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

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

木の同型性判定

できること

• 見た目の描き方は違っても、グラフとしての構造（つながり方）が本質的に同じかどうかを判定
• 定義: 2つの木 $T_1$ と $T_2$ の頂点の番号をうまく付け替えることで $T_1$ と $T_2$ を全く同じ形にできるとき、同型である
• 部分木の検索や木の圧縮、重複排除などに活用できる

計算量

$O(N \log N)$

使い方

bool is_same = tree_isomorphism(t, g);

Depends on

• 辺 (library/graph/base/edge.hpp)
• グラフ (library/graph/base/graph.hpp)
• 木の重心 (library/graph/tree/centroid.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"
#include "library/graph/tree/centroid.hpp"
bool tree_isomorphism(const Graph &a, const Graph &b) {
    if (a.size() != b.size()) return false;
    const int N = (int)a.size();
    using pvi = pair<vector<int>, vector<int>>;
    auto get_uku = [&](const Graph &t, int e) {
        stack<pair<int, int>> st;
        st.emplace(e, -1);
        vector<int> dep(N, -1), par(N);
        while (!st.empty()) {
            auto p = st.top();
            if (dep[p.first] == -1ll) {
                dep[p.first] = p.second == -1ll ? 0 : dep[p.second] + 1;
                for (auto &&[from, to, cost, idx] : t[p.first]) {
                    if (to != p.second) st.emplace(to, p.first);
                }
            } else {
                par[p.first] = p.second;
                st.pop();
            }
        }
        return make_pair(dep, par);
    };
    auto judge = [&](const pvi &latte, const pvi &malta) {
        int d = *max_element(begin(latte.first), end(latte.first));
        if (d != *max_element(begin(malta.first), end(malta.first))) {
            return false;
        }
        vector<vector<int>> latte_d(d + 1), malta_d(d + 1), latte_key(N),
            malta_key(N);
        for (int i = 0; i < N; i++) latte_d[latte.first[i]].emplace_back(i);
        for (int i = 0; i < N; i++) malta_d[malta.first[i]].emplace_back(i);
        for (int i = d; i >= 0ll; i--) {
            map<vector<int>, int> ord;
            for (auto &idx : latte_d[i]) {
                sort(begin(latte_key[idx]), end(latte_key[idx]));
                ord[latte_key[idx]]++;
            }
            for (auto &idx : malta_d[i]) {
                sort(begin(malta_key[idx]), end(malta_key[idx]));
                if (--ord[malta_key[idx]] < 0ll) return false;
            }
            if (i == 0ll) return ord.size() == 1ll;
            int ptr = 0;
            for (auto &p : ord) {
                if (p.second != 0ll) return false;
                p.second = ptr++;
            }
            for (auto &idx : latte_d[i]) {
                latte_key[latte.second[idx]].emplace_back(ord[latte_key[idx]]);
            }
            for (auto &idx : malta_d[i]) {
                malta_key[malta.second[idx]].emplace_back(ord[malta_key[idx]]);
            }
        }
        assert(0);
    };
    auto p = centroid(a), q = centroid(b);
    if (p.size() != q.size()) return false;
    auto a1 = get_uku(a, p[0]);
    auto b1 = get_uku(b, q[0]);
    if (judge(a1, b1)) return true;
    if (p.size() == 1ll) return false;
    auto a2 = get_uku(a, p[1]);
    return judge(a2, b1);
}

#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;
}
#line 4 "library/graph/tree/tree_isomorphism.hpp"
bool tree_isomorphism(const Graph &a, const Graph &b) {
    if (a.size() != b.size()) return false;
    const int N = (int)a.size();
    using pvi = pair<vector<int>, vector<int>>;
    auto get_uku = [&](const Graph &t, int e) {
        stack<pair<int, int>> st;
        st.emplace(e, -1);
        vector<int> dep(N, -1), par(N);
        while (!st.empty()) {
            auto p = st.top();
            if (dep[p.first] == -1ll) {
                dep[p.first] = p.second == -1ll ? 0 : dep[p.second] + 1;
                for (auto &&[from, to, cost, idx] : t[p.first]) {
                    if (to != p.second) st.emplace(to, p.first);
                }
            } else {
                par[p.first] = p.second;
                st.pop();
            }
        }
        return make_pair(dep, par);
    };
    auto judge = [&](const pvi &latte, const pvi &malta) {
        int d = *max_element(begin(latte.first), end(latte.first));
        if (d != *max_element(begin(malta.first), end(malta.first))) {
            return false;
        }
        vector<vector<int>> latte_d(d + 1), malta_d(d + 1), latte_key(N),
            malta_key(N);
        for (int i = 0; i < N; i++) latte_d[latte.first[i]].emplace_back(i);
        for (int i = 0; i < N; i++) malta_d[malta.first[i]].emplace_back(i);
        for (int i = d; i >= 0ll; i--) {
            map<vector<int>, int> ord;
            for (auto &idx : latte_d[i]) {
                sort(begin(latte_key[idx]), end(latte_key[idx]));
                ord[latte_key[idx]]++;
            }
            for (auto &idx : malta_d[i]) {
                sort(begin(malta_key[idx]), end(malta_key[idx]));
                if (--ord[malta_key[idx]] < 0ll) return false;
            }
            if (i == 0ll) return ord.size() == 1ll;
            int ptr = 0;
            for (auto &p : ord) {
                if (p.second != 0ll) return false;
                p.second = ptr++;
            }
            for (auto &idx : latte_d[i]) {
                latte_key[latte.second[idx]].emplace_back(ord[latte_key[idx]]);
            }
            for (auto &idx : malta_d[i]) {
                malta_key[malta.second[idx]].emplace_back(ord[malta_key[idx]]);
            }
        }
        assert(0);
    };
    auto p = centroid(a), q = centroid(b);
    if (p.size() != q.size()) return false;
    auto a1 = get_uku(a, p[0]);
    auto b1 = get_uku(b, q[0]);
    if (judge(a1, b1)) return true;
    if (p.size() == 1ll) return false;
    auto a2 = get_uku(a, p[1]);
    return judge(a2, b1);
}

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