Library

This documentation is automatically generated by online-judge-tools/verification-helper

View the Project on GitHub ebi-fly13/Library

:heavy_check_mark: Count Spanning Tree
(graph/count_spanning_tree.hpp)

説明

与えられたグラフについて、全域木の個数を数える。頂点数を $N$ として、 $O(N^3)$ 。

count_spanning_tree(std::vector<std::vector> g)

$g[i][j] := (i, j)の辺の本数$ とする。 全域木の個数を返す。

count_spanning_tree(Graph g)

無向グラフ $g$ の全域木の個数を返す。

count_directed_spanning_tree(std::vector<std::vector> g, int root, bool in)

$g[i][j] := i から j への辺の本数$ とする。 根を $root$ としたときの有向全域木の個数を返す。 $in$ が true のときは、全ての辺が根の方を向いている全域木を考える。 $in$ が false のときは、根から全頂点に到達できる全域木を考える。

count_directed_spanning_tree(Graph g, int root, bool in)

グラフ $g$ について、根を $root$ としたときの有向全域木の個数を返す。

Depends on

Required by

Verified with

Code

#pragma once

#include <cassert>
#include <vector>

#include "../graph/base.hpp"
#include "../matrix/base.hpp"
#include "../modint/base.hpp"

namespace ebi {

template <Modint mint>
mint count_spanning_tree(const std::vector<std::vector<int>> &g) {
    const int n = (int)g.size();
    if (n == 1) return 1;
    std::vector<int> deg(n, 0);
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            if (i == j) continue;
            assert(g[i][j] == g[j][i]);
            deg[i] += g[i][j];
        }
    }
    matrix<mint> L(n - 1, n - 1, 0);
    for (int i = 1; i < n; i++) {
        for (int j = 1; j < n; j++) {
            if (i == j)
                L[i - 1][j - 1] = deg[i];
            else
                L[i - 1][j - 1] -= g[i][j];
        }
    }
    return det(L);
}

template <Modint mint, class T> mint count_spanning_tree(const Graph<T> &g) {
    int n = g.node_number();
    if (n == 1) return 1;
    std::vector a(n, std::vector<int>(n, 0));
    for (int i = 0; i < n; i++) {
        for (auto e : g[i]) {
            a[i][e.to]++;
        }
    }
    return count_spanning_tree<mint>(a);
}

template <Modint mint>
mint count_directed_spanning_tree(const std::vector<std::vector<int>> &g,
                                  int root, bool in = false) {
    const int n = (int)g.size();
    if (n == 1) return 1;
    std::vector<int> d(n, 0);
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            if (i != j) d[i] += in ? g[i][j] : g[j][i];
        }
    }
    matrix<mint> L(n - 1, n - 1, 0);
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            int a = i, b = j;
            if (a == root || b == root) continue;
            if (root < a) a--;
            if (root < b) b--;
            if (a == b)
                L[a][b] = d[i];
            else
                L[a][b] = in ? -g[i][j] : -g[j][i];
        }
    }
    return det(L);
}

template <Modint mint, class T>
mint count_directed_spanning_tree(const Graph<T> &g, int root, bool in = false) {
    const int n = g.node_number();
    if (n == 1) return 1;
    std::vector d(n, std::vector<int>(n, 0));
    for (int i = 0; i < n; i++) {
        for (auto e : g[i]) {
            d[i][e.to]++;
        }
    }
    return count_directed_spanning_tree<mint>(d, root, in);
}

}  // namespace ebi
#line 2 "graph/count_spanning_tree.hpp"

#include <cassert>
#include <vector>

#line 2 "graph/base.hpp"

#line 4 "graph/base.hpp"
#include <iostream>
#include <ranges>
#line 7 "graph/base.hpp"

#line 2 "data_structure/simple_csr.hpp"

#line 4 "data_structure/simple_csr.hpp"
#include <utility>
#line 6 "data_structure/simple_csr.hpp"

namespace ebi {

template <class E> struct simple_csr {
    simple_csr() = default;

    simple_csr(int n, const std::vector<std::pair<int, E>>& elements)
        : start(n + 1, 0), elist(elements.size()) {
        for (auto e : elements) {
            start[e.first + 1]++;
        }
        for (auto i : std::views::iota(0, n)) {
            start[i + 1] += start[i];
        }
        auto counter = start;
        for (auto [i, e] : elements) {
            elist[counter[i]++] = e;
        }
    }

    simple_csr(const std::vector<std::vector<E>>& es)
        : start(es.size() + 1, 0) {
        int n = es.size();
        for (auto i : std::views::iota(0, n)) {
            start[i + 1] = (int)es[i].size() + start[i];
        }
        elist.resize(start.back());
        for (auto i : std::views::iota(0, n)) {
            std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]);
        }
    }

    int size() const {
        return (int)start.size() - 1;
    }

    const auto operator[](int i) const {
        return std::ranges::subrange(elist.begin() + start[i],
                                     elist.begin() + start[i + 1]);
    }
    auto operator[](int i) {
        return std::ranges::subrange(elist.begin() + start[i],
                                     elist.begin() + start[i + 1]);
    }

    const auto operator()(int i, int l, int r) const {
        return std::ranges::subrange(elist.begin() + start[i] + l,
                                     elist.begin() + start[i + 1] + r);
    }
    auto operator()(int i, int l, int r) {
        return std::ranges::subrange(elist.begin() + start[i] + l,
                                     elist.begin() + start[i + 1] + r);
    }

  private:
    std::vector<int> start;
    std::vector<E> elist;
};

}  // namespace ebi
#line 9 "graph/base.hpp"

namespace ebi {

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

template <class E> struct Graph {
    using cost_type = E;
    using edge_type = Edge<cost_type>;

    Graph(int n_) : n(n_) {}

    Graph() = default;

    void add_edge(int u, int v, cost_type c) {
        assert(!prepared && u < n && v < n);
        buff.emplace_back(u, edge_type{u, v, c, m});
        edges.emplace_back(edge_type{u, v, c, m++});
    }

    void add_undirected_edge(int u, int v, cost_type c) {
        assert(!prepared && u < n && v < n);
        buff.emplace_back(u, edge_type{u, v, c, m});
        buff.emplace_back(v, edge_type{v, u, c, m});
        edges.emplace_back(edge_type{u, v, c, m});
        m++;
    }

    void read_tree(int offset = 1, bool is_weighted = false) {
        read_graph(n - 1, offset, false, is_weighted);
    }

    void read_parents(int offset = 1) {
        for (auto i : std::views::iota(1, n)) {
            int p;
            std::cin >> p;
            p -= offset;
            add_undirected_edge(p, i, 1);
        }
        build();
    }

    void read_graph(int e, int offset = 1, bool is_directed = false,
                    bool is_weighted = false) {
        for (int i = 0; i < e; i++) {
            int u, v;
            std::cin >> u >> v;
            u -= offset;
            v -= offset;
            if (is_weighted) {
                cost_type c;
                std::cin >> c;
                if (is_directed) {
                    add_edge(u, v, c);
                } else {
                    add_undirected_edge(u, v, c);
                }
            } else {
                if (is_directed) {
                    add_edge(u, v, 1);
                } else {
                    add_undirected_edge(u, v, 1);
                }
            }
        }
        build();
    }

    void build() {
        assert(!prepared);
        csr = simple_csr<edge_type>(n, buff);
        buff.clear();
        prepared = true;
    }

    int size() const {
        return n;
    }

    int node_number() const {
        return n;
    }

    int edge_number() const {
        return m;
    }

    edge_type get_edge(int i) const {
        assert(prepared);
        return edges[i];
    }

    std::vector<edge_type> get_edges() const {
        assert(prepared);
        return edges;
    }

    const auto operator[](int i) const {
        assert(prepared);
        return csr[i];
    }
    auto operator[](int i) {
        assert(prepared);
        return csr[i];
    }

  private:
    int n, m = 0;

    std::vector<std::pair<int, edge_type>> buff;

    std::vector<edge_type> edges;
    simple_csr<edge_type> csr;
    bool prepared = false;
};

}  // namespace ebi
#line 2 "matrix/base.hpp"

#include <algorithm>
#line 8 "matrix/base.hpp"

namespace ebi {

template <class T> struct matrix;

template <class T> matrix<T> identify_matrix(int n) {
    matrix<T> a(n, n);
    for (int i = 0; i < n; i++) {
        a[i][i] = 1;
    }
    return a;
}

template <class T> struct matrix {
  private:
    using Self = matrix<T>;

  public:
    matrix(int n_, int m_, T init_val = 0)
        : n(n_), m(m_), data(n * m, init_val) {}

    matrix(const std::vector<std::vector<T>> &a)
        : n((int)a.size()), m((int)a[0].size()) {
        data = std::vector(n * m);
        for (int i = 0; i < n; i++) {
            std::copy(a[i].begin(), a[i].end(), data.begin() + i * m);
        }
    }

    Self operator+(Self &rhs) const noexcept {
        return Self(*this) += rhs;
    }

    Self operator-(Self &rhs) const noexcept {
        return Self(*this) -= rhs;
    }

    Self operator*(Self &rhs) const noexcept {
        return Self(*this) *= rhs;
    }

    Self operator/(Self &rhs) const noexcept {
        return Self(*this) /= rhs;
    }

    friend Self operator*(const T &lhs, const Self &rhs) {
        return Self(rhs) *= lhs;
    }

    friend Self operator*(const Self &lhs, const T &rhs) {
        return Self(lhs) *= rhs;
    }

    std::vector<T> operator*(const std::vector<T> &rhs) noexcept {
        assert(m == (int)rhs.size());
        std::vector<T> res(n, 0);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                res[i] += (*this)[i][j] * rhs[j];
            }
        }
        return res;
    }

    Self &operator+=(Self &rhs) noexcept {
        assert(this->size() == rhs.size());
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < m; ++j) {
                (*this)[i][j] += rhs[i][j];
            }
        }
        return *this;
    }

    Self &operator-=(Self &rhs) noexcept {
        assert(this->size() == rhs.size());
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < m; ++j) {
                (*this)[i][j] -= rhs[i][j];
            }
        }
        return *this;
    }

    Self &operator*=(Self &rhs) noexcept {
        int h = n, w = rhs.column_size();
        assert(m == rhs.row_size());
        Self ret(h, w);
        for (int i = 0; i < h; ++i) {
            for (int k = 0; k < m; ++k) {
                for (int j = 0; j < w; ++j) {
                    ret[i][j] += (*this)[i][k] * rhs[k][j];
                }
            }
        }
        return *this = ret;
    }

    Self &operator/=(const Self &rhs) noexcept {
        auto ret = rhs.inv();
        assert(ret);
        return *this *= ret.value();
    }

    Self &operator*=(const T &rhs) noexcept {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                (*this)[i][j] *= rhs;
            }
        }
        return *this;
    }

    const auto operator[](int i) const {
        return std::ranges::subrange(data.begin() + i * m,
                                     data.begin() + (i + 1) * m);
    }

    auto operator[](int i) {
        return std::ranges::subrange(data.begin() + i * m,
                                     data.begin() + (i + 1) * m);
    }

    void swap(int i, int j) {
        std::swap_ranges(data.begin() + i * m, data.begin() + (i + 1) * m,
                         data.begin() + j * m);
    }

    int rank() const;

    Self transposition() const {
        Self res(m, n);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                res[j][i] = (*this)[i][j];
            }
        }
        return res;
    }

    std::optional<Self> inv() const {
        assert(row_size() == column_size());
        Self a = *this;
        Self b = identify_matrix<T>(n);
        for (int r = 0; r < n; r++) {
            for (int i = r; i < n; i++) {
                if (a[i][r] != 0) {
                    a.swap(r, i);
                    b.swap(r, i);
                    break;
                }
            }
            if (a[r][r] == 0) return std::nullopt;
            T x = a[r][r].inv();
            for (int j = 0; j < n; j++) {
                if (r < j) a[r][j] *= x;
                b[r][j] *= x;
            }
            for (int i = 0; i < n; i++) {
                if (i == r) continue;
                for (int j = 0; j < n; j++) {
                    if (r < j) a[i][j] -= a[i][r] * a[r][j];
                    b[i][j] -= a[i][r] * b[r][j];
                }
            }
        }
        return b;
    }

    Self pow(long long k) const {
        assert(row_size() == column_size() && k >= 0);
        Self res = identify_matrix<T>(row_size());
        Self x = *this;
        while (k) {
            if (k & 1) res *= x;
            x *= x;
            k >>= 1;
        }
        return res;
    }

    int row_size() const {
        return n;
    }

    int column_size() const {
        return m;
    }

    std::pair<int, int> size() const {
        return {n, m};
    }

  private:
    int n, m;
    std::vector<T> data;
};

template <class T> T det(matrix<T> a) {
    assert(a.row_size() == a.column_size());
    T d = 1;
    int n = a.row_size();
    for (int r = 0; r < n; r++) {
        if (a[r][r] == 0) {
            for (int i = r + 1; i < n; i++) {
                if (a[i][r] != 0) {
                    a.swap(r, i);
                    d = -d;
                }
            }
        }
        if (a[r][r] == 0) return 0;
        d *= a[r][r];
        T inv = a[r][r].inv();
        for (int i = r + 1; i < n; i++) {
            T x = a[i][r] * inv;
            for (int j = r; j < n; j++) {
                a[i][j] -= x * a[r][j];
            }
        }
    }
    return d;
}

template <class T> std::istream &operator>>(std::istream &os, matrix<T> &a) {
    for (int i = 0; i < a.row_size(); i++) {
        for (int j = 0; j < a.column_size(); j++) {
            os >> a[i][j];
        }
    }
    return os;
}

template <class T>
std::ostream &operator<<(std::ostream &os, const matrix<T> &a) {
    for (int i = 0; i < a.row_size(); i++) {
        for (int j = 0; j < a.column_size(); j++) {
            os << a[i][j];
            if (j < a.column_size() - 1) os << ' ';
        }
        if (i < a.row_size() - 1) os << '\n';
    }
    return os;
}

}  // namespace ebi
#line 2 "modint/base.hpp"

#include <concepts>
#line 6 "modint/base.hpp"

namespace ebi {

template <class T>
concept Modint = requires(T a, T b) {
    a + b;
    a - b;
    a * b;
    a / b;
    a.inv();
    a.val();
    a.pow(std::declval<long long>());
    T::mod();
};

template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) {
    long long x;
    os >> x;
    a = x;
    return os;
}

template <Modint mint>
std::ostream &operator<<(std::ostream &os, const mint &a) {
    return os << a.val();
}

}  // namespace ebi
#line 9 "graph/count_spanning_tree.hpp"

namespace ebi {

template <Modint mint>
mint count_spanning_tree(const std::vector<std::vector<int>> &g) {
    const int n = (int)g.size();
    if (n == 1) return 1;
    std::vector<int> deg(n, 0);
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            if (i == j) continue;
            assert(g[i][j] == g[j][i]);
            deg[i] += g[i][j];
        }
    }
    matrix<mint> L(n - 1, n - 1, 0);
    for (int i = 1; i < n; i++) {
        for (int j = 1; j < n; j++) {
            if (i == j)
                L[i - 1][j - 1] = deg[i];
            else
                L[i - 1][j - 1] -= g[i][j];
        }
    }
    return det(L);
}

template <Modint mint, class T> mint count_spanning_tree(const Graph<T> &g) {
    int n = g.node_number();
    if (n == 1) return 1;
    std::vector a(n, std::vector<int>(n, 0));
    for (int i = 0; i < n; i++) {
        for (auto e : g[i]) {
            a[i][e.to]++;
        }
    }
    return count_spanning_tree<mint>(a);
}

template <Modint mint>
mint count_directed_spanning_tree(const std::vector<std::vector<int>> &g,
                                  int root, bool in = false) {
    const int n = (int)g.size();
    if (n == 1) return 1;
    std::vector<int> d(n, 0);
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            if (i != j) d[i] += in ? g[i][j] : g[j][i];
        }
    }
    matrix<mint> L(n - 1, n - 1, 0);
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            int a = i, b = j;
            if (a == root || b == root) continue;
            if (root < a) a--;
            if (root < b) b--;
            if (a == b)
                L[a][b] = d[i];
            else
                L[a][b] = in ? -g[i][j] : -g[j][i];
        }
    }
    return det(L);
}

template <Modint mint, class T>
mint count_directed_spanning_tree(const Graph<T> &g, int root, bool in = false) {
    const int n = g.node_number();
    if (n == 1) return 1;
    std::vector d(n, std::vector<int>(n, 0));
    for (int i = 0; i < n; i++) {
        for (auto e : g[i]) {
            d[i][e.to]++;
        }
    }
    return count_directed_spanning_tree<mint>(d, root, in);
}

}  // namespace ebi
Back to top page