Dominator Tree
(graph/dominator_tree.hpp)

View this file on GitHub
Last update: 2026-08-13 01:41:40+09:00
Include: #include "graph/dominator_tree.hpp"

Overview

In a directed graph rooted at root, vertex u dominates vertex v when every directed path from root to v passes through u.

dominator_tree(graph, root) computes immediate dominators with the Lengauer-Tarjan algorithm. The immediate dominator of v is the closest strict dominator of v; these edges form the dominator tree.

Only vertices reachable from root belong to the tree. Inactive edges are ignored.

Result

DominatorTree exposes:

Member	Description
`root`	Start vertex used for the computation.
`immediate_dominator[v]`	Immediate dominator of `v`; the root dominates itself, and unreachable vertices store `-1`.
`children[v]`	Children of `v` in the dominator tree.
`dfs_order`	Reachable vertices in the original graph’s DFS discovery order.
`tin`, `tout`	Euler intervals of the dominator tree; unreachable vertices store `-1`.

Methods:

Method	Description	Complexity
`size()`	Returns the original graph’s vertex count.	$O(1)$
`reachable(v)`	Returns whether `v` is reachable from the root.	$O(1)$
`dominates(u, v)`	Returns whether `u` dominates `v`.	$O(1)$

Complexity

The Lengauer-Tarjan algorithm runs in near-linear time, $O((N+M)\alpha(N,M))$, and uses $O(N+M)$ memory.

The graph traversal and dominator-tree Euler traversal are iterative, avoiding recursion-depth issues on long paths.

Example

#include "graph/dominator_tree.hpp"
#include "graph/graph.hpp"

#include <iostream>

int main() {
    m1une::graph::Graph<> graph(4);
    graph.add_directed_edge(0, 1);
    graph.add_directed_edge(0, 2);
    graph.add_directed_edge(1, 3);
    graph.add_directed_edge(2, 3);

    auto tree = m1une::graph::dominator_tree(graph, 0);
    std::cout << tree.immediate_dominator[3] << "\n"; // 0
    std::cout << tree.dominates(0, 3) << "\n";        // 1
}

Depends on

Graph (graph/graph.hpp)

Required by

Graph All (graph/all.hpp)

Verified with

Code

#ifndef M1UNE_GRAPH_DOMINATOR_TREE_HPP
#define M1UNE_GRAPH_DOMINATOR_TREE_HPP 1

#include <cassert>
#include <utility>
#include <vector>

#include "graph.hpp"

namespace m1une {
namespace graph {

struct DominatorTree {
    int root;
    std::vector<int> immediate_dominator;
    std::vector<std::vector<int>> children;
    std::vector<int> dfs_order;
    std::vector<int> tin;
    std::vector<int> tout;

    int size() const {
        return int(immediate_dominator.size());
    }

    bool reachable(int vertex) const {
        assert(0 <= vertex && vertex < size());
        return immediate_dominator[vertex] != -1;
    }

    bool dominates(int ancestor, int vertex) const {
        assert(0 <= ancestor && ancestor < size());
        assert(0 <= vertex && vertex < size());
        return
            reachable(ancestor) &&
            reachable(vertex) &&
            tin[ancestor] <= tin[vertex] &&
            tin[vertex] < tout[ancestor];
    }
};

// Lengauer-Tarjan immediate dominators from one start vertex.
template <class T>
DominatorTree dominator_tree(const Graph<T>& graph, int root) {
    int n = graph.size();
    assert(0 <= root && root < n);

    std::vector<int> dfs_index(n, -1);
    std::vector<int> vertex;
    std::vector<int> parent_vertex(n, -1);
    std::vector<std::pair<int, int>> stack;
    dfs_index[root] = 0;
    vertex.push_back(root);
    stack.emplace_back(root, 0);

    while (!stack.empty()) {
        int current = stack.back().first;
        int& edge_index = stack.back().second;
        if (edge_index == int(graph[current].size())) {
            stack.pop_back();
            continue;
        }
        const auto& edge = graph[current][edge_index++];
        if (!edge.alive || dfs_index[edge.to] != -1) continue;
        parent_vertex[edge.to] = current;
        dfs_index[edge.to] = int(vertex.size());
        vertex.push_back(edge.to);
        stack.emplace_back(edge.to, 0);
    }

    int reachable_count = int(vertex.size());
    std::vector<std::vector<int>> predecessor(reachable_count);
    for (int from : vertex) {
        for (const auto& edge : graph[from]) {
            if (!edge.alive || dfs_index[edge.to] == -1) continue;
            predecessor[dfs_index[edge.to]].push_back(dfs_index[from]);
        }
    }

    std::vector<int> parent(reachable_count, -1);
    for (int index = 1; index < reachable_count; ++index) {
        parent[index] = dfs_index[parent_vertex[vertex[index]]];
    }

    std::vector<int> semi(reachable_count);
    std::vector<int> idom(reachable_count, -1);
    std::vector<int> ancestor(reachable_count, -1);
    std::vector<int> label(reachable_count);
    std::vector<std::vector<int>> bucket(reachable_count);
    for (int index = 0; index < reachable_count; ++index) {
        semi[index] = index;
        label[index] = index;
    }

    auto compress = [&](int start) {
        std::vector<int> path;
        int current = start;
        while (
            ancestor[current] != -1 &&
            ancestor[ancestor[current]] != -1
        ) {
            path.push_back(current);
            current = ancestor[current];
        }
        for (int index = int(path.size()) - 1; index >= 0; --index) {
            int node = path[index];
            int parent_node = ancestor[node];
            if (semi[label[parent_node]] < semi[label[node]]) {
                label[node] = label[parent_node];
            }
            ancestor[node] = ancestor[parent_node];
        }
    };

    auto eval = [&](int node) {
        if (ancestor[node] == -1) return label[node];
        compress(node);
        int parent_node = ancestor[node];
        if (semi[label[parent_node]] < semi[label[node]]) {
            return label[parent_node];
        }
        return label[node];
    };

    for (int current = reachable_count - 1; current >= 1; --current) {
        for (int previous : predecessor[current]) {
            semi[current] = std::min(semi[current], semi[eval(previous)]);
        }
        bucket[semi[current]].push_back(current);
        ancestor[current] = parent[current];

        int parent_node = parent[current];
        for (int node : bucket[parent_node]) {
            int best = eval(node);
            idom[node] =
                semi[best] < semi[node] ? best : parent_node;
        }
        bucket[parent_node].clear();
    }

    for (int current = 1; current < reachable_count; ++current) {
        if (idom[current] != semi[current]) {
            idom[current] = idom[idom[current]];
        }
    }
    idom[0] = 0;

    DominatorTree result;
    result.root = root;
    result.immediate_dominator.assign(n, -1);
    result.children.assign(n, {});
    result.dfs_order = vertex;
    for (int index = 0; index < reachable_count; ++index) {
        int current = vertex[index];
        int dominator = vertex[idom[index]];
        result.immediate_dominator[current] = dominator;
        if (current != root) result.children[dominator].push_back(current);
    }

    result.tin.assign(n, -1);
    result.tout.assign(n, -1);
    int timer = 0;
    std::vector<std::pair<int, int>> tree_stack;
    tree_stack.emplace_back(root, 0);
    result.tin[root] = timer++;
    while (!tree_stack.empty()) {
        int current = tree_stack.back().first;
        int& child_index = tree_stack.back().second;
        if (child_index == int(result.children[current].size())) {
            result.tout[current] = timer;
            tree_stack.pop_back();
            continue;
        }
        int child = result.children[current][child_index++];
        result.tin[child] = timer++;
        tree_stack.emplace_back(child, 0);
    }
    return result;
}

}  // namespace graph
}  // namespace m1une

#endif  // M1UNE_GRAPH_DOMINATOR_TREE_HPP

#line 1 "graph/dominator_tree.hpp"



#include <cassert>
#include <utility>
#include <vector>

#line 1 "graph/graph.hpp"



#include <array>
#line 8 "graph/graph.hpp"

namespace m1une {
namespace graph {

template <class T = int>
struct Edge {
    using cost_type = T;

    int from;
    int to;
    T cost;
    int id;
    bool alive;

    Edge() : from(-1), to(-1), cost(T()), id(-1), alive(true) {}
    Edge(int from_, int to_, T cost_ = T(1), int id_ = -1, bool alive_ = true)
        : from(from_), to(to_), cost(cost_), id(id_), alive(alive_) {}

    int other(int v) const {
        assert(v == from || v == to);
        return from ^ to ^ v;
    }
};

template <class T = int>
struct Graph {
    using edge_type = Edge<T>;
    using cost_type = T;

   private:
    struct EdgePositions {
        std::array<std::pair<int, int>, 2> value{};
        int size = 0;

        void push_back(std::pair<int, int> position) {
            assert(size < 2);
            value[size++] = position;
        }
    };

    int _n;
    int _edge_count;
    std::vector<std::vector<edge_type>> _g;
    std::vector<EdgePositions> _edge_positions;

   public:
    Graph() : _n(0), _edge_count(0) {}
    explicit Graph(int n) : _n(n), _edge_count(0), _g(n) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    int edge_count() const {
        return _edge_count;
    }

    int add_vertex() {
        _g.emplace_back();
        return _n++;
    }

    int add_directed_edge(int from, int to, T cost = T(1)) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        int id = _edge_count++;
        int idx = int(_g[from].size());
        _g[from].push_back(edge_type(from, to, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({from, idx});
        return id;
    }

    int add_edge(int u, int v, T cost = T(1)) {
        assert(0 <= u && u < _n);
        assert(0 <= v && v < _n);
        int id = _edge_count++;
        int u_idx = int(_g[u].size());
        _g[u].push_back(edge_type(u, v, cost, id));
        int v_idx = int(_g[v].size());
        _g[v].push_back(edge_type(v, u, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({u, u_idx});
        _edge_positions.back().push_back({v, v_idx});
        return id;
    }

    void set_edge_alive(int id, bool alive) {
        assert(0 <= id && id < _edge_count);
        for (int i = 0; i < _edge_positions[id].size; ++i) {
            auto [v, idx] = _edge_positions[id].value[i];
            _g[v][idx].alive = alive;
        }
    }

    void erase_edge(int id) {
        set_edge_alive(id, false);
    }

    void revive_edge(int id) {
        set_edge_alive(id, true);
    }

    bool is_edge_alive(int id) const {
        assert(0 <= id && id < _edge_count);
        assert(_edge_positions[id].size != 0);
        auto [v, idx] = _edge_positions[id].value[0];
        return _g[v][idx].alive;
    }

    const std::vector<edge_type>& operator[](int v) const {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    std::vector<edge_type>& operator[](int v) {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    const std::vector<std::vector<edge_type>>& adjacency() const {
        return _g;
    }

    std::vector<std::vector<edge_type>>& adjacency() {
        return _g;
    }

    std::vector<edge_type> edges(bool include_inactive = false) const {
        std::vector<edge_type> result;
        result.reserve(_edge_count);
        std::vector<char> used(_edge_count, false);
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                if (!include_inactive && !e.alive) continue;
                if (0 <= e.id && e.id < _edge_count) {
                    if (used[e.id]) continue;
                    used[e.id] = true;
                }
                result.push_back(e);
            }
        }
        return result;
    }

    Graph reversed() const {
        Graph result(_n);
        result._edge_count = _edge_count;
        result._edge_positions.assign(_edge_count, {});
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                int idx = int(result._g[e.to].size());
                result._g[e.to].push_back(edge_type(e.to, e.from, e.cost, e.id, e.alive));
                if (0 <= e.id && e.id < _edge_count) result._edge_positions[e.id].push_back({e.to, idx});
            }
        }
        return result;
    }
};

}  // namespace graph
}  // namespace m1une


#line 9 "graph/dominator_tree.hpp"

namespace m1une {
namespace graph {

struct DominatorTree {
    int root;
    std::vector<int> immediate_dominator;
    std::vector<std::vector<int>> children;
    std::vector<int> dfs_order;
    std::vector<int> tin;
    std::vector<int> tout;

    int size() const {
        return int(immediate_dominator.size());
    }

    bool reachable(int vertex) const {
        assert(0 <= vertex && vertex < size());
        return immediate_dominator[vertex] != -1;
    }

    bool dominates(int ancestor, int vertex) const {
        assert(0 <= ancestor && ancestor < size());
        assert(0 <= vertex && vertex < size());
        return
            reachable(ancestor) &&
            reachable(vertex) &&
            tin[ancestor] <= tin[vertex] &&
            tin[vertex] < tout[ancestor];
    }
};

// Lengauer-Tarjan immediate dominators from one start vertex.
template <class T>
DominatorTree dominator_tree(const Graph<T>& graph, int root) {
    int n = graph.size();
    assert(0 <= root && root < n);

    std::vector<int> dfs_index(n, -1);
    std::vector<int> vertex;
    std::vector<int> parent_vertex(n, -1);
    std::vector<std::pair<int, int>> stack;
    dfs_index[root] = 0;
    vertex.push_back(root);
    stack.emplace_back(root, 0);

    while (!stack.empty()) {
        int current = stack.back().first;
        int& edge_index = stack.back().second;
        if (edge_index == int(graph[current].size())) {
            stack.pop_back();
            continue;
        }
        const auto& edge = graph[current][edge_index++];
        if (!edge.alive || dfs_index[edge.to] != -1) continue;
        parent_vertex[edge.to] = current;
        dfs_index[edge.to] = int(vertex.size());
        vertex.push_back(edge.to);
        stack.emplace_back(edge.to, 0);
    }

    int reachable_count = int(vertex.size());
    std::vector<std::vector<int>> predecessor(reachable_count);
    for (int from : vertex) {
        for (const auto& edge : graph[from]) {
            if (!edge.alive || dfs_index[edge.to] == -1) continue;
            predecessor[dfs_index[edge.to]].push_back(dfs_index[from]);
        }
    }

    std::vector<int> parent(reachable_count, -1);
    for (int index = 1; index < reachable_count; ++index) {
        parent[index] = dfs_index[parent_vertex[vertex[index]]];
    }

    std::vector<int> semi(reachable_count);
    std::vector<int> idom(reachable_count, -1);
    std::vector<int> ancestor(reachable_count, -1);
    std::vector<int> label(reachable_count);
    std::vector<std::vector<int>> bucket(reachable_count);
    for (int index = 0; index < reachable_count; ++index) {
        semi[index] = index;
        label[index] = index;
    }

    auto compress = [&](int start) {
        std::vector<int> path;
        int current = start;
        while (
            ancestor[current] != -1 &&
            ancestor[ancestor[current]] != -1
        ) {
            path.push_back(current);
            current = ancestor[current];
        }
        for (int index = int(path.size()) - 1; index >= 0; --index) {
            int node = path[index];
            int parent_node = ancestor[node];
            if (semi[label[parent_node]] < semi[label[node]]) {
                label[node] = label[parent_node];
            }
            ancestor[node] = ancestor[parent_node];
        }
    };

    auto eval = [&](int node) {
        if (ancestor[node] == -1) return label[node];
        compress(node);
        int parent_node = ancestor[node];
        if (semi[label[parent_node]] < semi[label[node]]) {
            return label[parent_node];
        }
        return label[node];
    };

    for (int current = reachable_count - 1; current >= 1; --current) {
        for (int previous : predecessor[current]) {
            semi[current] = std::min(semi[current], semi[eval(previous)]);
        }
        bucket[semi[current]].push_back(current);
        ancestor[current] = parent[current];

        int parent_node = parent[current];
        for (int node : bucket[parent_node]) {
            int best = eval(node);
            idom[node] =
                semi[best] < semi[node] ? best : parent_node;
        }
        bucket[parent_node].clear();
    }

    for (int current = 1; current < reachable_count; ++current) {
        if (idom[current] != semi[current]) {
            idom[current] = idom[idom[current]];
        }
    }
    idom[0] = 0;

    DominatorTree result;
    result.root = root;
    result.immediate_dominator.assign(n, -1);
    result.children.assign(n, {});
    result.dfs_order = vertex;
    for (int index = 0; index < reachable_count; ++index) {
        int current = vertex[index];
        int dominator = vertex[idom[index]];
        result.immediate_dominator[current] = dominator;
        if (current != root) result.children[dominator].push_back(current);
    }

    result.tin.assign(n, -1);
    result.tout.assign(n, -1);
    int timer = 0;
    std::vector<std::pair<int, int>> tree_stack;
    tree_stack.emplace_back(root, 0);
    result.tin[root] = timer++;
    while (!tree_stack.empty()) {
        int current = tree_stack.back().first;
        int& child_index = tree_stack.back().second;
        if (child_index == int(result.children[current].size())) {
            result.tout[current] = timer;
            tree_stack.pop_back();
            continue;
        }
        int child = result.children[current][child_index++];
        result.tin[child] = timer++;
        tree_stack.emplace_back(child, 0);
    }
    return result;
}

}  // namespace graph
}  // namespace m1une

m1une's library

This documentation is automatically generated by online-judge-tools/verification-helper

Dominator Tree
(graph/dominator_tree.hpp)

Overview

Result

Complexity

Example

Depends on

Required by

Verified with

Code

Dominator Tree (graph/dominator_tree.hpp)

Overview

Result

Complexity

Example

Depends on

Required by

Verified with

Code

Dominator Tree
(graph/dominator_tree.hpp)