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:heavy_check_mark: DP on Static Top Tree
(tree/dp_on_static_top_tree.hpp)

説明

$1$ 点更新・木DP計算が可能なデータ構造である。

必要な関数は以下である。

参考文献

AtCoder Beginner Contest 351 解説

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Verified with

Code

#pragma once

#include "../tree/static_top_tree.hpp"

namespace ebi {

template <class T, class Path, class Point, class Compress, class Rake,
          class Add_edge, class Add_vertex>
struct dp_on_static_top_tree {
  private:
    void dfs(int v) {
        auto [lch, rch] = stt.child(v);
        if (lch != -1) dfs(lch);
        if (rch != -1) dfs(rch);
        update_(v);
    }

    void update_(int v) {
        if (stt.type(v) == Type::Vertex) {
            path[v] = vertex[v];
        } else if (stt.type(v) == Type::Compress) {
            path[v] =
                compress(path[stt.left_child(v)], path[stt.right_child(v)]);
        } else if (stt.type(v) == Type::Rake) {
            point[v] =
                rake(point[stt.left_child(v)], point[stt.right_child(v)]);
        } else if (stt.type(v) == Type::AddEdge) {
            point[v] = add_edge(path[stt.left_child(v)]);
        } else if (stt.type(v) == Type::AddVertex) {
            path[v] = add_vertex(point[stt.left_child(v)], vertex[v]);
        }
    }

    void update(int v) {
        while (v != -1) {
            update_(v);
            v = stt.parent(v);
        }
    }

  public:
    dp_on_static_top_tree(const Graph<T> &g, int root,
                          const std::vector<Path> &vertex_,
                          const Compress &compress_, const Rake &rake_,
                          const Add_edge &add_edge_,
                          const Add_vertex &add_vertex_)
        : stt(g, root),
          n(stt.node_num()),
          path(n),
          point(n),
          vertex(vertex_),
          compress(compress_),
          rake(rake_),
          add_edge(add_edge_),
          add_vertex(add_vertex_) {
        dfs(stt.root());
    }

    Path get() const {
        return path[stt.root()];
    }

    void set(int v, Path x) {
        vertex[v] = x;
        update(v);
    }

  private:
    static_top_tree<T> stt;

    int n;
    std::vector<Path> path;
    std::vector<Point> point;
    std::vector<Path> vertex;
    const Compress compress;
    const Rake rake;
    const Add_edge add_edge;
    const Add_vertex add_vertex;
};

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

#line 2 "tree/static_top_tree.hpp"

#line 2 "graph/base.hpp"

#include <cassert>
#include <iostream>
#include <ranges>
#include <vector>

#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) {
        assert(!prepared && u < n && v < n);
        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) {
        assert(!prepared && u < n && v < n);
        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 {
        assert(prepared);
        return edges[i];
    }

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

    const auto operator[](int i) const {
        assert(prepared);
        return csr[i];
    }
    auto operator[](int i) {
        assert(prepared);
        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 4 "tree/static_top_tree.hpp"

namespace ebi {

enum Type { Vertex, Compress, Rake, AddEdge, AddVertex };

template <class T> struct static_top_tree {
  private:
    struct Node {
        int par = -1, lch = -1, rch = -1;
        Type ty = Type::Vertex;
    };

    void dfs_sz(int v) {
        for (auto &e : g[v]) {
            if (e.to == par[v]) continue;
            par[e.to] = v;
            dfs_sz(e.to);
            sz[v] += sz[e.to];
            if (sz[e.to] > sz[g[v][0].to] || g[v][0].to == par[v]) {
                std::swap(e, g[v][0]);
            }
        }
    }

    int new_node(int k, int l, int r, Type t) {
        if (k == -1) {
            k = (int)stt.size();
            stt.emplace_back(-1, l, r, t);
        } else {
            stt[k].lch = l;
            stt[k].rch = r;
            stt[k].ty = t;
        }
        if (l != -1) stt[l].par = k;
        if (r != -1) stt[r].par = k;
        return k;
    }

    std::pair<int, int> merge(const std::vector<std::pair<int, int>> &a,
                              Type t) {
        if (a.size() == 1) {
            return a[0];
        }
        int sum = 0;
        for (auto [v_, s] : a) sum += s;
        std::vector<std::pair<int, int>> b, c;
        for (auto [i, s] : a) {
            if (sum > s)
                b.emplace_back(i, s);
            else
                c.emplace_back(i, s);
            sum -= 2 * s;
        }
        auto [i, si] = merge(b, t);
        auto [j, sj] = merge(c, t);
        return {new_node(-1, i, j, t), si + sj};
    }

    std::pair<int, int> compress(int v) {
        std::vector<std::pair<int, int>> path{add_vertex(v)};
        while (g[v][0].to != par[v]) {
            path.emplace_back(add_vertex(v = g[v][0].to));
        }
        return merge(path, Type::Compress);
    }

    std::pair<int, int> rake(int v) {
        std::vector<std::pair<int, int>> ch;
        for (int i = 1; i < (int)g[v].size(); i++) {
            if (g[v][i].to == par[v]) continue;
            ch.emplace_back(add_edge(g[v][i].to));
        }
        return ch.empty() ? std::pair<int, int>{-1, 0} : merge(ch, Type::Rake);
    }

    std::pair<int, int> add_edge(int v) {
        auto [i, si] = compress(v);
        return {new_node(-1, i, -1, Type::AddEdge), si};
    }

    std::pair<int, int> add_vertex(int v) {
        auto [i, si] = rake(v);
        return {new_node(v, i, -1, i == -1 ? Type::Vertex : Type::AddVertex),
                si + 1};
    }

  public:
    static_top_tree(Graph<T> g_, int root = 0)
        : n(g_.size()), g(g_), par(n, -1), sz(n, 1), stt(n) {
        if (n == 1) {
            stt_root = 0;
            return;
        }
        dfs_sz(root);
        stt_root = compress(root).first;
    }

    int node_num() const {
        return (int)stt.size();
    }

    int parent(int v) const {
        return stt[v].par;
    }

    std::pair<int, int> child(int v) const {
        return {stt[v].lch, stt[v].rch};
    }

    int left_child(int v) const {
        return stt[v].lch;
    }

    int right_child(int v) const {
        return stt[v].rch;
    }

    Type type(int v) const {
        return stt[v].ty;
    }

    int root() const {
        return stt_root;
    }

  private:
    int n;
    Graph<T> g;
    std::vector<int> par, sz;
    std::vector<Node> stt;
    int stt_root;
};

}  // namespace ebi
#line 4 "tree/dp_on_static_top_tree.hpp"

namespace ebi {

template <class T, class Path, class Point, class Compress, class Rake,
          class Add_edge, class Add_vertex>
struct dp_on_static_top_tree {
  private:
    void dfs(int v) {
        auto [lch, rch] = stt.child(v);
        if (lch != -1) dfs(lch);
        if (rch != -1) dfs(rch);
        update_(v);
    }

    void update_(int v) {
        if (stt.type(v) == Type::Vertex) {
            path[v] = vertex[v];
        } else if (stt.type(v) == Type::Compress) {
            path[v] =
                compress(path[stt.left_child(v)], path[stt.right_child(v)]);
        } else if (stt.type(v) == Type::Rake) {
            point[v] =
                rake(point[stt.left_child(v)], point[stt.right_child(v)]);
        } else if (stt.type(v) == Type::AddEdge) {
            point[v] = add_edge(path[stt.left_child(v)]);
        } else if (stt.type(v) == Type::AddVertex) {
            path[v] = add_vertex(point[stt.left_child(v)], vertex[v]);
        }
    }

    void update(int v) {
        while (v != -1) {
            update_(v);
            v = stt.parent(v);
        }
    }

  public:
    dp_on_static_top_tree(const Graph<T> &g, int root,
                          const std::vector<Path> &vertex_,
                          const Compress &compress_, const Rake &rake_,
                          const Add_edge &add_edge_,
                          const Add_vertex &add_vertex_)
        : stt(g, root),
          n(stt.node_num()),
          path(n),
          point(n),
          vertex(vertex_),
          compress(compress_),
          rake(rake_),
          add_edge(add_edge_),
          add_vertex(add_vertex_) {
        dfs(stt.root());
    }

    Path get() const {
        return path[stt.root()];
    }

    void set(int v, Path x) {
        vertex[v] = x;
        update(v);
    }

  private:
    static_top_tree<T> stt;

    int n;
    std::vector<Path> path;
    std::vector<Point> point;
    std::vector<Path> vertex;
    const Compress compress;
    const Rake rake;
    const Add_edge add_edge;
    const Add_vertex add_vertex;
};

}  // namespace ebi
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