that every edge has one endpoint in either set. A matching in a graph is a subset of its edges with no common vertices. A maximum matching is a matching with the maximum number of edges. A perfect matching is a matching which matches all vertices in the graph. A vertex cover in a graph is a subset of its vertices such that every edge is incident to at least one vertex. A minimum vertex cover is a vertex cover with the minimum number of vertices. By Konig's theorem, in any bipartite graph, the maximum number of edges in matching equals the minimum number of vertices in a vertex cover. The maximum matching problem in nonbipartite graphs is also important, but all known algorithms for this more general problem are substantially more complicated.

    This implementation uses the Hopcroft-Karp algorithm. The order of growth of the running time in the worst case is (E + V) sqrt(V), where E is the number of edges and V is the number of vertices in the graph. It uses extra space (not including the graph) proportional to V.

    See also BipartiteMatching, which solves the problem in O(E V) time using the alternating path algorithm and BipartiteMatchingToMaxflow, which solves the problem in O(E V) time via a reduction to the maxflow problem.

    For additional documentation, see Section 6.5 Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne.

    Author:
    Robert Sedgewick, Kevin Wayne

    • Constructor Summary

      Constructors 
      Constructor Description
      HopcroftKarp​(Graph G)
      Determines a maximum matching (and a minimum vertex cover) in a bipartite graph.

    • Method Summary

      All Methods Static Methods Instance Methods Concrete Methods 
      Modifier and Type Method Description
      boolean inMinVertexCover​(int v)
      Returns true if the specified vertex is in the minimum vertex cover computed by the algorithm.
      boolean isMatched​(int v)
      Returns true if the specified vertex is matched in the maximum matching computed by the algorithm.
      boolean isPerfect()
      Returns true if the graph contains a perfect matching.
      static void main​(String[] args)
      Unit tests the HopcroftKarp data type.
      int mate​(int v)
      Returns the vertex to which the specified vertex is matched in the maximum matching computed by the algorithm.
      int size()
      Returns the number of edges in any maximum matching.

    • Constructor Detail

      • HopcroftKarp

        public HopcroftKarp​(Graph G)
        Determines a maximum matching (and a minimum vertex cover) in a bipartite graph.
        Parameters:
        G - the bipartite graph
        Throws:
        IllegalArgumentException - if G is not bipartite

    • Method Detail

      • mate

        public int mate​(int v)
        Returns the vertex to which the specified vertex is matched in the maximum matching computed by the algorithm.
        Parameters:
        v - the vertex
        Returns:
        the vertex to which vertex v is matched in the maximum matching; -1 if the vertex is not matched
        Throws:
        IllegalArgumentException - unless 0 <= v < V

      • isMatched

        public boolean isMatched​(int v)
        Returns true if the specified vertex is matched in the maximum matching computed by the algorithm.
        Parameters:
        v - the vertex
        Returns:
        true if vertex v is matched in maximum matching; false otherwise
        Throws:
        IllegalArgumentException - unless 0 <= v < V

      • size

        public int size()
        Returns the number of edges in any maximum matching.
        Returns:
        the number of edges in any maximum matching

      • isPerfect

        public boolean isPerfect()
        Returns true if the graph contains a perfect matching. That is, the number of edges in a maximum matching is equal to one half of the number of vertices in the graph (so that every vertex is matched).
        Returns:
        true if the graph contains a perfect matching; false otherwise

      • inMinVertexCover

        public boolean inMinVertexCover​(int v)
        Returns true if the specified vertex is in the minimum vertex cover computed by the algorithm.
        Parameters:
        v - the vertex
        Returns:
        true if vertex v is in the minimum vertex cover; false otherwise
        Throws:
        IllegalArgumentException - unless 0 <= v < V

      • main

        public static void main​(String[] args)
        Unit tests the HopcroftKarp data type. Takes three command-line arguments V1, V2, and E; creates a random bipartite graph with V1 + V2 vertices and E edges; computes a maximum matching and minimum vertex cover; and prints the results.
        Parameters:
        args - the command-line arguments