木の重心
(library/graph/tree/centroid.hpp)

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• Last update: 2026-01-21 11:49:22+09:00
• Include: #include "library/graph/tree/centroid.hpp"

木の重心

できること

• 木の重心を求める
• 重心 $\cdots$ その頂点を取り除いたときにできる部分木たちの頂点数が、全て半分以下になる
• 重心はは1個か2個存在する

計算量

$O(V)$

使い方

int N;
Graph G(N);
vector<int> C = centroid(G);

Depends on

• 辺 (library/graph/base/edge.hpp)
• グラフ (library/graph/base/graph.hpp)

Required by

• 木の同型性判定 (library/graph/tree/tree_isomorphism.hpp)

Verified with

• 木 - 木の同型性判定のテスト (tests/graph.tree.tree_isomorphism.test.cpp)

Code

#pragma once
#include "library/graph/base/graph.hpp"
vector<int> centroid(const Graph &G) {
    const int N = (int)G.size();
    stack<pair<int, int>> st;
    st.emplace(0ll, -1ll);
    vector<int> sz(N), par(N);
    while (!st.empty()) {
        auto p = st.top();
        if (sz[p.first] == 0ll) {
            sz[p.first] = 1;
            for (auto &&[from, to, cost, idx] : G[p.first]) {
                if (to != p.second) st.emplace(to, p.first);
            }
        } else {
            for (auto &&[from, to, cost, idx] : G[p.first]) {
                if (to != p.second) sz[p.first] += sz[to];
            }
            par[p.first] = p.second;
            st.pop();
        }
    }
    vector<int> ret;
    int size = N;
    for (int i = 0; i < N; ++i) {
        int val = N - sz[i];
        for (auto &&[from, to, cost, idx] : G[i]) {
            if (to != par[i]) val = max(val, sz[to]);
        }
        if (val < size) size = val, ret.clear();
        if (val == size) ret.emplace_back(i);
    }
    return ret;
}

#line 2 "library/graph/base/edge.hpp"
struct Edge {
    int from, to;
    long long cost;
    int idx;
    Edge(int from, int to, long long cost = 1, int idx = -1)
        : from(from), to(to), cost(cost), idx(idx) {}
};
#line 3 "library/graph/base/graph.hpp"
struct Graph {
    int N;
    vector<vector<Edge>> G;
    int es;
    Graph() = default;
    Graph(int N) : N(N), G(N), es(0) {}
    const vector<Edge> &operator[](int v) const { return G[v]; }
    int size() const { return N; }
    void add(int from, int to, long long cost = 1) {
        G[from].push_back(Edge(from, to, cost, es++));
    }
    void add_both(int from, int to, long long cost = 1) {
        G[from].push_back(Edge(from, to, cost, es));
        G[to].push_back(Edge(to, from, cost, es++));
    }
    void read(int M, int padding = -1, bool weighted = false,
              bool directed = false) {
        for (int i = 0; i < M; i++) {
            int u, v;
            cin >> u >> v;
            u += padding, v += padding;
            long long cost = 1ll;
            if (weighted) cin >> cost;
            if (directed) {
                add(u, v, cost);
            } else {
                add_both(u, v, cost);
            }
        }
    }
};
#line 3 "library/graph/tree/centroid.hpp"
vector<int> centroid(const Graph &G) {
    const int N = (int)G.size();
    stack<pair<int, int>> st;
    st.emplace(0ll, -1ll);
    vector<int> sz(N), par(N);
    while (!st.empty()) {
        auto p = st.top();
        if (sz[p.first] == 0ll) {
            sz[p.first] = 1;
            for (auto &&[from, to, cost, idx] : G[p.first]) {
                if (to != p.second) st.emplace(to, p.first);
            }
        } else {
            for (auto &&[from, to, cost, idx] : G[p.first]) {
                if (to != p.second) sz[p.first] += sz[to];
            }
            par[p.first] = p.second;
            st.pop();
        }
    }
    vector<int> ret;
    int size = N;
    for (int i = 0; i < N; ++i) {
        int val = N - sz[i];
        for (auto &&[from, to, cost, idx] : G[i]) {
            if (to != par[i]) val = max(val, sz[to]);
        }
        if (val < size) size = val, ret.clear();
        if (val == size) ret.emplace_back(i);
    }
    return ret;
}

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