Fu_L's Library

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:heavy_check_mark: counting_spanning_tree_undirected
(src/graph/counting_spanning_tree_undirected.hpp)

counting_spanning_tree_undirected

T counting_spanning_tree_undirected(Graph<U> g)

無向グラフ $G$ の全域木の個数を返します．

制約

計算量

$G$ の頂点数と辺数をそれぞれ $N, M$ として，

Depends on

Verified with

Code

#pragma once
#include "../template/template.hpp"
#include "../graph/graph_template.hpp"
#include "../matrix/matrix.hpp"
#include "../matrix/gauss_elimination.hpp"
template <typename T, typename U>
T counting_spanning_tree_undirected(const Graph<U>& g) {
    const int n = g.size();
    Matrix<T> mat(n, n);
    for(int i = 0; i < n; ++i) {
        mat[i][i] = (int)g[i].size();
        for(const auto& e : g[i]) {
            --mat[i][e.to];
        }
    }
    Matrix<T> lap(n - 1, n - 1);
    for(int i = 0; i < n - 1; ++i) {
        for(int j = 0; j < n - 1; ++j) {
            lap[i][j] = mat[i][j];
        }
    }
    return gauss_elimination(lap).second;
}
#line 2 "src/template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using P = pair<long long, long long>;
#define rep(i, a, b) for(long long i = (a); i < (b); ++i)
#define rrep(i, a, b) for(long long i = (a); i >= (b); --i)
constexpr long long inf = 4e18;
struct SetupIO {
    SetupIO() {
        ios::sync_with_stdio(0);
        cin.tie(0);
        cout << fixed << setprecision(30);
    }
} setup_io;
#line 3 "src/graph/graph_template.hpp"
template <typename T>
struct Edge {
    int from, to;
    T cost;
    int idx;
    Edge()
        : from(-1), to(-1), cost(-1), idx(-1) {}
    Edge(const int from, const int to, const T& cost = 1, const int idx = -1)
        : from(from), to(to), cost(cost), idx(idx) {}
    operator int() const {
        return to;
    }
};
template <typename T>
struct Graph {
    Graph(const int N)
        : n(N), es(0), g(N) {}
    int size() const {
        return n;
    }
    int edge_size() const {
        return es;
    }
    void add_edge(const int from, const int to, const T& cost = 1) {
        assert(0 <= from and from < n);
        assert(0 <= to and to < n);
        g[from].emplace_back(from, to, cost, es);
        g[to].emplace_back(to, from, cost, es++);
    }
    void add_directed_edge(const int from, const int to, const T& cost = 1) {
        assert(0 <= from and from < n);
        assert(0 <= to and to < n);
        g[from].emplace_back(from, to, cost, es++);
    }
    inline vector<Edge<T>>& operator[](const int& k) {
        assert(0 <= k and k < n);
        return g[k];
    }
    inline const vector<Edge<T>>& operator[](const int& k) const {
        assert(0 <= k and k < n);
        return g[k];
    }

   private:
    int n, es;
    vector<vector<Edge<T>>> g;
};
template <typename T>
using Edges = vector<Edge<T>>;
#line 3 "src/matrix/matrix.hpp"
template <typename T>
struct Matrix {
    Matrix(const int h, const int w, const T& val = 0)
        : h(h), w(w), A(h, vector<T>(w, val)) {}
    int H() const {
        return h;
    }
    int W() const {
        return w;
    }
    const vector<T>& operator[](const int i) const {
        assert(0 <= i and i < h);
        return A[i];
    }
    vector<T>& operator[](const int i) {
        assert(0 <= i and i < h);
        return A[i];
    }
    static Matrix I(const int n) {
        Matrix mat(n, n);
        for(int i = 0; i < n; ++i) mat[i][i] = 1;
        return mat;
    }
    Matrix& operator+=(const Matrix& B) {
        assert(h == B.h and w == B.w);
        for(int i = 0; i < h; ++i) {
            for(int j = 0; j < w; ++j) {
                (*this)[i][j] += B[i][j];
            }
        }
        return (*this);
    }
    Matrix& operator-=(const Matrix& B) {
        assert(h == B.h and w == B.w);
        for(int i = 0; i < h; ++i) {
            for(int j = 0; j < w; ++j) {
                (*this)[i][j] -= B[i][j];
            }
        }
        return (*this);
    }
    Matrix& operator*=(const Matrix& B) {
        assert(w == B.h);
        vector<vector<T>> C(h, vector<T>(B.w, 0));
        for(int i = 0; i < h; ++i) {
            for(int k = 0; k < w; ++k) {
                for(int j = 0; j < B.w; ++j) {
                    C[i][j] += (*this)[i][k] * B[k][j];
                }
            }
        }
        A.swap(C);
        return (*this);
    }
    Matrix& pow(long long t) {
        assert(h == w);
        assert(t >= 0);
        Matrix B = Matrix::I(h);
        while(t > 0) {
            if(t & 1ll) B *= (*this);
            (*this) *= (*this);
            t >>= 1ll;
        }
        A.swap(B.A);
        return (*this);
    }
    Matrix operator+(const Matrix& B) const {
        return (Matrix(*this) += B);
    }
    Matrix operator-(const Matrix& B) const {
        return (Matrix(*this) -= B);
    }
    Matrix operator*(const Matrix& B) const {
        return (Matrix(*this) *= B);
    }
    bool operator==(const Matrix& B) const {
        assert(h == B.H() and w == B.W());
        for(int i = 0; i < h; ++i) {
            for(int j = 0; j < w; ++j) {
                if(A[i][j] != B[i][j]) return false;
            }
        }
        return true;
    }
    bool operator!=(const Matrix& B) const {
        assert(h == B.H() and w == B.W());
        for(int i = 0; i < h; ++i) {
            for(int j = 0; j < w; ++j) {
                if(A[i][j] != B[i][j]) return true;
            }
        }
        return false;
    }

   private:
    int h, w;
    vector<vector<T>> A;
};
#line 4 "src/matrix/gauss_elimination.hpp"
template <typename T>
pair<int, T> gauss_elimination(Matrix<T>& a, int pivot_end = -1) {
    const int h = a.H(), w = a.W();
    int rank = 0;
    assert(-1 <= pivot_end and pivot_end <= w);
    if(pivot_end == -1) pivot_end = w;
    T det = 1;
    for(int j = 0; j < pivot_end; ++j) {
        int idx = -1;
        for(int i = rank; i < h; ++i) {
            if(a[i][j] != T(0)) {
                idx = i;
                break;
            }
        }
        if(idx == -1) {
            det = 0;
            continue;
        }
        if(rank != idx) det = -det, swap(a[rank], a[idx]);
        det *= a[rank][j];
        if(a[rank][j] != T(1)) {
            const T coeff = T(1) / a[rank][j];
            for(int k = j; k < w; ++k) a[rank][k] *= coeff;
        }
        for(int i = 0; i < h; ++i) {
            if(i == rank) continue;
            if(a[i][j] != T(0)) {
                const T coeff = a[i][j] / a[rank][j];
                for(int k = j; k < w; ++k) a[i][k] -= a[rank][k] * coeff;
            }
        }
        ++rank;
    }
    return {rank, det};
}
#line 6 "src/graph/counting_spanning_tree_undirected.hpp"
template <typename T, typename U>
T counting_spanning_tree_undirected(const Graph<U>& g) {
    const int n = g.size();
    Matrix<T> mat(n, n);
    for(int i = 0; i < n; ++i) {
        mat[i][i] = (int)g[i].size();
        for(const auto& e : g[i]) {
            --mat[i][e.to];
        }
    }
    Matrix<T> lap(n - 1, n - 1);
    for(int i = 0; i < n - 1; ++i) {
        for(int j = 0; j < n - 1; ++j) {
            lap[i][j] = mat[i][j];
        }
    }
    return gauss_elimination(lap).second;
}
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