Merge pull request #337 from hitonanode/static-top-tree

hitonanode · web-flow · commit 8d9072409e10 · 2024-05-05T14:25:21.000+09:00
Add static top tree
diff --git a/data_structure/static_toptree.hpp b/data_structure/static_toptree.hpp
@@ -0,0 +1,213 @@
+#pragma once
+#include <cassert>
+#include <utility>
+#include <vector>
+
+// Structure of static top tree
+// https://atcoder.jp/contests/abc351/submissions/52777033
+struct static_toptree_structure {
+    enum NodeType {
+        Vertex,
+        Compress,
+        Rake,
+        AddEdge,
+        AddVertex,
+    };
+
+    const std::vector<std::vector<int>> &to;
+    std::vector<int> par;
+    std::vector<int> heavy_child; // heavy_child[i] = child of i on heavy path
+
+    // toptree data
+    int stt_root = -1;
+    std::vector<int> P, L, R;
+    std::vector<NodeType> T;
+
+private:
+    int hld_dfs(int now, int prv) {
+        int sz = 1, max_sz = 0;
+        for (int nxt : to.at(now)) {
+            if (nxt == prv) continue;
+            par.at(nxt) = now;
+            int sub_sz = hld_dfs(nxt, now);
+            sz += sub_sz;
+            if (max_sz < sub_sz) max_sz = sub_sz, heavy_child.at(now) = nxt;
+        }
+        return sz;
+    }
+
+    int create(int k, int l, int r, NodeType t) {
+        if (k == -1) {
+            k = P.size();
+            P.push_back(-1), L.push_back(l), R.push_back(r), T.push_back(t);
+        } else {
+            P.at(k) = -1, L.at(k) = l, R.at(k) = r, T.at(k) = t;
+        }
+
+        if (l != -1) P.at(l) = k;
+        if (r != -1) P.at(r) = k;
+
+        return k;
+    }
+
+    std::pair<int, int> merge(const std::vector<std::pair<int, int>> &a, NodeType t) {
+        if (a.size() == 1) return a.front();
+
+        int u = 0;
+        for (auto &[idx, sz] : a) u += sz;
+
+        std::vector<std::pair<int, int>> left, right;
+        for (const auto &[idx, sz] : a) {
+            (u > sz ? left : right).emplace_back(idx, sz), u -= sz * 2;
+        }
+
+        auto [left_idx, left_sz] = merge(left, t);
+        auto [right_idx, right_sz] = merge(right, t);
+
+        return {create(-1, left_idx, right_idx, t), left_sz + right_sz};
+    }
+
+    std::pair<int, int> compress(int i) {
+        std::vector<std::pair<int, int>> chs{add_vertex(i)};
+        while (heavy_child.at(i) != -1) {
+            i = heavy_child.at(i);
+            chs.push_back(add_vertex(i));
+        }
+
+        return merge(chs, Compress);
+    }
+
+    std::pair<int, int> rake(int i) {
+        std::vector<std::pair<int, int>> chs;
+        for (int j : to.at(i)) {
+            if (j == par.at(i) or j == heavy_child.at(i)) continue;
+            chs.push_back(add_edge(j));
+        }
+
+        return chs.empty() ? std::make_pair(-1, 0) : merge(chs, Rake);
+    }
+
+    std::pair<int, int> add_edge(int i) {
+        auto [c, sz] = compress(i);
+        return {create(-1, c, -1, AddEdge), sz};
+    }
+
+    std::pair<int, int> add_vertex(int i) {
+        auto [c, sz] = rake(i);
+        return {create(i, c, -1, c == -1 ? Vertex : AddVertex), sz + 1};
+    }
+
+public:
+    static_toptree_structure(const std::vector<std::vector<int>> &to, int root) : to(to) {
+
+        const int n = to.size();
+
+        par.assign(n, -1), heavy_child.assign(n, -1);
+
+        hld_dfs(root, -1);
+
+        P.assign(n, -1), L.assign(n, -1), R.assign(n, -1), T.assign(n, Vertex);
+
+        stt_root = compress(root).first;
+    }
+
+    int size() const { return P.size(); }
+
+    // Top tree の帰りがけ順に f() を呼ぶ
+    // データの初期化などに利用可能
+    template <class Callback> void dfs_postorder(Callback f) const {
+        auto rec = [&](auto &&self, int now) -> void {
+            if (L.at(now) != -1) self(self, L.at(now));
+            if (R.at(now) != -1) self(self, R.at(now));
+            f(now);
+        };
+        rec(rec, stt_root);
+    }
+
+    // Top tree の v から根（!= もとの木の根）までのパス上で f() を呼ぶ
+    // 一点更新などに利用可能
+    template <class Callback> void path_to_root(int v, Callback f) const {
+        while (v != -1) f(v), v = P.at(v);
+    }
+};
+
+// Static top tree
+template <class TreeDP> struct static_toptree {
+
+    using Point = typename TreeDP::Point;
+    using Path = typename TreeDP::Path;
+
+    const static_toptree_structure &stts;
+    TreeDP &tree_dp;
+
+    std::vector<Point> point;
+    std::vector<Path> path;
+
+private:
+    void _update(int i) {
+        if (stts.T.at(i) == static_toptree_structure::Vertex) {
+            path.at(i) = tree_dp.vertex(i);
+        } else if (stts.T.at(i) == static_toptree_structure::Compress) {
+            path.at(i) = tree_dp.compress(path.at(stts.L.at(i)), path.at(stts.R.at(i)));
+        } else if (stts.T.at(i) == static_toptree_structure::Rake) {
+            point.at(i) = tree_dp.rake(point.at(stts.L.at(i)), point.at(stts.R.at(i)));
+        } else if (stts.T.at(i) == static_toptree_structure::AddEdge) {
+            point.at(i) = tree_dp.add_edge(path.at(stts.L.at(i)));
+        } else if (stts.T.at(i) == static_toptree_structure::AddVertex) {
+            path.at(i) = tree_dp.add_vertex(point.at(stts.L.at(i)), i);
+        } else {
+            assert(false);
+        }
+    }
+
+public:
+    static_toptree(const static_toptree_structure &stts, TreeDP &tree_dp)
+        : stts(stts), tree_dp(tree_dp), point(stts.size()), path(stts.size()) {
+        stts.dfs_postorder([&](int k) { _update(k); });
+    }
+
+    void set(int i) {
+        stts.path_to_root(i, [&](int k) { _update(k); });
+    }
+
+    Path all_prod() const { return path.at(stts.stt_root); }
+
+    // Not verified yet
+    Path get_subtree(int i) const {
+        Path res = path.at(i);
+        while (true) {
+            const int p = stts.P.at(i);
+            if (p == -1 or stts.T.at(p) != static_toptree_structure::Compress) break;
+
+            if (stts.L.at(p) == i) res = tree_dp.compress(res, path.at(stts.R.at(p)));
+            i = p;
+        }
+
+        return res;
+    }
+};
+
+/*
+struct tree_dp {
+    struct Point;  // Point Cluster
+    struct Path;  // Path Cluster
+
+    Path vertex(int i);
+
+    Path compress(const Path &p, const Path &c);
+
+    Point rake(const Point &l, const Point &r);
+
+    Point add_edge(const Path &d);
+
+    Path add_vertex(const Point &d, int i);
+};
+
+vector<vector<int>> to(n);
+int root;
+
+const static_toptree_structure stts(to, root);
+
+tree_dp dp;
+static_toptree tree(stts, dp);
+*/
diff --git a/data_structure/static_toptree.md b/data_structure/static_toptree.md
@@ -0,0 +1,69 @@
+---
+title: Static top tree
+documentation_of: ./static_toptree.hpp
+---
+
+いわゆる Static top tree を扱う．根付き木の部分木に関する各種演算をクエリあたり $O(n \log n)$ で行える．
+
+## 原理
+
+Static top tree のしくみについては [解説 - AtCoder Beginner Contest 351](https://atcoder.jp/contests/abc351/editorial/9868) 等も併せて参照されたい．
+
+Static top tree は，根付き木をもとに平衡二分木を構築する．平衡二分木の各頂点には `PointCluster` と `PathCluster` の 2 種類のデータ構造のいずれかが載る．
+
+例えば，入力として以下の根付き木（ $0$ が根）を与えた場合に構築される static top tree を下に示す．ここで入力の木の辺のうち二重線は heavy-light decomposition (HLD) の heavy edge, 破線は light edge を示す．
+
+![入力](https://gist.githubusercontent.com/hitonanode/330e4c27e0fb231f82cafaba541bbe19/raw/d9ce4a3b4fdda25e8cb9380c65ead5475cce18e6/static-top-tree-example-input-graph.svg)
+
+![出力](https://gist.githubusercontent.com/hitonanode/330e4c27e0fb231f82cafaba541bbe19/raw/d9ce4a3b4fdda25e8cb9380c65ead5475cce18e6/static-top-tree-example-output-graph.svg)
+
+出力の辺のうち実線は PathCluster, 破線は PointCluster が親頂点に伝播されることをそれぞれ意味する．
+
+図から分かるように，HLD の heavy path は PathCluster として管理され， compress 処理によってマージされる．また， light edge で親と繋がる子頂点たちは PointCluster として管理され， rake 処理によってマージされる．これらの処理を問題に応じてうまく考案・実装してやる必要がある．
+
+## 使用方法
+
+まず，以下のように `Point` クラス・ `Path` クラスと `vertex()` / `compress()` / `rake()` / `add_edge()` / `add_vertex()` メソッドを持つクラスを定義する（ `static` はなくても可）．
+
+```cpp
+struct tree_dp {
+    // Point Cluster
+    struct Point {};
+
+    // Path Cluster
+    struct Path {};
+
+    Path vertex(int i);
+
+    static Path compress(const Path &parent, const Path &child);
+
+    static Point rake(const Point &l, const Point &r);
+
+    static Point add_edge(const Path &d);
+
+    Path add_vertex(const Point &d, int i);
+};
+```
+
+その後，以下の手順で構築・利用する．
+
+```cpp
+// 前準備
+vector<vector<int>> to;  // 隣接リスト
+int root = 0;
+
+// 構築
+const static_toptree_structure stts(to, root);
+tree_dp dp;
+static_toptree tree(stts, dp);
+
+// 利用
+tree.set(u); // 頂点 u に更新があった場合に呼ぶ
+
+auto p = tree.all_prod();  // 根に関して値取得
+```
+
+## 問題例
+
+- [AtCoder Beginner Contest 351 G - Hash on Tree](https://atcoder.jp/contests/abc351/tasks/abc351_g)
+- [Library Checker: Point Set Tree Path Composite Sum (Fixed Root)](https://judge.yosupo.jp/problem/point_set_tree_path_composite_sum_fixed_root)
diff --git a/data_structure/test/static_toptree.test.cpp b/data_structure/test/static_toptree.test.cpp
@@ -0,0 +1,108 @@
+#define PROBLEM "https://judge.yosupo.jp/problem/point_set_tree_path_composite_sum_fixed_root"
+
+#include "../static_toptree.hpp"
+
+#include <iostream>
+#include <vector>
+using namespace std;
+
+#include <atcoder/modint>
+using mint = atcoder::modint998244353;
+
+struct tree_dp {
+    vector<int> A;
+    vector<mint> parB;
+    vector<mint> parC;
+
+    // Point Cluster
+    struct Point {
+        mint n;
+        mint sum;
+    };
+
+    // Path Cluster
+    struct Path {
+        mint n;
+        mint sum;
+
+        mint b;
+        mint c;
+    };
+
+    Path vertex(int i) { return {1, A.at(i) * parB.at(i) + parC.at(i), parB.at(i), parC.at(i)}; }
+
+    static Path compress(const Path &p, const Path &c) {
+        return {p.n + c.n, p.sum + c.sum * p.b + c.n * p.c, c.b * p.b, p.c + c.c * p.b};
+    }
+
+    static Point rake(const Point &l, const Point &r) { return {l.n + r.n, l.sum + r.sum}; }
+
+    static Point add_edge(const Path &d) { return {d.n, d.sum}; }
+
+    Path add_vertex(const Point &d, int i) {
+        return {d.n + 1, (d.sum + A.at(i)) * parB.at(i) + (d.n + 1) * parC.at(i), parB.at(i),
+                parC.at(i)};
+    }
+};
+
+int main() {
+    cin.tie(nullptr), ios::sync_with_stdio(false);
+
+    int N, Q;
+    cin >> N >> Q;
+
+    vector<int> A(N);
+    for (auto &a : A) cin >> a;
+
+    vector<int> U(N - 1), V(N - 1), B(N - 1), C(N - 1);
+    vector<vector<int>> to(N);
+
+    for (int e = 0; e < N - 1; ++e) {
+        cin >> U.at(e) >> V.at(e) >> B.at(e) >> C.at(e);
+        to.at(U.at(e)).push_back(V.at(e));
+        to.at(V.at(e)).push_back(U.at(e));
+    }
+
+    const static_toptree_structure stts(to, 0);
+
+    vector<mint> parB(N, 1);
+    vector<mint> parC(N, 0);
+
+    for (int e = 0; e < N - 1; ++e) {
+        int u = U.at(e), v = V.at(e);
+        if (stts.par.at(u) != v) swap(u, v);
+
+        assert(stts.par.at(u) == v);
+
+        parB.at(u) = B.at(e);
+        parC.at(u) = C.at(e);
+    }
+
+    tree_dp dp{A, parB, parC};
+
+    static_toptree tree(stts, dp);
+
+    while (Q--) {
+        int tp;
+        cin >> tp;
+        if (tp == 0) {
+            int w, x;
+            cin >> w >> x;
+            dp.A.at(w) = x;
+            tree.set(w);
+        } else {
+            int e, y, z;
+            cin >> e >> y >> z;
+
+            int u = U.at(e), v = V.at(e);
+            if (stts.par.at(u) != v) swap(u, v);
+            assert(stts.par.at(u) == v);
+
+            dp.parB.at(u) = y;
+            dp.parC.at(u) = z;
+            tree.set(u);
+        }
+
+        cout << tree.all_prod().sum.val() << '\n';
+    }
+}
