Library

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:heavy_check_mark: Lowest Common Ancestor
(tree/lowest_common_ancestor.hpp)

説明

根付き木の最小共通祖先を、構築 $O(N\log N)$ / クエリ $O(1)$ で求める。

lca(u, v)

uとvの最小共通祖先を返す。 $O(1)$

distance(u, v)

uとvの距離を返す。

Depends on

Verified with

Code

#pragma once

#include <vector>


#include "../data_structure/sparse_table.hpp"

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


namespace ebi {

namespace internal_lca {
std::pair<int, int> op(std::pair<int, int> a, std::pair<int, int> b) {
    return a.second < b.second ? a : b;
}
}  // namespace internal_lca


template <class T> struct lowest_common_ancestor {
  public:
    lowest_common_ancestor(const Graph<T> &gh, int root = 0)
        : n(gh.size()), id(n), dist(n, 0) {
        auto dfs = [&](auto &&self, int v, int par = -1, T d = 0) -> void {
            id[v] = int(vs.size());
            vs.emplace_back(v, d);
            for (const auto &e : gh[v])
                if (e.to != par) {
                    dist[e.to] = dist[v] + e.cost;
                    self(self, e.to, v, d + 1);
                    vs.emplace_back(v, d);
                }
        };
        dfs(dfs, root);
        st.build(vs);
    }

    int lca(int u, int v) {
        int l = id[u], r = id[v];
        if (r < l) std::swap(l, r);
        return st.fold(l, r + 1).first;
    }

    T distance(int u, int v) {
        int w = lca(u, v);
        return dist[u] + dist[v] - 2 * dist[w];
    }

  private:
    int n;
    std::vector<int> id;
    std::vector<T> dist;
    std::vector<std::pair<int, int>> vs;
    sparse_table<std::pair<int, int>, internal_lca::op> st;
};

}  // namespace ebi
#line 2 "tree/lowest_common_ancestor.hpp"

#include <vector>


#line 2 "data_structure/sparse_table.hpp"

#line 4 "data_structure/sparse_table.hpp"

/*
    reference: https://scrapbox.io/data-structures/Sparse_Table
*/

namespace ebi {

template <class Band, Band (*op)(Band, Band)> struct sparse_table {
  public:
    sparse_table() = default;

    sparse_table(const std::vector<Band> &a) : n(a.size()) {
        table = std::vector(std::__lg(n) + 1, std::vector<Band>(n));
        for (int i = 0; i < n; i++) {
            table[0][i] = a[i];
        }
        for (int k = 1; (1 << k) <= n; k++) {
            for (int i = 0; i + (1 << k) <= n; i++) {
                table[k][i] =
                    op(table[k - 1][i], table[k - 1][i + (1 << (k - 1))]);
            }
        }
    }

    void build(const std::vector<Band> &a) {
        n = (int)a.size();
        table = std::vector(std::__lg(n) + 1, std::vector<Band>(n));
        for (int i = 0; i < n; i++) {
            table[0][i] = a[i];
        }
        for (int k = 1; (1 << k) <= n; k++) {
            for (int i = 0; i + (1 << k) <= n; i++) {
                table[k][i] =
                    op(table[k - 1][i], table[k - 1][i + (1 << (k - 1))]);
            }
        }
    }

    // [l, r)

    Band fold(int l, int r) {
        int k = std::__lg(r - l);
        return op(table[k][l], table[k][r - (1 << k)]);
    }

  private:
    int n;
    std::vector<std::vector<Band>> table;
};

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

#include <cassert>
#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) {
        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) {
        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 {
        return edges[i];
    }

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

    const auto operator[](int i) const {
        return csr[i];
    }
    auto operator[](int i) {
        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 7 "tree/lowest_common_ancestor.hpp"

namespace ebi {

namespace internal_lca {
std::pair<int, int> op(std::pair<int, int> a, std::pair<int, int> b) {
    return a.second < b.second ? a : b;
}
}  // namespace internal_lca


template <class T> struct lowest_common_ancestor {
  public:
    lowest_common_ancestor(const Graph<T> &gh, int root = 0)
        : n(gh.size()), id(n), dist(n, 0) {
        auto dfs = [&](auto &&self, int v, int par = -1, T d = 0) -> void {
            id[v] = int(vs.size());
            vs.emplace_back(v, d);
            for (const auto &e : gh[v])
                if (e.to != par) {
                    dist[e.to] = dist[v] + e.cost;
                    self(self, e.to, v, d + 1);
                    vs.emplace_back(v, d);
                }
        };
        dfs(dfs, root);
        st.build(vs);
    }

    int lca(int u, int v) {
        int l = id[u], r = id[v];
        if (r < l) std::swap(l, r);
        return st.fold(l, r + 1).first;
    }

    T distance(int u, int v) {
        int w = lca(u, v);
        return dist[u] + dist[v] - 2 * dist[w];
    }

  private:
    int n;
    std::vector<int> id;
    std::vector<T> dist;
    std::vector<std::pair<int, int>> vs;
    sparse_table<std::pair<int, int>, internal_lca::op> st;
};

}  // namespace ebi
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