heavy_light_decomposition.hpp

#pragma once
#include <algorithm>
#include <cassert>
#include <functional>
#include <queue>
#include <stack>
#include <utility>
#include <vector>

// Heavy-Light Decomposition of trees
// Based on http://beet-aizu.hatenablog.com/entry/2017/12/12/235950
struct HeavyLightDecomposition {
    int V;
    int k;
    int nb_heavy_path;
    std::vector<std::vector<int>> e;
    std::vector<int> par;         // par[i] = parent of vertex i (Default: -1)
    std::vector<int> depth;       // depth[i] = distance between root and vertex i
    std::vector<int> subtree_sz;  // subtree_sz[i] = size of subtree whose root is i
    std::vector<int> heavy_child; // heavy_child[i] = child of vertex i on heavy path (Default: -1)
    std::vector<int> tree_id;     // tree_id[i] = id of tree vertex i belongs to
    std::vector<int> aligned_id,
        aligned_id_inv;    // aligned_id[i] =  aligned id for vertex i (consecutive on heavy edges)
    std::vector<int> head; // head[i] = id of vertex on heavy path of vertex i, nearest to root
    std::vector<int> head_ids;      // consist of head vertex id's
    std::vector<int> heavy_path_id; // heavy_path_id[i] = heavy_path_id for vertex [i]

    HeavyLightDecomposition(int sz = 0)
        : V(sz), k(0), nb_heavy_path(0), e(sz), par(sz), depth(sz), subtree_sz(sz), heavy_child(sz),
          tree_id(sz, -1), aligned_id(sz), aligned_id_inv(sz), head(sz), heavy_path_id(sz, -1) {}
    void add_edge(int u, int v) {
        e[u].emplace_back(v);
        e[v].emplace_back(u);
    }

    void _build_dfs(int root) {
        std::stack<std::pair<int, int>> st;
        par[root] = -1;
        depth[root] = 0;
        st.emplace(root, 0);
        while (!st.empty()) {
            int now = st.top().first;
            int &i = st.top().second;
            if (i < (int)e[now].size()) {
                int nxt = e[now][i++];
                if (nxt == par[now]) continue;
                par[nxt] = now;
                depth[nxt] = depth[now] + 1;
                st.emplace(nxt, 0);
            } else {
                st.pop();
                int max_sub_sz = 0;
                subtree_sz[now] = 1;
                heavy_child[now] = -1;
                for (auto nxt : e[now]) {
                    if (nxt == par[now]) continue;
                    subtree_sz[now] += subtree_sz[nxt];
                    if (max_sub_sz < subtree_sz[nxt])
                        max_sub_sz = subtree_sz[nxt], heavy_child[now] = nxt;
                }
            }
        }
    }

    void _build_bfs(int root, int tree_id_now) {
        std::queue<int> q({root});
        while (!q.empty()) {
            int h = q.front();
            q.pop();
            head_ids.emplace_back(h);
            for (int now = h; now != -1; now = heavy_child[now]) {
                tree_id[now] = tree_id_now;
                aligned_id[now] = k++;
                aligned_id_inv[aligned_id[now]] = now;
                heavy_path_id[now] = nb_heavy_path;
                head[now] = h;
                for (int nxt : e[now])
                    if (nxt != par[now] and nxt != heavy_child[now]) q.push(nxt);
            }
            nb_heavy_path++;
        }
    }

    void build(std::vector<int> roots = {0}) {
        int tree_id_now = 0;
        for (auto r : roots) _build_dfs(r), _build_bfs(r, tree_id_now++);
    }

    template <class T> std::vector<T> segtree_rearrange(const std::vector<T> &data) const {
        assert(int(data.size()) == V);
        std::vector<T> ret;
        ret.reserve(V);
        for (int i = 0; i < V; i++) ret.emplace_back(data[aligned_id_inv[i]]);
        return ret;
    }

    // query for vertices on path [u, v] (INCLUSIVE)
    void
    for_each_vertex(int u, int v, const std::function<void(int ancestor, int descendant)> &f) const {
        while (true) {
            if (aligned_id[u] > aligned_id[v]) std::swap(u, v);
            f(std::max(aligned_id[head[v]], aligned_id[u]), aligned_id[v]);
            if (head[u] == head[v]) break;
            v = par[head[v]];
        }
    }

    void for_each_vertex_noncommutative(
        int from, int to, const std::function<void(int ancestor, int descendant)> &fup,
        const std::function<void(int ancestor, int descendant)> &fdown) const {
        int u = from, v = to;
        const int lca = lowest_common_ancestor(u, v), dlca = depth[lca];
        while (u >= 0 and depth[u] > dlca) {
            const int p = (depth[head[u]] > dlca ? head[u] : lca);
            fup(aligned_id[p] + (p == lca), aligned_id[u]), u = par[p];
        }
        static std::vector<std::pair<int, int>> lrs;
        int sz = 0;
        while (v >= 0 and depth[v] >= dlca) {
            const int p = (depth[head[v]] >= dlca ? head[v] : lca);
            if (int(lrs.size()) == sz) lrs.emplace_back(0, 0);
            lrs.at(sz++) = {p, v}, v = par.at(p);
        }
        while (sz--) fdown(aligned_id[lrs.at(sz).first], aligned_id[lrs.at(sz).second]);
    }

    // query for edges on path [u, v]
    void for_each_edge(int u, int v, const std::function<void(int, int)> &f) const {
        while (true) {
            if (aligned_id[u] > aligned_id[v]) std::swap(u, v);
            if (head[u] != head[v]) {
                f(aligned_id[head[v]], aligned_id[v]);
                v = par[head[v]];
            } else {
                if (u != v) f(aligned_id[u] + 1, aligned_id[v]);
                break;
            }
        }
    }

    // lowest_common_ancestor: O(log V)
    int lowest_common_ancestor(int u, int v) const {
        assert(tree_id[u] == tree_id[v] and tree_id[u] >= 0);
        while (true) {
            if (aligned_id[u] > aligned_id[v]) std::swap(u, v);
            if (head[u] == head[v]) return u;
            v = par[head[v]];
        }
    }

    int distance(int u, int v) const {
        assert(tree_id[u] == tree_id[v] and tree_id[u] >= 0);
        return depth[u] + depth[v] - 2 * depth[lowest_common_ancestor(u, v)];
    }

    // Level ancestor, O(log V)
    // if k-th parent is out of range, return -1
    int kth_parent(int v, int k) const {
        if (k < 0) return -1;
        while (v >= 0) {
            int h = head.at(v), len = depth.at(v) - depth.at(h);
            if (k <= len) return aligned_id_inv.at(aligned_id.at(v) - k);
            k -= len + 1, v = par.at(h);
        }
        return -1;
    }

    // Jump on tree, O(log V)
    int s_to_t_by_k_steps(int s, int t, int k) const {
        if (k < 0) return -1;
        if (k == 0) return s;
        int lca = lowest_common_ancestor(s, t);
        if (k <= depth.at(s) - depth.at(lca)) return kth_parent(s, k);
        return kth_parent(t, depth.at(s) + depth.at(t) - depth.at(lca) * 2 - k);
    }
};