Merge pull request #367 from hitonanode/update-euler-tour

hitonanode · web-flow · commit 6afb86cdf363 · 2025-01-03T10:18:09.000+09:00
Update euler tour
diff --git a/graph/extended_block_cut_trees.md b/graph/extended_block_cut_trees.md
@@ -51,5 +51,5 @@ hld.build();
 
 - [Block-cut tree - ei1333の日記](https://ei1333.hateblo.jp/entry/2020/03/25/010057)
 - [My Algorithm : kopricky アルゴリズムライブラリ](https://kopricky.github.io/code/GraphDecomposition/articulation.html)
-- [拡張 Block Cut Tree | cp_library](https://ssrs-cp.github.io/cp_library/graph/extended_block_cut_tree.hpp.html)
+- [拡張 Block Cut Tree \| cp_library](https://ssrs-cp.github.io/cp_library/graph/extended_block_cut_tree.hpp.html)
 - [Xユーザーの熨斗袋さん: 「G = (V, E) に対する Block Cut Tree、G の2-点連結成分全体を C として...](https://x.com/noshi91/status/1529858538650374144)
diff --git a/other_algorithms/mos_algorithm.md b/other_algorithms/mos_algorithm.md
@@ -29,8 +29,10 @@ for (auto x : ret) cout << x << '\n';
 ## 問題例
 
 - [Codeforces Round #221 (Div. 1) D. Tree and Queries](https://codeforces.com/contest/375/submission/114665433)
-  - 部分木クエリにも使用可能．
+  - 部分木クエリにも使用可能（ただし DSU on tree でもより良い最悪計算量で同様のことが可能）．
 
 ## Links
 
 - [Codeforces blog のコメント](https://codeforces.com/blog/entry/20032?#comment-248430) : "Mo's algorithm" の起源に関する議論．
+- [Mo's algorithm - ei1333の日記](https://ei1333.hateblo.jp/entry/2017/09/11/211011)
+  - 木の部分木クエリやパスクエリに対する Mo's algorithm の応用にも触れられている．
diff --git a/tree/euler_tour.hpp b/tree/euler_tour.hpp
@@ -0,0 +1,64 @@
+#pragma once
+#include <cassert>
+#include <utility>
+#include <vector>
+
+// Euler tour
+// https://maspypy.com/euler-tour-%E3%81%AE%E3%81%8A%E5%8B%89%E5%BC%B7
+struct euler_tour {
+    int n;
+    int root;
+
+    std::vector<std::pair<int, int>> edges; // (parent, child)
+
+    // - 頂点 v に関する部分木に含まれる辺は， [begins[v], ends[v]) に 2 回ずつ登場
+    // - [begins[u], begins[v]) (begins[u] <= begins[v]) の半開区間に u-v パスを構成する辺が奇数回登場
+    std::vector<int> begins;
+    std::vector<int> ends;
+
+    std::vector<int> par_eid;
+    std::vector<std::pair<int, bool>> tour; // (edge_id, flg) flg=true: down, false: up
+
+    void _build_dfs(int now, int prv_eid, const std::vector<std::vector<std::pair<int, int>>> &to) {
+        tour.emplace_back(prv_eid, true);
+        begins[now] = tour.size();
+
+        for (auto [nxt, eid] : to[now]) {
+            if (eid == prv_eid) continue;
+            par_eid[nxt] = eid;
+            if (edges[eid].first == nxt) std::swap(edges[eid].first, edges[eid].second);
+            _build_dfs(nxt, eid, to);
+        }
+
+        ends[now] = tour.size();
+        tour.emplace_back(prv_eid, false);
+    }
+
+    euler_tour() = default;
+
+    euler_tour(int n, const std::vector<std::pair<int, int>> &edges_, int root)
+        : n(n), root(root), edges(edges_), begins(n, -1), ends(n, -1), par_eid(n, -1) {
+        std::vector<std::vector<std::pair<int, int>>> to(n);
+        for (int eid = 0; eid < (int)edges.size(); ++eid) {
+            auto [u, v] = edges[eid];
+            assert(u != v);
+            to.at(u).emplace_back(v, eid);
+            to.at(v).emplace_back(u, eid);
+        }
+
+        _build_dfs(root, -1, to);
+    }
+
+    // 頂点 v の部分木の頂点数
+    int subtree_size(int v) const { return (ends.at(v) - begins.at(v)) / 2 + 1; }
+
+    int par(int v) const {
+        int eid = par_eid.at(v);
+        return eid == -1 ? -1 : edges[eid].first;
+    }
+
+    int tour_child(int idx) const {
+        int eid = tour.at(idx).first;
+        return eid < 0 ? root : edges[eid].second;
+    }
+};
diff --git a/tree/euler_tour.md b/tree/euler_tour.md
@@ -0,0 +1,38 @@
+---
+title: Euler tour （木のオイラーツアー）
+documentation_of: ./euler_tour.hpp
+---
+
+木のオイラーツアーを構築．パスクエリをデータ構造に載せて高速に解く場合や， Mo's algorithm を適用する場合に有用．
+
+## 使用方法
+
+以下で構築できる．
+
+```cpp
+int n, root;
+vector<pair<int, int>> edges;
+
+const euler_tour et(n, edges, root);
+```
+
+木上の $u$-$v$ パスに着目したい場合，以下の処理を行うと，当該パスを構成する辺が半開区間 `[begins, ends)` に丁度奇数回登場する．
+
+```cpp
+int begins = et.begins.at(u), ends = et.begins.at(v);
+
+for (int i = begins; i < ends; ++i) {
+    // Insert or erase et.tour_child(i);
+}
+```
+
+半開区間 `[begins, ends)` たちに対して Mo's algorithm などを適用することで，クエリたちが効率的に処理できることがある．
+
+## 問題例
+
+- [Primitive Queries \| CodeChef](https://www.codechef.com/problems/DISTNUM3)
+  - 木のパス上の頂点集合に関するクエリを Mo's algorithm で解く．
+
+## 参考文献・リンク
+
+- [Mo's algorithm - ei1333の日記](https://ei1333.hateblo.jp/entry/2017/09/11/211011)
diff --git a/tree/eulertour.hpp b/tree/eulertour.hpp
diff --git a/tree/preorder_tree_dfs.hpp b/tree/preorder_tree_dfs.hpp
@@ -0,0 +1,30 @@
+#pragma once
+#include <cassert>
+#include <vector>
+
+// Preorder tree DFS
+// 木を DFS して行きがけ順に頂点を保持する．
+// 木を区間に変換する．部分木を構成する頂点は連続して現れるので，部分木の頂点クエリ等に有用．
+// heavy_light_decomposition が上位互換
+struct preorder_tree_dfs {
+    int n; // # of vertices of tree
+    int root;
+    std::vector<int> subtree_begin, subtree_end;
+    std::vector<int> vis_order;
+
+    void _build_dfs(int now, const std::vector<std::vector<int>> &to) {
+        subtree_begin[now] = vis_order.size();
+        vis_order.push_back(now);
+        for (auto nxt : to[now]) {
+            if (subtree_begin[nxt] == -1) _build_dfs(nxt, to);
+        }
+        subtree_end[now] = vis_order.size();
+    }
+
+    preorder_tree_dfs() = default;
+
+    preorder_tree_dfs(const std::vector<std::vector<int>> &to, int root)
+        : n(to.size()), root(root), subtree_begin(n, -1), subtree_end(n, -1) {
+        _build_dfs(root, to);
+    }
+};
diff --git a/tree/preorder_tree_dfs.md b/tree/preorder_tree_dfs.md
@@ -0,0 +1,22 @@
+---
+title: Preorder tree DFS （木の行きがけ順 DFS）
+documentation_of: ./preorder_tree_dfs.hpp
+---
+
+与えられた木に対して，行きがけ順に頂点を列挙した列を構築する．部分木を構成する頂点は連続して現れるので，部分木の頂点クエリ等に有用．
+
+## 使用方法
+
+```cpp
+int N;
+vector<vector<int>> to(N);
+int root;
+
+preorder_tree_dfs tour(to, root);
+
+// 頂点 v の部分木を構成する頂点は半開区間 [tour.subtree_begin[v], tour.subtree_end[v]) に現れる
+```
+
+## 問題例
+
+- [Library Checker: Vertex Add Subtree Sum](https://judge.yosupo.jp/problem/vertex_add_subtree_sum)
diff --git a/tree/test/preorder_tree_dfs.test.cpp b/tree/test/preorder_tree_dfs.test.cpp
@@ -0,0 +1,39 @@
+#define PROBLEM "https://judge.yosupo.jp/problem/vertex_add_subtree_sum"
+#include "../preorder_tree_dfs.hpp"
+#include <cassert>
+#include <iostream>
+#include <atcoder/fenwicktree>
+using namespace std;
+
+int main() {
+    int N, Q;
+    cin >> N >> Q;
+    vector<long long> A(N);
+    vector<vector<int>> to(N);
+    for (auto &a : A) cin >> a;
+    for (int i = 1; i < N; i++) {
+        int p;
+        cin >> p;
+        to.at(p).push_back(i);
+        to.at(i).push_back(p);
+    }
+
+    preorder_tree_dfs tour(to, 0);
+
+    atcoder::fenwick_tree<long long> ft(N);
+    for (int i = 0; i < N; i++) ft.add(tour.subtree_begin.at(i), A.at(i));
+
+    while (Q--) {
+        int q;
+        cin >> q;
+        if (q) {
+            int u;
+            cin >> u;
+            printf("%lld\n", ft.sum(tour.subtree_begin[u], tour.subtree_end[u]));
+        } else {
+            int u, x;
+            cin >> u >> x;
+            ft.add(tour.subtree_begin[u], x);
+        }
+    }
+}
diff --git a/tree/test/vertex-add-subtree-sum.test.cpp b/tree/test/vertex-add-subtree-sum.test.cpp
@@ -1,41 +1,42 @@
-#include "../../segmenttree/point-update-range-get_nonrecursive.hpp"
-#include "../eulertour.hpp"
 #define PROBLEM "https://judge.yosupo.jp/problem/vertex_add_subtree_sum"
+#include "../heavy_light_decomposition.hpp"
 #include <cassert>
 #include <iostream>
+#include <atcoder/fenwicktree>
 using namespace std;
 
 int main() {
     int N, Q;
     cin >> N >> Q;
     vector<long long> A(N);
-    vector<vector<int>> to(N);
+    vector<pair<int, int>> edges;
     for (auto &a : A) cin >> a;
     for (int i = 1; i < N; i++) {
         int p;
         cin >> p;
-        to[p].push_back(i);
-        to[i].push_back(p);
+        edges.emplace_back(p, i);
     }
 
-    PreorderEulerTour tour(to, 0);
+    heavy_light_decomposition tour(N, edges);
+    tour.build();
+
     vector<long long> v;
     for (auto i : tour.vis_order) v.push_back(A[i]);
     assert(int(v.size()) == N);
-    PointUpdateRangeSum<long long> rsq(v, 0);
+    atcoder::fenwick_tree<long long> ft(N);
+    for (int i = 0; i < N; i++) ft.add(i, v[i]);
 
     while (Q--) {
         int q;
         cin >> q;
         if (q) {
             int u;
             cin >> u;
-            printf("%lld\n", rsq.get(tour.subtree_begin[u], tour.subtree_end[u]));
+            printf("%lld\n", ft.sum(tour.subtree_begin[u], tour.subtree_end[u]));
         } else {
             int u, x;
             cin >> u >> x;
-            A[u] += x;
-            rsq.update(tour.subtree_begin[u], A[u]);
+            ft.add(tour.subtree_begin[u], x);
         }
     }
 }
