Travelling salesman problem
(Mylib/Graph/travelling_salesman_problem.cpp)

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• Last update: 2021-04-23 23:44:44+09:00
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Operations

• travelling_salesman_problem(g, int src)
  • src始点の全頂点を丁度一度だけ通る最小閉路の距離を返す。
  • Time complexity $O(n^2 2^n)$

Requirements

Notes

Problems

• AOJ DPL_2_A Traveling Salesman Problem

References

• https://ja.wikipedia.org/wiki/%E5%B7%A1%E5%9B%9E%E3%82%BB%E3%83%BC%E3%83%AB%E3%82%B9%E3%83%9E%E3%83%B3%E5%95%8F%E9%A1%8C

Depends on

• Basic graph (Mylib/Graph/Template/graph.cpp)

Verified with

• test/aoj/DPL_2_A/main.test.cpp

Code

#pragma once
#include <algorithm>
#include <optional>
#include <vector>
#include "Mylib/Graph/Template/graph.cpp"

namespace haar_lib {
  template <typename T>
  std::optional<T> travelling_salesman_problem(const graph<T> &g, int src) {
    const int n = g.size();

    std::vector<std::vector<std::optional<T>>> dp(n, std::vector<std::optional<T>>(1 << n));

    for (auto &e : g[src]) {
      if (not dp[e.to][1 << e.to]) {
        dp[e.to][1 << e.to] = e.cost;
      } else {
        dp[e.to][1 << e.to] = std::min(*dp[e.to][1 << e.to], e.cost);
      }
    }

    for (int s = 1; s < (1 << n); ++s) {
      for (int i = 0; i < n; ++i) {
        if (not(s & (1 << i))) continue;

        for (auto &e : g[i]) {
          if (s & (1 << e.to)) continue;

          if (dp[i][s]) {
            if (not dp[e.to][s | (1 << e.to)]) {
              dp[e.to][s | (1 << e.to)] = *dp[i][s] + e.cost;
            } else {
              dp[e.to][s | (1 << e.to)] = std::min(*dp[e.to][s | (1 << e.to)], *dp[i][s] + e.cost);
            }
          }
        }
      }
    }

    return dp[src][(1 << n) - 1];
  }
}  // namespace haar_lib

#line 2 "Mylib/Graph/travelling_salesman_problem.cpp"
#include <algorithm>
#include <optional>
#include <vector>
#line 2 "Mylib/Graph/Template/graph.cpp"
#include <iostream>
#line 4 "Mylib/Graph/Template/graph.cpp"

namespace haar_lib {
  template <typename T>
  struct edge {
    int from, to;
    T cost;
    int index = -1;
    edge() {}
    edge(int from, int to, T cost) : from(from), to(to), cost(cost) {}
    edge(int from, int to, T cost, int index) : from(from), to(to), cost(cost), index(index) {}
  };

  template <typename T>
  struct graph {
    using weight_type = T;
    using edge_type   = edge<T>;

    std::vector<std::vector<edge<T>>> data;

    auto& operator[](size_t i) { return data[i]; }
    const auto& operator[](size_t i) const { return data[i]; }

    auto begin() const { return data.begin(); }
    auto end() const { return data.end(); }

    graph() {}
    graph(int N) : data(N) {}

    bool empty() const { return data.empty(); }
    int size() const { return data.size(); }

    void add_edge(int i, int j, T w, int index = -1) {
      data[i].emplace_back(i, j, w, index);
    }

    void add_undirected(int i, int j, T w, int index = -1) {
      add_edge(i, j, w, index);
      add_edge(j, i, w, index);
    }

    template <size_t I, bool DIRECTED = true, bool WEIGHTED = true>
    void read(int M) {
      for (int i = 0; i < M; ++i) {
        int u, v;
        std::cin >> u >> v;
        u -= I;
        v -= I;
        T w = 1;
        if (WEIGHTED) std::cin >> w;
        if (DIRECTED)
          add_edge(u, v, w, i);
        else
          add_undirected(u, v, w, i);
      }
    }
  };

  template <typename T>
  using tree = graph<T>;
}  // namespace haar_lib
#line 6 "Mylib/Graph/travelling_salesman_problem.cpp"

namespace haar_lib {
  template <typename T>
  std::optional<T> travelling_salesman_problem(const graph<T> &g, int src) {
    const int n = g.size();

    std::vector<std::vector<std::optional<T>>> dp(n, std::vector<std::optional<T>>(1 << n));

    for (auto &e : g[src]) {
      if (not dp[e.to][1 << e.to]) {
        dp[e.to][1 << e.to] = e.cost;
      } else {
        dp[e.to][1 << e.to] = std::min(*dp[e.to][1 << e.to], e.cost);
      }
    }

    for (int s = 1; s < (1 << n); ++s) {
      for (int i = 0; i < n; ++i) {
        if (not(s & (1 << i))) continue;

        for (auto &e : g[i]) {
          if (s & (1 << e.to)) continue;

          if (dp[i][s]) {
            if (not dp[e.to][s | (1 << e.to)]) {
              dp[e.to][s | (1 << e.to)] = *dp[i][s] + e.cost;
            } else {
              dp[e.to][s | (1 << e.to)] = std::min(*dp[e.to][s | (1 << e.to)], *dp[i][s] + e.cost);
            }
          }
        }
      }
    }

    return dp[src][(1 << n) - 1];
  }
}  // namespace haar_lib

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