Virtual Tree (graph/tree/virtual-tree.hpp)

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• Last update: 2025-08-13 22:03:35+02:00
• Include: #include "graph/tree/virtual-tree.hpp"

Virtual Tree

wiki: https://oi-wiki.org/graph/virtual-tree/

Virtual tree only connects the edge for the important points without losing the position retation of these important points.

I implement the method 1 in the wiki.

(todo: Not verified)

Depends on

• graph/graph-template.hpp
• graph/tree/doubling-lowest-common-ancestor.hpp

Code

#include "../graph-template.hpp"
#include "./doubling-lowest-common-ancestor.hpp"

template <typename T>
struct VirtualTree {
   public:
    VirtualTree() {}

    Graph<T> build(const Graph<T>& g, const vector<int>& points, int root = 0) {
        DoublingLowestCommonAncestor<T> LCA = DoublingLowestCommonAncestor(g, root);

        timestamp = 0;
        int n = g.size();
        stt.resize(n);
        dfs(root, -1);

        auto p = points;
        sort(p.begin(), p.end(), [&](int x, int y) {
            return stt[x] < stt[y];
        });
        int sz = points.size();
        for (int i = 0; i < sz - 1; i++) {
            p.push_back(LCA.lca(p[i], p[i + 1]));
        }

        sort(p.begin(), p.end(), [&](int x, int y) {
            return stt[x] < stt[y];
        });
        p.resize(unique(p.begin(), p.end()) - p.begin());

        sz = p.size();
        Graph<T> g2(n);
        for (int i = 0; i < sz - 1; i++) {
            int lc = LCA.lca(p[i], p[i + 1]);
            g2.add_edge(lc, p[i + 1]);
        }
        return g2;
    }

   private:
    vector<int> stt;
    int timestamp;
    void dfs(int u, int pre) {
        stt[u] = timestamp++;
        for (auto v : g[u]) {
            if (v == pre) continue;
            dfs(v, u);
        }
    }
};

#line 2 "graph/graph-template.hpp"
#include <bits/stdc++.h>
using namespace std;

template <typename T = int>
struct Edge {
    int from, to;
    T cost;
    int idx;

    Edge() = default;

    Edge(int from, int to, T cost = 1, int idx = -1)
        : from(from), to(to), cost(cost), idx(idx) {}

    operator int() const { return to; }
};

template <typename T = int>
struct Graph {
    vector<vector<Edge<T> > > g;
    int es;

    Graph() = default;

    explicit Graph(int n) : g(n), es(0) {}

    size_t size() const { return g.size(); }

    virtual void add_directed_edge(int from, int to, T cost = 1) {
        g[from].emplace_back(from, to, cost, es++);
    }

    // virtual 可以被重载，实现多态
    virtual void add_edge(int from, int to, T cost = 1) {
        g[from].emplace_back(from, to, cost, es);
        g[to].emplace_back(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 a, b;
            cin >> a >> b;
            a += padding;
            b += padding;
            T c = T(1);
            if (weighted) cin >> c;
            if (directed)
                add_directed_edge(a, b, c);
            else
                add_edge(a, b, c);
        }
    }

    inline vector<Edge<T> > &operator[](const int &k) { return g[k]; }

    inline const vector<Edge<T> > &operator[](const int &k) const { return g[k]; }
};

template <typename T = int>
using Edges = vector<Edge<T> >;
#line 3 "graph/tree/doubling-lowest-common-ancestor.hpp"
/*
树上倍增找 lca
*/

template <typename T = int>
struct DoublingLowestCommonAncestor {
    explicit DoublingLowestCommonAncestor() {}

    explicit DoublingLowestCommonAncestor(const Graph<T> &g, int root = 0) : g(g) {
        build(root);
    }

    void build(int root = 0) {
        LOG = 32 - __builtin_clz(g.size());
        dep.assign(g.size(), 0);
        sum.assign(g.size(), 0);
        table.assign(LOG, vector<int>(g.size(), -1));
        dfs(root, -1, 0);
        for (int k = 0; k + 1 < LOG; k++) {
            for (int i = 0; i < (int)table[k].size(); i++) {
                if (table[k][i] == -1)
                    table[k + 1][i] = -1;
                else
                    table[k + 1][i] = table[k][table[k][i]];
            }
        }
    }

    int lca(int u, int v) {
        if (dep[u] > dep[v]) swap(u, v);
        v = climb(v, dep[v] - dep[u]);
        if (u == v) return u;
        for (int i = LOG - 1; i >= 0; i--) {
            if (table[i][u] != table[i][v]) {
                u = table[i][u];
                v = table[i][v];
            }
        }
        return table[0][u];
    }

    int climb(int u, int k) {
        if (dep[u] < k) return -1;
        for (int i = LOG - 1; i >= 0; i--) {
            if ((k >> i) & 1) u = table[i][u];
        }
        return u;
    }

    T dist(int u, int v) { return sum[u] + sum[v] - 2 * sum[lca(u, v)]; }

   private:
    vector<int> dep;
    vector<T> sum;
    vector<vector<int> > table;
    int LOG;
    const Graph<T> &g;

    void dfs(int idx, int par, int d) {
        table[0][idx] = par;
        dep[idx] = d;
        for (auto &to : g[idx]) {
            if (to != par) {
                sum[to] = sum[idx] + to.cost;
                dfs(to, idx, d + 1);
            }
        }
    }
};
#line 3 "graph/tree/virtual-tree.hpp"

template <typename T>
struct VirtualTree {
   public:
    VirtualTree() {}

    Graph<T> build(const Graph<T>& g, const vector<int>& points, int root = 0) {
        DoublingLowestCommonAncestor<T> LCA = DoublingLowestCommonAncestor(g, root);

        timestamp = 0;
        int n = g.size();
        stt.resize(n);
        dfs(root, -1);

        auto p = points;
        sort(p.begin(), p.end(), [&](int x, int y) {
            return stt[x] < stt[y];
        });
        int sz = points.size();
        for (int i = 0; i < sz - 1; i++) {
            p.push_back(LCA.lca(p[i], p[i + 1]));
        }

        sort(p.begin(), p.end(), [&](int x, int y) {
            return stt[x] < stt[y];
        });
        p.resize(unique(p.begin(), p.end()) - p.begin());

        sz = p.size();
        Graph<T> g2(n);
        for (int i = 0; i < sz - 1; i++) {
            int lc = LCA.lca(p[i], p[i + 1]);
            g2.add_edge(lc, p[i + 1]);
        }
        return g2;
    }

   private:
    vector<int> stt;
    int timestamp;
    void dfs(int u, int pre) {
        stt[u] = timestamp++;
        for (auto v : g[u]) {
            if (v == pre) continue;
            dfs(v, u);
        }
    }
};

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