結果

問題 No.114 遠い未来
ユーザー ruthen71ruthen71
提出日時 2024-07-28 03:01:52
言語 C++23(gcc13)
(gcc 13.2.0 + boost 1.83.0)
結果
TLE  
実行時間 -
コード長 10,000 bytes
コンパイル時間 2,167 ms
コンパイル使用メモリ 149,996 KB
実行使用メモリ 13,312 KB
最終ジャッジ日時 2024-07-28 03:02:11
合計ジャッジ時間 17,712 ms
ジャッジサーバーID
（参考情報） 		judge3 / judge5
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 16 ms
6,816 KB
testcase_01 AC 539 ms
13,312 KB
testcase_02 AC 381 ms
6,940 KB
testcase_03 AC 160 ms
6,940 KB
testcase_04 AC 2 ms
6,940 KB
testcase_05 AC 5 ms
6,944 KB
testcase_06 AC 2,800 ms
6,940 KB
testcase_07 AC 2 ms
6,944 KB
testcase_08 AC 2 ms
6,940 KB
testcase_09 AC 8 ms
6,944 KB
testcase_10 AC 82 ms
6,940 KB
testcase_11 AC 191 ms
8,448 KB
testcase_12 AC 527 ms
13,184 KB
testcase_13 AC 528 ms
13,312 KB
testcase_14 AC 2,987 ms
6,940 KB
testcase_15 TLE -
testcase_16 -- -
testcase_17 -- -
testcase_18 -- -
testcase_19 -- -
testcase_20 -- -
testcase_21 -- -
testcase_22 -- -
testcase_23 -- -
testcase_24 -- -
testcase_25 -- -
testcase_26 -- -
testcase_27 -- -
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ソースコード

diff # 

// #include <iostream>
// 
// #include "graph/read_graph.hpp"
// #include "graph/minimum_steriner_tree.hpp"
// 
// int main() {
//     int N, M, T;
//     std::cin >> N >> M >> T;
//     auto g = read_graph<long long>(N, M, true);
//     std::vector<int> terminals(T);
//     for (int i = 0; i < T; i++) {
//         std::cin >> terminals[i];
//         terminals[i]--;
//     }
//     if (T <= 15) {
//         auto dp = minimum_steiner_tree(g, terminals, 1'000'000'000'000'000'000LL);
//         std::cout << dp.back()[terminals[0]] << '\n';
//     } else {
//         std::cout << minimum_steiner_tree_mst(g, terminals, 1'000'000'000'000'000'000LL) << '\n';
//     }
//     return 0;
// }
#include <iostream>

#include <string>

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

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

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

template <class T> using Edges = std::vector<Edge<T>>;
template <class T> using Graph = std::vector<std::vector<Edge<T>>>;

template <class T> Graph<T> read_graph(const int n, const int m, const bool weight = false, const bool directed = false, const int offset = 1) {
    Graph<T> g(n);
    for (int i = 0; i < m; i++) {
        int a, b;
        std::cin >> a >> b;
        a -= offset, b -= offset;
        if (weight) {
            T c;
            std::cin >> c;
            if (!directed) g[b].push_back(Edge(b, a, c, i));
            g[a].push_back(Edge(a, b, c, i));
        } else {
            // c = 1
            if (!directed) g[b].push_back(Edge(b, a, T(1), i));
            g[a].push_back(Edge(a, b, T(1), i));
        }
    }
    return g;
}

template <class T> Graph<T> read_parent(const int n, const bool weight = false, const bool directed = false, const int offset = 1) {
    Graph<T> g(n);
    for (int i = 1; i < n; i++) {
        int p;
        std::cin >> p;
        p -= offset;
        if (weight) {
            T c;
            std::cin >> c;
            if (!directed) g[i].push_back(Edge(i, p, c, i - 1));
            g[p].push_back(Edge(p, i, c, i - 1));
        } else {
            // c = 1
            if (!directed) g[i].push_back(Edge(i, p, T(1), i - 1));
            g[p].push_back(Edge(p, i, T(1), i - 1));
        }
    }
    return g;
}

std::tuple<Graph<int>, std::vector<std::vector<int>>, std::vector<std::pair<int, int>>> read_grid(const int h, const int w, std::string rel = ".#") {
    std::vector<std::string> s(h);
    std::vector id(h, std::vector<int>(w, -1));
    std::vector<std::pair<int, int>> loc;
    int n = 0;
    for (int i = 0; i < h; i++) {
        std::cin >> s[i];
        for (int j = 0; j < w; j++) {
            if (s[i][j] == rel[1]) {
                id[i][j] = n++;
                loc.emplace_back(i, j);
            }
        }
    }
    int m = 0;
    Graph<int> g(n);
    for (int i = 0; i < h; i++) {
        for (int j = 0; j < w; j++) {
            if (s[i][j] == rel[1]) {
                if (i + 1 < h and s[i + 1][j] == rel[1]) {
                    g[id[i][j]].push_back(Edge(id[i][j], id[i + 1][j], 1, m));
                    g[id[i + 1][j]].push_back(Edge(id[i + 1][j], id[i][j], 1, m++));
                }
                if (j + 1 < w and s[i][j + 1] == rel[1]) {
                    g[id[i][j]].push_back(Edge(id[i][j], id[i][j + 1], 1, m));
                    g[id[i][j + 1]].push_back(Edge(id[i][j + 1], id[i][j], 1, m++));
                }
            }
        }
    }
    return {g, id, loc};
}

#include <queue>
#include <set>
#include <algorithm>
#include <cassert>

template <class T> Edges<T> get_edges(Graph<T>& G) {
    int N = (int)G.size(), M = 0;
    for (int i = 0; i < N; i++) {
        for (auto&& e : G[i]) {
            M = std::max(M, e.id + 1);
        }
    }
    Edges<T> es(M);
    for (int i = N - 1; i >= 0; i--) {
        for (auto&& e : G[i]) {
            es[e.id] = e;
        }
    }
    return es;
}

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
};

// 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, 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} m \log n)
// https://yukicoder.me/problems/no/114/editorial
// n - k <= 20, n <= 64
template <class T> T minimum_steiner_tree_mst(Graph<T>& g, std::vector<int>& terminals, const T inf) {
    const int n = (int)(g.size());
    const int k = (int)(terminals.size());
    assert(n <= 64);

    // ターミナルに含まれない点集合 (others) を取得
    unsigned long long st = 0;
    for (int i = 0; i < k; i++) st |= 1LL << terminals[i];
    std::vector<int> others;
    for (int i = 0; i < n; i++)
        if ((st >> i & 1) == 0) others.emplace_back(i);

    // 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
        unsigned long long subv = st;
        for (int i = 0; i < n - k; i++) {
            if (bit >> i & 1) {
                subv |= 1LL << others[i];
            }
        }

        // subv に対する g の誘導部分グラフ
        std::vector<Edge<T>> edges;
        for (int v = 0; v < n; v++) {
            if (subv >> v & 1) {
                for (auto&& e : g[v]) {
                    if (subv >> e.to & 1) {
                        edges.push_back(e);
                    }
                }
            }
        }

        std::sort(edges.begin(), edges.end(), [&](Edge<T>& a, Edge<T>& b) -> bool { return a.cost < b.cost; });
        UnionFind uf(n);

        // Minimum Spanning Tree を計算
        T cur = 0;
        int connected = 0;
        for (auto&& e : edges) {
            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);
    }
    return ans;
}

int main() {
    int N, M, T;
    std::cin >> N >> M >> T;
    auto g = read_graph<long long>(N, M, true);
    std::vector<int> terminals(T);
    for (int i = 0; i < T; i++) {
        std::cin >> terminals[i];
        terminals[i]--;
    }
    if (T <= 15) {
        auto dp = minimum_steiner_tree(g, terminals, 1'000'000'000'000'000'000LL);
        std::cout << dp.back()[terminals[0]] << '\n';
    } else {
        std::cout << minimum_steiner_tree_mst(g, terminals, 1'000'000'000'000'000'000LL) << '\n';
    }
    return 0;
}
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