Library

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:heavy_check_mark: Chromatic Number (彩色数)
(graph/chromatic_number.hpp)

説明

グラフの彩色数を求める。 $O(2^N N)$

高速化はほとんど行っていないため、より高速な実装が欲しい場合は Nachiaさんの記事 を見てください。

Depends on

Verified with

Code

#pragma once

#include <cassert>
#include <ranges>
#include <vector>

#include "../convolution/or_convolution.hpp"
#include "../graph/base.hpp"

namespace ebi {

template <class T> int chromatic_number(const Graph<T> &g) {
    int n = g.node_number();
    std::vector<int> es(n, 0);
    for (auto e : g.get_edges()) {
        es[e.from] |= 1 << e.to;
    }
    std::vector<int> dp1(1 << n, 1);
    for (int s : std::views::iota(0, 1 << n)) {
        for (int v : std::views::iota(0, n)) {
            if (((s >> v) & 1) && (s & es[v])) {
                dp1[s] = 0;
            }
        }
    }
    if (dp1.back() == 1) return 1;
    auto now = dp1;
    for (int k : std::views::iota(2, n + 1)) {
        int sz = now.size();
        now = or_convolution(dp1, now);
        if (now.back() > 0) {
            return k;
        }
        for (int i : std::views::iota(0, sz / 2)) {
            now[i] = now[i + sz / 2] > 0;
        }
        now.resize(sz / 2);
        dp1.resize(sz / 2);
    }
    assert(0);
}

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

#include <cassert>
#include <ranges>
#include <vector>

#line 2 "convolution/or_convolution.hpp"

#line 2 "set_function/subset_transform.hpp"

#include <bit>
#line 6 "set_function/subset_transform.hpp"

namespace ebi {

template <class T> std::vector<T> subset_zeta(const std::vector<T> &a) {
    int n = std::bit_width(a.size()) - 1;
    assert((1 << n) == (int)a.size());
    std::vector<T> ra = a;
    for (int i = 0; i < n; i++) {
        int w = 1 << i;
        for (int p = 0; p < (1 << n); p += 2 * w) {
            for (int s = p; s < p + w; s++) {
                int t = s | w;
                ra[t] += ra[s];
            }
        }
    }
    return ra;
}

template <class T> std::vector<T> subset_mobius(const std::vector<T> &ra) {
    int n = std::bit_width(ra.size()) - 1;
    assert((1 << n) == (int)ra.size());
    std::vector<T> a = ra;
    for (int i = 0; i < n; i++) {
        int w = 1 << i;
        for (int p = 0; p < (1 << n); p += 2 * w) {
            for (int s = p; s < p + w; s++) {
                int t = s | w;
                a[t] -= a[s];
            }
        }
    }
    return a;
}

}  // namespace ebi
#line 4 "convolution/or_convolution.hpp"

namespace ebi {

template <class T>
std::vector<T> or_convolution(const std::vector<T> &a,
                              const std::vector<T> &b) {
    int n = a.size();
    auto ra = subset_zeta(a);
    auto rb = subset_zeta(b);
    for (int i = 0; i < n; i++) {
        ra[i] *= rb[i];
    }
    return subset_mobius(ra);
}

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

#line 4 "graph/base.hpp"
#include <iostream>
#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 9 "graph/chromatic_number.hpp"

namespace ebi {

template <class T> int chromatic_number(const Graph<T> &g) {
    int n = g.node_number();
    std::vector<int> es(n, 0);
    for (auto e : g.get_edges()) {
        es[e.from] |= 1 << e.to;
    }
    std::vector<int> dp1(1 << n, 1);
    for (int s : std::views::iota(0, 1 << n)) {
        for (int v : std::views::iota(0, n)) {
            if (((s >> v) & 1) && (s & es[v])) {
                dp1[s] = 0;
            }
        }
    }
    if (dp1.back() == 1) return 1;
    auto now = dp1;
    for (int k : std::views::iota(2, n + 1)) {
        int sz = now.size();
        now = or_convolution(dp1, now);
        if (now.back() > 0) {
            return k;
        }
        for (int i : std::views::iota(0, sz / 2)) {
            now[i] = now[i + sz / 2] > 0;
        }
        now.resize(sz / 2);
        dp1.resize(sz / 2);
    }
    assert(0);
}

}  // namespace ebi
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