結果

問題 No.1775 Love Triangle 2
ユーザー hitonanodehitonanode
提出日時 2021-11-28 14:12:47
言語 C++17
(gcc 12.3.0 + boost 1.83.0)
結果
AC  
実行時間 76 ms / 8,000 ms
コード長 7,450 bytes
コンパイル時間 1,923 ms
コンパイル使用メモリ 117,664 KB
実行使用メモリ 6,944 KB
最終ジャッジ日時 2024-07-03 21:19:50
合計ジャッジ時間 10,309 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge2
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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 2 ms
6,812 KB
testcase_01 AC 2 ms
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testcase_02 AC 2 ms
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testcase_03 AC 2 ms
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testcase_04 AC 48 ms
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testcase_05 AC 48 ms
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testcase_06 AC 48 ms
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testcase_07 AC 46 ms
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testcase_08 AC 47 ms
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testcase_09 AC 70 ms
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testcase_10 AC 72 ms
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testcase_11 AC 72 ms
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testcase_12 AC 73 ms
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testcase_13 AC 72 ms
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testcase_14 AC 74 ms
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testcase_15 AC 73 ms
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testcase_16 AC 75 ms
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testcase_17 AC 75 ms
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testcase_18 AC 76 ms
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testcase_19 AC 72 ms
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testcase_20 AC 72 ms
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testcase_21 AC 73 ms
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testcase_22 AC 72 ms
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testcase_23 AC 74 ms
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testcase_24 AC 74 ms
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testcase_25 AC 74 ms
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testcase_26 AC 72 ms
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testcase_27 AC 72 ms
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testcase_28 AC 71 ms
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testcase_29 AC 76 ms
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testcase_30 AC 75 ms
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testcase_31 AC 76 ms
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testcase_32 AC 74 ms
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testcase_33 AC 73 ms
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testcase_34 AC 73 ms
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testcase_35 AC 73 ms
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testcase_36 AC 74 ms
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testcase_37 AC 72 ms
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testcase_38 AC 71 ms
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testcase_39 AC 74 ms
6,940 KB
testcase_40 AC 73 ms
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testcase_41 AC 74 ms
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testcase_42 AC 73 ms
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testcase_43 AC 73 ms
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testcase_44 AC 74 ms
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testcase_45 AC 74 ms
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testcase_46 AC 72 ms
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testcase_47 AC 72 ms
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testcase_48 AC 71 ms
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testcase_49 AC 74 ms
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testcase_50 AC 73 ms
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testcase_51 AC 74 ms
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testcase_52 AC 73 ms
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testcase_53 AC 74 ms
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testcase_54 AC 72 ms
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testcase_55 AC 72 ms
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testcase_56 AC 74 ms
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testcase_57 AC 74 ms
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testcase_58 AC 73 ms
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testcase_59 AC 76 ms
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testcase_60 AC 74 ms
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testcase_61 AC 73 ms
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testcase_62 AC 74 ms
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testcase_63 AC 72 ms
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testcase_64 AC 74 ms
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testcase_65 AC 72 ms
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testcase_66 AC 73 ms
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testcase_67 AC 73 ms
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testcase_68 AC 73 ms
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testcase_69 AC 73 ms
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testcase_70 AC 73 ms
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testcase_71 AC 72 ms
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testcase_72 AC 71 ms
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testcase_73 AC 72 ms
6,944 KB
testcase_74 AC 73 ms
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testcase_75 AC 73 ms
6,940 KB
testcase_76 AC 71 ms
6,940 KB
testcase_77 AC 70 ms
6,940 KB
testcase_78 AC 70 ms
6,940 KB
testcase_79 AC 73 ms
6,940 KB
testcase_80 AC 73 ms
6,944 KB
testcase_81 AC 72 ms
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testcase_82 AC 70 ms
6,940 KB
testcase_83 AC 72 ms
6,940 KB
testcase_84 AC 73 ms
6,940 KB
testcase_85 AC 74 ms
6,940 KB
testcase_86 AC 72 ms
6,944 KB
testcase_87 AC 73 ms
6,944 KB
testcase_88 AC 73 ms
6,944 KB
testcase_89 AC 73 ms
6,940 KB
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ソースコード

diff # 

// O(n^3) weighted-matroid-parity-based solution
#include <algorithm>
#include <cassert>
#include <iostream>
#include <numeric>
#include <tuple>
#include <type_traits>
#include <utility>
#include <vector>
using namespace std;
#include <atcoder/modint>
using mint = atcoder::static_modint<(1 << 30) + 3>;


uint32_t rand_int() {
    static uint32_t x = 123456789, y = 362436069, z = 521288629, w = 88675123;
    uint32_t t = x ^ (x << 11);
    x = y;
    y = z;
    z = w;
    return w = (w ^ (w >> 19)) ^ (t ^ (t >> 8));
}

tuple<int, int, int, int, vector<vector<int>>> read_input() {
    int N, M, X, Y, Z;
    cin >> N >> M >> X >> Y >> Z;
    --X, --Y, --Z;
    vector mat(N, vector<int>(N));
    for (int i = 0; i < N; ++i) {
        for (int j = 0; j < N; ++j) mat[i][j] = (i != j);
    }
    while (M--) {
        int a, b;
        cin >> a >> b;
        --a, --b;
        mat[a][b] = mat[b][a] = 0;
    }
    vector<vector<int>> to(N);
    for (int i = 0; i < N; ++i) {
        for (int j = 0; j < N; ++j) {
            if (mat[i][j]) to[i].push_back(j);
        }
    }
    return {N, X, Y, Z, to};
}

// Upper Hessenberg reduction of square matrices
// Complexity: O(n^3)
// Reference:
// http://www.phys.uri.edu/nigh/NumRec/bookfpdf/f11-5.pdf
template <class Tp> void hessenberg_reduction(std::vector<std::vector<Tp>> &M) {
    assert(M.size() == M[0].size());
    const int N = M.size();
    for (int r = 0; r < N - 2; r++) {
        int piv = -1;
        for (int j = r + 1; j < N; ++j) if (M[j][r] != 0) {
            piv = j;
            break;
        }
        if (piv < 0) continue;
        for (int i = 0; i < N; i++) std::swap(M[r + 1][i], M[piv][i]);
        for (int i = 0; i < N; i++) std::swap(M[i][r + 1], M[i][piv]);

        const auto rinv = Tp(1) / M[r + 1][r];
        for (int i = r + 2; i < N; i++) {
            const auto n = M[i][r] * rinv;
            for (int j = 0; j < N; j++) M[i][j] -= M[r + 1][j] * n;
            for (int j = 0; j < N; j++) M[j][r + 1] += M[j][i] * n;
        }
    }
}

// Characteristic polynomial of matrix M (|xI - M|)
// Complexity: O(n^3)
// R. Rehman, I. C. Ipsen, "La Budde's Method for Computing Characteristic Polynomials," 2011.
template <class Tp> std::vector<Tp> characteristic_poly(std::vector<std::vector<Tp>> &M) {
    hessenberg_reduction(M);
    const int N = M.size();
    std::vector<std::vector<Tp>> p(N + 1); // p[i + 1] = (Characteristic polynomial of i-th leading principal minor)
    p[0] = {1};
    for (int i = 0; i < N; i++) {
        p[i + 1].assign(i + 2, 0);
        for (int j = 0; j < i + 1; j++) p[i + 1][j + 1] += p[i][j];
        for (int j = 0; j < i + 1; j++) p[i + 1][j] -= p[i][j] * M[i][i];
        Tp betas = 1;
        for (int j = i - 1; j >= 0; j--) {
            betas *= M[j + 1][j];
            Tp hb = -M[j][i] * betas;
            for (int k = 0; k < j + 1; k++) p[i + 1][k] += hb * p[j][k];
        }
    }
    return p[N];
}




int main() {
    cin.tie(nullptr), ios::sync_with_stdio(false);

    auto Graph = read_input();
    int X = get<1>(Graph), Y = get<2>(Graph), Z = get<3>(Graph);
    auto to = get<4>(Graph);
    const int N = to.size();

    const int V = to.size() + 5, r = V * 2;

    auto Label = [&](int i) -> int {
        if (i == X or i == N) return 1;
        if (i == Y or i == N + 1) return 2;
        if (i == Z or i == N + 2) return 3;
        if (i == N + 3) return 4;
        if (i == N + 4) return 5;
        return 0;
    };

    auto gen_edge_vec = [&](int i, int j) -> pair<vector<pair<int, mint>>, vector<pair<int, mint>>> {
        vector<mint> b(r), c(r);
        b[i * 2] = 1, b[j * 2] = -1;
        c[i * 2 + 1] = 1, c[j * 2 + 1] = -1;
        if (Label(i)) {
            b[i * 2] -= 1;
            b[i * 2 + 1] -= Label(i);
        }
        if (Label(j)) {
            b[j * 2] += 1;
            b[j * 2 + 1] += Label(j);
        }
        vector<pair<int, mint>> bret, cret;
        for (int i = 0; i < int(b.size()); ++i) {
            if (b[i] != 0) bret.emplace_back(i, b[i]);
            if (c[i] != 0) cret.emplace_back(i, c[i]);
        }
        return {bret, cret};
    };

    vector<tuple<vector<pair<int, mint>>, vector<pair<int, mint>>, int>> bcws;
    to.resize(N + 3);
    for (auto i : to[X]) {
        to[N].push_back(i);
        to[i].push_back(N);
    }
    for (auto i : to[Y]) {
        to[N + 1].push_back(i);
        to[i].push_back(N + 1);
    }
    for (auto i : to[Z]) {
        to[N + 2].push_back(i);
        to[i].push_back(N + 2);
    }

    for (int i = 0; i < N + 3; ++i) {
        for (int j : to[i]) {
            if (i > j) continue;
            auto [b, c] = gen_edge_vec(i, j);
            bcws.emplace_back(b, c, 1);
        }

        if (!Label(i)) {
            auto [b, c] = gen_edge_vec(i, N + 3);
            bcws.emplace_back(b, c, 0);
        }
    }

    {
        auto [b, c] = gen_edge_vec(N + 3, N + 4);
        bcws.emplace_back(b, c, 0);
    }

    vector mat0(r, vector<mint>(r));
    vector mat1(r, vector<mint>(r));
    for (const auto &[b, c, w] : bcws) {
        mint x = rand_int() % mint::mod();
        mint y = rand_int() % mint::mod();
        for (auto [i, bi] : b) {
            for (auto [j, cj] : c) {
                auto v = bi * cj * x;
                if (w == 0) {
                    mat0[i][j] += v;
                    mat0[j][i] -= v;
                    mat1[i][j] += v * y;
                    mat1[j][i] -= v * y;
                }
                if (w == 1) {
                    mat1[i][j] += v;
                    mat1[j][i] -= v;
                }
            }
        }
    }

    for (int t = 0; t < 8; ++t) {
        std::vector<mint> b(r), c(r);
        for (auto &x : b) x = rand_int() % mint::mod();
        for (auto &x : c) x = rand_int() % mint::mod();
        mint y = rand_int() % mint::mod();

        for (int i = 0; i < r; ++i) {
            for (int j = 0; j < r; ++j) {
                mat0[i][j] += b[i] * c[j] * y;
                mat1[i][j] += b[i] * c[j];
            }
        }
    }

    // det(x mat1 + mat0) を x の多項式として求めたい
    // mat1 を掃き出して det(\lambda I - A) の形に帰着させる
    for (int i = 0; i < r; ++i) {
        int piv = -1;
        for (int h = i; h < r; ++h) {
            if (mat1[h][i] != 0) piv = h;
        }
        if (piv < 0) {
            cout << "-1\n";
            return 0;
        }
        assert(piv >= i);
        swap(mat0[i], mat0[piv]);
        swap(mat1[i], mat1[piv]);

        if (i != piv) {
            for (int w = 0; w < r; ++w) {
                mat0[i][w] *= -1;
                mat1[i][w] *= -1;
            }
        }

        mint inv = mat1[i][i].inv();
        for (int w = 0; w < r; ++w) {
            mat0[i][w] *= inv;
            mat1[i][w] *= inv;
        }
        for (int h = 0; h < r; ++h) {
            if (h == i) continue;
            if (mat1[h][i] == 0) continue;
            const mint coeff = mat1[h][i];
            for (int w = 0; w < r; ++w) {
                mat1[h][w] -= coeff * mat1[i][w];
                mat0[h][w] -= coeff * mat0[i][w];
            }
        }
    }

    for(auto &v : mat0) for (auto &x : v) x = -x;

    auto det_poly = characteristic_poly<mint>(mat0);

    int ret = 0;
    while (ret < int(det_poly.size()) and det_poly[ret] == 0) ++ret;

    if (ret < int(det_poly.size())) {
        cout << ret / 2 << '\n';
    } else {
        cout << "-1\n";
    }
}
0