Minimum Steiner Tree (最小シュタイナー木)
(graph/minimum_steiner_tree.hpp)

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• Last update: 2024-08-01 13:43:30+09:00
• Include: #include "graph/minimum_steiner_tree.hpp"
• Link: https://atcoder.jp/contests/abc364/editorial/10547
• Link: https://kopricky.github.io/code/Academic/steiner_tree.html
• Link: https://www.slideshare.net/wata_orz/ss-12131479#50
• Link: https://yukicoder.me/problems/no/114/editorial

使い方

Graph<T> g;
std::vector<int> terminals;
auto dp = minimum_steiner_tree(g, terminals, INF);
auto wt = minimum_steiner_tree_mst(g, terminals, INF);

• minimum_steiner_tree(g, terminals, inf)
  • $ O(3 ^ k n + 2 ^ k m \log m) $
  • 戻り値は dp の配列
• minimum_steiner_tree_mst(g, terminals, inf)
  • ほとんどの頂点がターミナル点であるときに残りの点のうち使うものを全探索することで解く方法
  • $ O(2 ^ {n - k} (n + m)) $
  • 戻り値は最小シュタイナー木の辺の重みの総和

参考文献

Depends on

• UnionFind (data_structure/unionfind.hpp)
• グラフテンプレート (graph/graph_template.hpp)

Verified with

• verify/graph/minimum_steiner_tree.test.cpp

Code

#pragma once

#include "graph/graph_template.hpp"
#include "data_structure/unionfind.hpp"

#include <vector>
#include <queue>
#include <algorithm>
#include <cassert>

// minimum steiner tree
// O(3 ^ k n + 2 ^ k m \log m) (n = |V|, m = |E|, k = |terminals|)
// https://www.slideshare.net/wata_orz/ss-12131479#50
// https://kopricky.github.io/code/Academic/steiner_tree.html
// https://atcoder.jp/contests/abc364/editorial/10547
template <class T> std::vector<std::vector<T>> minimum_steiner_tree(Graph<T>& g, const std::vector<int>& terminals, const T inf) {
    const int n = (int)(g.size());
    const int k = (int)(terminals.size());
    const int k2 = 1 << k;

    // dp[bit][v] = ターミナルの部分集合が bit (0 ~ k - 1 に圧縮), 加えて頂点 v も含まれる最小シュタイナー木
    std::vector dp(k2, std::vector<T>(n, inf));
    for (int i = 0; i < k; i++) dp[1 << i][terminals[i]] = T(0);

    for (int bit = 0; bit < (1 << k); bit++) {
        // dp[bit][v] = min(dp[bit][v], dp[sub][v] + dp[bit ^ sub][v])
        // 通常の実装
        // for (int sub = bit; sub > 0; sub = (sub - 1) & bit) {
        // 定数倍高速化
        // bit の中で 1 要素だけ sub と bit ^ sub のどちらに属するか決める
        int bit2 = bit ^ (bit & -bit);
        for (int sub = bit2; sub > 0; sub = (sub - 1) & bit2) {
            for (int v = 0; v < n; v++) {
                dp[bit][v] = std::min(dp[bit][v], dp[sub][v] + dp[bit ^ sub][v]);
            }
        }
        // dp[bit][v] = min(dp[bit][v], dp[bit][u] + cost(u, v))
        using tp = std::pair<T, int>;
        std::priority_queue<tp, std::vector<tp>, std::greater<tp>> que;
        for (int u = 0; u < n; u++) que.emplace(dp[bit][u], u);
        while (!que.empty()) {
            auto [d, u] = que.top();
            que.pop();
            if (dp[bit][u] != d) continue;
            for (auto&& e : g[u]) {
                if (dp[bit][e.to] > d + e.cost) {
                    dp[bit][e.to] = d + e.cost;
                    que.emplace(dp[bit][e.to], e.to);
                }
            }
        }
    }
    // dp[k2 - 1][i] = ターミナルと頂点 i を含む最小シュタイナー木
    // dp[k2 - 1][terminals[0]] が基本的な答えになる
    return dp;
}

// O(2 ^ {n - k} (n + m)) (n = |V|, m = |E|, k = |terminals|)
// https://yukicoder.me/problems/no/114/editorial
// n - k <= 20
template <class T> T minimum_steiner_tree_mst(Graph<T>& g, const std::vector<int>& terminals, const T inf) {
    const int n = (int)(g.size());
    const int k = (int)(terminals.size());

    // ターミナルに含まれない点集合 (others) を取得
    std::vector<int> used(n, 0);
    for (int i = 0; i < k; i++) used[terminals[i]] = 1;
    std::vector<int> others;
    for (int i = 0; i < n; i++) {
        if (used[i] == 0) others.push_back(i);
    }

    // 辺のリスト
    std::vector<Edge<T>> edges;
    for (int v = 0; v < n; v++) {
        for (auto&& e : g[v]) {
            if (e.from < e.to) edges.push_back(e);
        }
    }
    std::sort(edges.begin(), edges.end(), [&](Edge<T>& a, Edge<T>& b) -> bool { return a.cost < b.cost; });

    // ターミナル + others の組合せを全列挙 -> Minimum Spanning Tree を求める
    T ans = inf;
    for (int bit = 0; bit < (1 << (n - k)); bit++) {
        // 使う頂点集合 (used) を計算
        for (int i = 0; i < n - k; i++) used[others[i]] = bit >> i & 1;

        // Minimum Spanning Tree を計算
        UnionFind uf(n);
        T cur = 0;
        int connected = 0;
        for (auto&& e : edges) {
            // subv に対する g の誘導部分グラフに含まれる辺のみ試す
            if (!(used[e.from] and used[e.to])) continue;
            if (!uf.same(e.from, e.to)) {
                uf.merge(e.from, e.to);
                cur += e.cost;
                connected++;
            }
        }

        // 全域木が作れたか判定
        if (connected + 1 == k + __builtin_popcount(bit)) ans = std::min(ans, cur);

        // used をもとに戻す
        for (int i = 0; i < n - k; i++) used[others[i]] = 0;
    }
    return ans;
}

#line 2 "graph/minimum_steiner_tree.hpp"

#line 2 "graph/graph_template.hpp"

#include <vector>
#include <cassert>

template <class T> struct Edge {
    int from, to;
    T cost;
    int id;

    Edge() = default;
    Edge(const int from, const int to, const T cost = T(1), const int id = -1) : from(from), to(to), cost(cost), id(id) {}

    friend bool operator<(const Edge<T>& a, const Edge<T>& b) { return a.cost < b.cost; }

    friend std::ostream& operator<<(std::ostream& os, const Edge<T>& e) {
        // output format: {id: cost(from, to) = cost}
        return os << "{" << e.id << ": cost(" << e.from << ", " << e.to << ") = " << e.cost << "}";
    }
};
template <class T> using Edges = std::vector<Edge<T>>;

template <class T> struct Graph {
    struct EdgeIterators {
       public:
        using Iterator = typename std::vector<Edge<T>>::iterator;
        EdgeIterators() = default;
        EdgeIterators(const Iterator& begit, const Iterator& endit) : begit(begit), endit(endit) {}
        Iterator begin() const { return begit; }
        Iterator end() const { return endit; }
        size_t size() const { return std::distance(begit, endit); }
        Edge<T>& operator[](int i) const { return begit[i]; }

       private:
        Iterator begit, endit;
    };

    int n, m;
    bool is_build, is_directed;
    std::vector<Edge<T>> edges;

    // CSR (Compressed Row Storage) 形式用
    std::vector<int> start;
    std::vector<Edge<T>> csr_edges;

    Graph() = default;
    Graph(const int n, const bool directed = false) : n(n), m(0), is_build(false), is_directed(directed), start(n + 1, 0) {}

    // 辺を追加し, その辺が何番目に追加されたかを返す
    int add_edge(const int from, const int to, const T cost = T(1), int id = -1) {
        assert(!is_build);
        assert(0 <= from and from < n);
        assert(0 <= to and to < n);
        if (id == -1) id = m;
        edges.emplace_back(from, to, cost, id);
        return m++;
    }

    // CSR 形式でグラフを構築
    void build() {
        assert(!is_build);
        for (auto&& e : edges) {
            start[e.from + 1]++;
            if (!is_directed) start[e.to + 1]++;
        }
        for (int v = 0; v < n; v++) start[v + 1] += start[v];
        auto counter = start;
        csr_edges.resize(start.back() + 1);
        for (auto&& e : edges) {
            csr_edges[counter[e.from]++] = e;
            if (!is_directed) csr_edges[counter[e.to]++] = Edge(e.to, e.from, e.cost, e.id);
        }
        is_build = true;
    }

    EdgeIterators operator[](int i) {
        if (!is_build) build();
        return EdgeIterators(csr_edges.begin() + start[i], csr_edges.begin() + start[i + 1]);
    }

    size_t size() const { return (size_t)(n); }

    friend std::ostream& operator<<(std::ostream& os, Graph<T>& g) {
        os << "[";
        for (int i = 0; i < (int)(g.size()); i++) {
            os << "[";
            for (int j = 0; j < (int)(g[i].size()); j++) {
                os << g[i][j];
                if (j + 1 != (int)(g[i].size())) os << ", ";
            }
            os << "]";
            if (i + 1 != (int)(g.size())) os << ", ";
        }
        return os << "]";
    }
};
#line 2 "data_structure/unionfind.hpp"

#line 4 "data_structure/unionfind.hpp"
#include <algorithm>

struct UnionFind {
    int n;
    std::vector<int> parents;

    UnionFind() {}
    UnionFind(int n) : n(n), parents(n, -1) {}

    int leader(int x) { return parents[x] < 0 ? x : parents[x] = leader(parents[x]); }

    bool merge(int x, int y) {
        x = leader(x), y = leader(y);
        if (x == y) return false;
        if (parents[x] > parents[y]) std::swap(x, y);
        parents[x] += parents[y];
        parents[y] = x;
        return true;
    }

    bool same(int x, int y) { return leader(x) == leader(y); }

    int size(int x) { return -parents[leader(x)]; }

    std::vector<std::vector<int>> groups() {
        std::vector<int> leader_buf(n), group_size(n);
        for (int i = 0; i < n; i++) {
            leader_buf[i] = leader(i);
            group_size[leader_buf[i]]++;
        }
        std::vector<std::vector<int>> result(n);
        for (int i = 0; i < n; i++) {
            result[i].reserve(group_size[i]);
        }
        for (int i = 0; i < n; i++) {
            result[leader_buf[i]].push_back(i);
        }
        result.erase(std::remove_if(result.begin(), result.end(), [&](const std::vector<int>& v) { return v.empty(); }), result.end());
        return result;
    }

    void init(int n) { parents.assign(n, -1); }  // reset
};
#line 5 "graph/minimum_steiner_tree.hpp"

#line 7 "graph/minimum_steiner_tree.hpp"
#include <queue>
#line 10 "graph/minimum_steiner_tree.hpp"

// minimum steiner tree
// O(3 ^ k n + 2 ^ k m \log m) (n = |V|, m = |E|, k = |terminals|)
// https://www.slideshare.net/wata_orz/ss-12131479#50
// https://kopricky.github.io/code/Academic/steiner_tree.html
// https://atcoder.jp/contests/abc364/editorial/10547
template <class T> std::vector<std::vector<T>> minimum_steiner_tree(Graph<T>& g, const std::vector<int>& terminals, const T inf) {
    const int n = (int)(g.size());
    const int k = (int)(terminals.size());
    const int k2 = 1 << k;

    // dp[bit][v] = ターミナルの部分集合が bit (0 ~ k - 1 に圧縮), 加えて頂点 v も含まれる最小シュタイナー木
    std::vector dp(k2, std::vector<T>(n, inf));
    for (int i = 0; i < k; i++) dp[1 << i][terminals[i]] = T(0);

    for (int bit = 0; bit < (1 << k); bit++) {
        // dp[bit][v] = min(dp[bit][v], dp[sub][v] + dp[bit ^ sub][v])
        // 通常の実装
        // for (int sub = bit; sub > 0; sub = (sub - 1) & bit) {
        // 定数倍高速化
        // bit の中で 1 要素だけ sub と bit ^ sub のどちらに属するか決める
        int bit2 = bit ^ (bit & -bit);
        for (int sub = bit2; sub > 0; sub = (sub - 1) & bit2) {
            for (int v = 0; v < n; v++) {
                dp[bit][v] = std::min(dp[bit][v], dp[sub][v] + dp[bit ^ sub][v]);
            }
        }
        // dp[bit][v] = min(dp[bit][v], dp[bit][u] + cost(u, v))
        using tp = std::pair<T, int>;
        std::priority_queue<tp, std::vector<tp>, std::greater<tp>> que;
        for (int u = 0; u < n; u++) que.emplace(dp[bit][u], u);
        while (!que.empty()) {
            auto [d, u] = que.top();
            que.pop();
            if (dp[bit][u] != d) continue;
            for (auto&& e : g[u]) {
                if (dp[bit][e.to] > d + e.cost) {
                    dp[bit][e.to] = d + e.cost;
                    que.emplace(dp[bit][e.to], e.to);
                }
            }
        }
    }
    // dp[k2 - 1][i] = ターミナルと頂点 i を含む最小シュタイナー木
    // dp[k2 - 1][terminals[0]] が基本的な答えになる
    return dp;
}

// O(2 ^ {n - k} (n + m)) (n = |V|, m = |E|, k = |terminals|)
// https://yukicoder.me/problems/no/114/editorial
// n - k <= 20
template <class T> T minimum_steiner_tree_mst(Graph<T>& g, const std::vector<int>& terminals, const T inf) {
    const int n = (int)(g.size());
    const int k = (int)(terminals.size());

    // ターミナルに含まれない点集合 (others) を取得
    std::vector<int> used(n, 0);
    for (int i = 0; i < k; i++) used[terminals[i]] = 1;
    std::vector<int> others;
    for (int i = 0; i < n; i++) {
        if (used[i] == 0) others.push_back(i);
    }

    // 辺のリスト
    std::vector<Edge<T>> edges;
    for (int v = 0; v < n; v++) {
        for (auto&& e : g[v]) {
            if (e.from < e.to) edges.push_back(e);
        }
    }
    std::sort(edges.begin(), edges.end(), [&](Edge<T>& a, Edge<T>& b) -> bool { return a.cost < b.cost; });

    // ターミナル + others の組合せを全列挙 -> Minimum Spanning Tree を求める
    T ans = inf;
    for (int bit = 0; bit < (1 << (n - k)); bit++) {
        // 使う頂点集合 (used) を計算
        for (int i = 0; i < n - k; i++) used[others[i]] = bit >> i & 1;

        // Minimum Spanning Tree を計算
        UnionFind uf(n);
        T cur = 0;
        int connected = 0;
        for (auto&& e : edges) {
            // subv に対する g の誘導部分グラフに含まれる辺のみ試す
            if (!(used[e.from] and used[e.to])) continue;
            if (!uf.same(e.from, e.to)) {
                uf.merge(e.from, e.to);
                cur += e.cost;
                connected++;
            }
        }

        // 全域木が作れたか判定
        if (connected + 1 == k + __builtin_popcount(bit)) ans = std::min(ans, cur);

        // used をもとに戻す
        for (int i = 0; i < n - k; i++) used[others[i]] = 0;
    }
    return ans;
}

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