Matching on bipartite graph (tools/bipartite_matching.hpp)

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• Last update: 2022-11-12 14:09:50+09:00
• Include: #include "tools/bipartite_matching.hpp"

It calculates the maximum matching on a given bipartite graph.

License

• CC0

Author

• anqooqie

Constructor

bipartite_matching graph(std::size_t n1, std::size_t n2);

It creates an bipartite graph with $0$ edges. Vertices of the graph can be divided into two disjoint and independent sets $U$ and $V$. $U$ consists of $n_1$ vertices and $V$ consists of $n_2$ vertices.

Constraints

• None

Time Complexity

• $O(n_1 + n_2)$

size1

std::size_t graph.size1();

It returns $n_1$.

Constraints

• None

Time Complexity

• $O(1)$

size2

std::size_t graph.size2();

It returns $n_2$.

Constraints

• None

Time Complexity

• $O(1)$

add_edge

std::size_t graph.add_edge(std::size_t a, std::size_t b);

It adds an edge between $a \in U$ and $b \in V$. It returns an integer $k$ such that this is the $k$-th edge that is added.

Constraints

• $0 \leq a < n_1$
• $0 \leq b < n_2$

Time Complexity

• $O(1)$ amortized

get_edge

struct edge {
  std::size_t id;
  std::size_t from;
  std::size_t to;
};
edge graph.get_edge(std::size_t k);

It returns the $k$-th edge.

Constraints

• $0 \leq k < |E|$ where $|E|$ is the number of edges

Time Complexity

• $O(1)$

edges

std::vector<edge> graph.edges();

It returns all the edges in the graph. The edges are ordered in the same order as added by add_edge.

Constraints

• None

Time Complexity

• $O(1)$

query

std::vector<std::size_t> graph.query();

It returns the indices of the edges which can reach the maximum matching.

Constraints

• None

Time Complexity

• $O\left((n_1 + n_2)^\frac{3}{2} + \min\left((n_1 + n_2)^\frac{2}{3} m, m^\frac{3}{2}\right)\right)$ where $m$ is the number of edges

Verified with

• tests/bipartite_matching.test.cpp

Code

#ifndef TOOLS_BIPARTITE_MATCHING_HPP
#define TOOLS_BIPARTITE_MATCHING_HPP

#include <cstddef>
#include <vector>
#include <cassert>
#include "atcoder/maxflow.hpp"

namespace tools {

  class bipartite_matching {
  public:
    struct edge {
      ::std::size_t id;
      ::std::size_t from;
      ::std::size_t to;
    };

  private:
    ::std::size_t m_size1;
    ::std::size_t m_size2;
    ::atcoder::mf_graph<int> m_graph;
    ::std::vector<edge> m_edges;

  public:
    bipartite_matching() = default;
    bipartite_matching(const ::tools::bipartite_matching&) = default;
    bipartite_matching(::tools::bipartite_matching&&) = default;
    ~bipartite_matching() = default;
    ::tools::bipartite_matching& operator=(const ::tools::bipartite_matching&) = default;
    ::tools::bipartite_matching& operator=(::tools::bipartite_matching&&) = default;

    bipartite_matching(const ::std::size_t size1, const ::std::size_t size2) :
      m_size1(size1), m_size2(size2), m_graph(size1 + size2 + 2) {
      for (::std::size_t i = 0; i < size1; ++i) {
        this->m_graph.add_edge(size1 + size2, i, 1);
      }
      for (::std::size_t i = 0; i < size2; ++i) {
        this->m_graph.add_edge(size1 + i, size1 + size2 + 1, 1);
      }
    }

    ::std::size_t size1() const {
      return this->m_size1;
    }

    ::std::size_t size2() const {
      return this->m_size2;
    }

    ::std::size_t add_edge(const ::std::size_t i, const ::std::size_t j) {
      assert(i < this->size1());
      assert(j < this->size2());
      this->m_graph.add_edge(i, this->m_size1 + j, 1);
      this->m_edges.emplace_back(edge({this->m_edges.size(), i, j}));
      return this->m_edges.size() - 1;
    }

    const edge& get_edge(const ::std::size_t k) const {
      assert(k < this->m_edges.size());
      return this->m_edges[k];
    }

    const ::std::vector<edge>& edges() const {
      return this->m_edges;
    }

    ::std::vector<::std::size_t> query() {
      ::std::vector<::std::size_t> edges;

      this->m_graph.flow(this->m_size1 + this->m_size2, this->m_size1 + this->m_size2 + 1);
      for (::std::size_t i = 0; i < this->m_edges.size(); ++i) {
        if (this->m_graph.get_edge(this->m_size1 + this->m_size2 + i).flow == 1) {
          edges.push_back(i);
        }
      }

      return edges;
    }
  };
}

#endif

#line 1 "tools/bipartite_matching.hpp"



#include <cstddef>
#include <vector>
#include <cassert>
#line 1 "lib/ac-library/atcoder/maxflow.hpp"



#include <algorithm>
#line 6 "lib/ac-library/atcoder/maxflow.hpp"
#include <limits>
#include <queue>
#line 9 "lib/ac-library/atcoder/maxflow.hpp"

#line 1 "lib/ac-library/atcoder/internal_queue.hpp"



#line 5 "lib/ac-library/atcoder/internal_queue.hpp"

namespace atcoder {

namespace internal {

template <class T> struct simple_queue {
    std::vector<T> payload;
    int pos = 0;
    void reserve(int n) { payload.reserve(n); }
    int size() const { return int(payload.size()) - pos; }
    bool empty() const { return pos == int(payload.size()); }
    void push(const T& t) { payload.push_back(t); }
    T& front() { return payload[pos]; }
    void clear() {
        payload.clear();
        pos = 0;
    }
    void pop() { pos++; }
};

}  // namespace internal

}  // namespace atcoder


#line 11 "lib/ac-library/atcoder/maxflow.hpp"

namespace atcoder {

template <class Cap> struct mf_graph {
  public:
    mf_graph() : _n(0) {}
    explicit mf_graph(int n) : _n(n), g(n) {}

    int add_edge(int from, int to, Cap cap) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        assert(0 <= cap);
        int m = int(pos.size());
        pos.push_back({from, int(g[from].size())});
        int from_id = int(g[from].size());
        int to_id = int(g[to].size());
        if (from == to) to_id++;
        g[from].push_back(_edge{to, to_id, cap});
        g[to].push_back(_edge{from, from_id, 0});
        return m;
    }

    struct edge {
        int from, to;
        Cap cap, flow;
    };

    edge get_edge(int i) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        auto _e = g[pos[i].first][pos[i].second];
        auto _re = g[_e.to][_e.rev];
        return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
    }
    std::vector<edge> edges() {
        int m = int(pos.size());
        std::vector<edge> result;
        for (int i = 0; i < m; i++) {
            result.push_back(get_edge(i));
        }
        return result;
    }
    void change_edge(int i, Cap new_cap, Cap new_flow) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        assert(0 <= new_flow && new_flow <= new_cap);
        auto& _e = g[pos[i].first][pos[i].second];
        auto& _re = g[_e.to][_e.rev];
        _e.cap = new_cap - new_flow;
        _re.cap = new_flow;
    }

    Cap flow(int s, int t) {
        return flow(s, t, std::numeric_limits<Cap>::max());
    }
    Cap flow(int s, int t, Cap flow_limit) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);

        std::vector<int> level(_n), iter(_n);
        internal::simple_queue<int> que;

        auto bfs = [&]() {
            std::fill(level.begin(), level.end(), -1);
            level[s] = 0;
            que.clear();
            que.push(s);
            while (!que.empty()) {
                int v = que.front();
                que.pop();
                for (auto e : g[v]) {
                    if (e.cap == 0 || level[e.to] >= 0) continue;
                    level[e.to] = level[v] + 1;
                    if (e.to == t) return;
                    que.push(e.to);
                }
            }
        };
        auto dfs = [&](auto self, int v, Cap up) {
            if (v == s) return up;
            Cap res = 0;
            int level_v = level[v];
            for (int& i = iter[v]; i < int(g[v].size()); i++) {
                _edge& e = g[v][i];
                if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
                Cap d =
                    self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
                if (d <= 0) continue;
                g[v][i].cap += d;
                g[e.to][e.rev].cap -= d;
                res += d;
                if (res == up) return res;
            }
            level[v] = _n;
            return res;
        };

        Cap flow = 0;
        while (flow < flow_limit) {
            bfs();
            if (level[t] == -1) break;
            std::fill(iter.begin(), iter.end(), 0);
            Cap f = dfs(dfs, t, flow_limit - flow);
            if (!f) break;
            flow += f;
        }
        return flow;
    }

    std::vector<bool> min_cut(int s) {
        std::vector<bool> visited(_n);
        internal::simple_queue<int> que;
        que.push(s);
        while (!que.empty()) {
            int p = que.front();
            que.pop();
            visited[p] = true;
            for (auto e : g[p]) {
                if (e.cap && !visited[e.to]) {
                    visited[e.to] = true;
                    que.push(e.to);
                }
            }
        }
        return visited;
    }

  private:
    int _n;
    struct _edge {
        int to, rev;
        Cap cap;
    };
    std::vector<std::pair<int, int>> pos;
    std::vector<std::vector<_edge>> g;
};

}  // namespace atcoder


#line 8 "tools/bipartite_matching.hpp"

namespace tools {

  class bipartite_matching {
  public:
    struct edge {
      ::std::size_t id;
      ::std::size_t from;
      ::std::size_t to;
    };

  private:
    ::std::size_t m_size1;
    ::std::size_t m_size2;
    ::atcoder::mf_graph<int> m_graph;
    ::std::vector<edge> m_edges;

  public:
    bipartite_matching() = default;
    bipartite_matching(const ::tools::bipartite_matching&) = default;
    bipartite_matching(::tools::bipartite_matching&&) = default;
    ~bipartite_matching() = default;
    ::tools::bipartite_matching& operator=(const ::tools::bipartite_matching&) = default;
    ::tools::bipartite_matching& operator=(::tools::bipartite_matching&&) = default;

    bipartite_matching(const ::std::size_t size1, const ::std::size_t size2) :
      m_size1(size1), m_size2(size2), m_graph(size1 + size2 + 2) {
      for (::std::size_t i = 0; i < size1; ++i) {
        this->m_graph.add_edge(size1 + size2, i, 1);
      }
      for (::std::size_t i = 0; i < size2; ++i) {
        this->m_graph.add_edge(size1 + i, size1 + size2 + 1, 1);
      }
    }

    ::std::size_t size1() const {
      return this->m_size1;
    }

    ::std::size_t size2() const {
      return this->m_size2;
    }

    ::std::size_t add_edge(const ::std::size_t i, const ::std::size_t j) {
      assert(i < this->size1());
      assert(j < this->size2());
      this->m_graph.add_edge(i, this->m_size1 + j, 1);
      this->m_edges.emplace_back(edge({this->m_edges.size(), i, j}));
      return this->m_edges.size() - 1;
    }

    const edge& get_edge(const ::std::size_t k) const {
      assert(k < this->m_edges.size());
      return this->m_edges[k];
    }

    const ::std::vector<edge>& edges() const {
      return this->m_edges;
    }

    ::std::vector<::std::size_t> query() {
      ::std::vector<::std::size_t> edges;

      this->m_graph.flow(this->m_size1 + this->m_size2, this->m_size1 + this->m_size2 + 1);
      for (::std::size_t i = 0; i < this->m_edges.size(); ++i) {
        if (this->m_graph.get_edge(this->m_size1 + this->m_size2 + i).flow == 1) {
          edges.push_back(i);
        }
      }

      return edges;
    }
  };
}

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