Sparse Table LCA
(graph/tree/sparse_table_lca.hpp)

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Last update: 2026-08-13 01:41:40+09:00
Include: #include "graph/tree/sparse_table_lca.hpp"

Overview

m1une::tree::SparseTableLca<T> preprocesses an undirected rooted tree so that lca(u, v) is answered in $O(1)$ time.

It uses the standard Euler-tour reduction:

Do a DFS from root.
Record a vertex every time the DFS enters a vertex or returns to a parent.
For each vertex v, store first[v], its first position in this Euler tour.
The LCA of u and v is the minimum-depth vertex in the Euler interval between first[u] and first[v].

That minimum-depth query is static RMQ, so this structure uses ds::SparseTable internally.

Use this when you need many LCA queries and want $O(1)$ per query. If you also need kth_ancestor or jump, use RootedTree instead.

The graph should be an undirected tree built with add_edge. Inactive edges are ignored. If the graph is disconnected, only the component reachable from root gets valid LCA data.

Public Members

Member	Type	What is stored
`root`	`int`	The root used for the DFS, or `-1` for an empty graph.
`parent[v]`	`int`	Parent of `v` in the rooted tree. `parent[root] == -1`.
`parent_edge[v]`	`int`	Edge id connecting `parent[v]` to `v`, or `-1` at the root.
`depth[v]`	`int`	Number of edges from `root` to `v`.
`dist[v]`	`T`	Sum of edge costs from `root` to `v`.
`subtree_size[v]`	`int`	Number of vertices in the rooted subtree of `v`.
`tin[v]`, `tout[v]`	`int`	DFS preorder subtree interval `[tin[v], tout[v])`.
`order[i]`	`int`	Vertex at preorder index `i`.
`first[v]`	`int`	First position of `v` in the Euler tour used for RMQ.
`euler[i]`	`int`	Vertex recorded at Euler-tour position `i`.

Do not confuse tin[v] and first[v]:

tin[v] is a preorder index used for subtree intervals.
first[v] is an Euler-tour index used for LCA RMQ.

Methods

Method	Description	Complexity
`SparseTableLca(g, root)`	Builds the Euler tour and sparse table.	$O(N \log N)$
`void build(g, root)`	Rebuilds the structure.	$O(N \log N)$
`int size()`	Returns the number of vertices in the source graph.	$O(1)$
`bool empty()`	Returns whether the source graph is empty.	$O(1)$
`bool is_ancestor(u, v)`	Returns whether `u` is an ancestor of `v`.	$O(1)$
`bool in_subtree(v, u)`	Returns whether `v` is in the subtree of `u`.	$O(1)$
`int lca(u, v)`	Returns the lowest common ancestor.	$O(1)$
`int dist_edges(u, v)`	Returns the number of edges on the path.	$O(1)$
`T dist_cost(u, v)`	Returns the sum of edge costs on the path.	$O(1)$
`std::pair<int, int> subtree_range(v)`	Returns `[tin[v], tout[v])`.	$O(1)$

Example

#include "graph/graph.hpp"
#include "graph/tree/sparse_table_lca.hpp"
#include <iostream>

int main() {
    m1une::graph::Graph<long long> g(5);
    g.add_edge(0, 1, 2);
    g.add_edge(0, 2, 3);
    g.add_edge(1, 3, 4);
    g.add_edge(1, 4, 5);

    m1une::tree::SparseTableLca<long long> lca(g, 0);

    std::cout << lca.lca(3, 4) << "\n";       // 1
    std::cout << lca.dist_cost(3, 2) << "\n"; // 4 + 2 + 3 = 9
}

Depends on

Required by

Verified with

Code

#ifndef M1UNE_TREE_SPARSE_TABLE_LCA_HPP
#define M1UNE_TREE_SPARSE_TABLE_LCA_HPP 1

#include <algorithm>
#include <cassert>
#include <limits>
#include <utility>
#include <vector>

#include "../../ds/range_query/sparse_table.hpp"
#include "../graph.hpp"

namespace m1une {
namespace tree {

template <class T = int>
struct SparseTableLca {
    using cost_type = T;
    using edge_type = m1une::graph::Edge<T>;

    int root;
    std::vector<int> parent;
    std::vector<int> parent_edge;
    std::vector<int> depth;
    std::vector<T> dist;
    std::vector<int> subtree_size;
    std::vector<int> tin;
    std::vector<int> tout;
    std::vector<int> order;
    std::vector<int> first;
    std::vector<int> euler;

   private:
    struct RmqNode {
        int depth;
        int vertex;
    };

    struct RmqMonoid {
        using value_type = RmqNode;

        static value_type id() {
            return {std::numeric_limits<int>::max(), -1};
        }

        static value_type op(const value_type& a, const value_type& b) {
            if (a.depth != b.depth) return a.depth < b.depth ? a : b;
            return a.vertex < b.vertex ? a : b;
        }
    };

    int _n;
    m1une::ds::SparseTable<RmqMonoid> _st;

    void check_vertex(int v) const {
        assert(0 <= v && v < _n);
        assert(first[v] != -1);
    }

   public:
    SparseTableLca() : root(-1), _n(0) {}
    explicit SparseTableLca(const m1une::graph::Graph<T>& g, int root_ = 0) {
        build(g, root_);
    }

    void build(const m1une::graph::Graph<T>& g, int root_ = 0) {
        _n = g.size();
        root = _n == 0 ? -1 : root_;
        parent.assign(_n, -2);
        parent_edge.assign(_n, -1);
        depth.assign(_n, 0);
        dist.assign(_n, T(0));
        subtree_size.assign(_n, 0);
        tin.assign(_n, -1);
        tout.assign(_n, -1);
        order.clear();
        order.reserve(_n);
        first.assign(_n, -1);
        euler.clear();
        euler.reserve(std::max(0, 2 * _n - 1));
        _st = m1une::ds::SparseTable<RmqMonoid>();

        if (_n == 0) return;
        assert(0 <= root && root < _n);

        std::vector<int> it(_n, 0);
        std::vector<char> visited(_n, false);
        std::vector<int> stack = {root};
        visited[root] = true;
        parent[root] = -1;

        int timer = 0;
        tin[root] = timer++;
        order.push_back(root);
        first[root] = 0;
        euler.push_back(root);

        while (!stack.empty()) {
            int v = stack.back();
            if (it[v] < int(g[v].size())) {
                const auto& e = g[v][it[v]++];
                if (!e.alive) continue;
                if (visited[e.to]) continue;
                visited[e.to] = true;
                parent[e.to] = v;
                parent_edge[e.to] = e.id;
                depth[e.to] = depth[v] + 1;
                dist[e.to] = dist[v] + e.cost;
                tin[e.to] = timer++;
                order.push_back(e.to);
                first[e.to] = int(euler.size());
                euler.push_back(e.to);
                stack.push_back(e.to);
            } else {
                subtree_size[v]++;
                if (parent[v] != -1) subtree_size[parent[v]] += subtree_size[v];
                tout[v] = timer;
                stack.pop_back();
                if (!stack.empty()) euler.push_back(stack.back());
            }
        }

        std::vector<RmqNode> rmq;
        rmq.reserve(euler.size());
        for (int v : euler) rmq.push_back({depth[v], v});
        _st = m1une::ds::SparseTable<RmqMonoid>(std::move(rmq));
    }

    int size() const {
        return _n;
    }

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

    bool is_ancestor(int u, int v) const {
        check_vertex(u);
        check_vertex(v);
        return tin[u] <= tin[v] && tout[v] <= tout[u];
    }

    bool in_subtree(int v, int u) const {
        return is_ancestor(u, v);
    }

    int lca(int u, int v) const {
        check_vertex(u);
        check_vertex(v);
        int l = first[u], r = first[v];
        if (l > r) std::swap(l, r);
        return _st.prod(l, r + 1).vertex;
    }

    int dist_edges(int u, int v) const {
        int w = lca(u, v);
        return depth[u] + depth[v] - 2 * depth[w];
    }

    T dist_cost(int u, int v) const {
        int w = lca(u, v);
        return dist[u] + dist[v] - dist[w] - dist[w];
    }

    std::pair<int, int> subtree_range(int v) const {
        check_vertex(v);
        return {tin[v], tout[v]};
    }
};

}  // namespace tree
}  // namespace m1une

#endif  // M1UNE_TREE_SPARSE_TABLE_LCA_HPP

#line 1 "graph/tree/sparse_table_lca.hpp"



#include <algorithm>
#include <cassert>
#include <limits>
#include <utility>
#include <vector>

#line 1 "ds/range_query/sparse_table.hpp"



#include <bit>
#line 6 "ds/range_query/sparse_table.hpp"
#include <concepts>
#line 9 "ds/range_query/sparse_table.hpp"

#line 1 "monoid/concept.hpp"



#line 5 "monoid/concept.hpp"

namespace m1une {
namespace monoid {

// Concept to check if a type satisfies the requirements of a Monoid.
// A Monoid must have a `value_type`, an identity element `id()`, and an associative binary operation `op()`.
template <typename M>
concept IsMonoid = requires(typename M::value_type a, typename M::value_type b) {
    // 1. Must define `value_type`
    typename M::value_type;

    // 2. Must have a static method `id()` returning `value_type`
    { M::id() } -> std::same_as<typename M::value_type>;

    // 3. Must have a static method `op(a, b)` returning `value_type`
    { M::op(a, b) } -> std::same_as<typename M::value_type>;
};

// Concept for groups. A type satisfying this concept must also obey the group
// laws; concepts can check the interface but not the algebraic properties.
template <typename M>
concept IsGroup = IsMonoid<M> && requires(typename M::value_type a) {
    { M::inv(a) } -> std::same_as<typename M::value_type>;
};

// Concept for commutative groups. Commutativity is a semantic requirement and
// cannot be checked by a C++ concept.
template <typename M>
concept IsCommutativeGroup = IsGroup<M>;

}  // namespace monoid
}  // namespace m1une


#line 11 "ds/range_query/sparse_table.hpp"

namespace m1une {
namespace ds {

// A Sparse Table utilizing C++20 Concepts for type safety.
// It requires a Monoid struct that satisfies `m1une::monoid::IsMonoid`.
// [IMPORTANT] For O(1) range queries to work correctly, the monoid operation MUST be idempotent.
// i.e., Monoid::op(x, x) == x must hold (e.g., Min, Max, GCD, Bitwise AND/OR).
template <m1une::monoid::IsMonoid Monoid>
struct SparseTable {
    using T = typename Monoid::value_type;

   private:
    int _n;
    std::vector<std::vector<T>> _st;

   public:
    // Constructs an empty sparse table.
    SparseTable() : _n(0) {}

    // Constructs a sparse table from an existing vector in O(N log N) time.
    explicit SparseTable(const std::vector<T>& v) : _n(int(v.size())) {
        if (_n == 0) return;

        // Compute the maximum power of 2 needed
        int max_log = std::bit_width((unsigned int)_n);
        _st.assign(max_log, std::vector<T>(_n));

        // Initialize the base level
        for (int i = 0; i < _n; i++) {
            _st[0][i] = v[i];
        }

        // Build the sparse table
        for (int k = 1; k < max_log; k++) {
            for (int i = 0; i + (1 << k) <= _n; i++) {
                _st[k][i] = Monoid::op(_st[k - 1][i], _st[k - 1][i + (1 << (k - 1))]);
            }
        }
    }
    explicit SparseTable(std::vector<T>&& v) : _n(int(v.size())) {
        if (_n == 0) return;

        int max_log = std::bit_width((unsigned int)_n);
        _st.assign(max_log, std::vector<T>(_n));

        for (int i = 0; i < _n; i++) {
            _st[0][i] = std::move(v[i]);
        }

        for (int k = 1; k < max_log; k++) {
            for (int i = 0; i + (1 << k) <= _n; i++) {
                _st[k][i] = Monoid::op(_st[k - 1][i], _st[k - 1][i + (1 << (k - 1))]);
            }
        }
    }

    // Constructs a sparse table from a vector of a different type U.
    // It automatically adapts to the Monoid's initialization requirements:
    // 1. Monoid::make(val) if it exists.
    // 2. Monoid::make(val, index) if the monoid requires global indices.
    // 3. static_cast<T>(val) as a fallback for simple monoids.
    template <typename U>
    requires (!std::same_as<U, T>) && (
        requires(U x) { Monoid::make(x); } ||
        requires(U x, int i) { Monoid::make(x, i); } ||
        std::convertible_to<U, T>
    )
    explicit SparseTable(const std::vector<U>& v) : _n(int(v.size())) {
        if (_n == 0) return;

        int max_log = std::bit_width((unsigned int)_n);
        _st.assign(max_log, std::vector<T>(_n));

        // Compile-time branching based on the available make() signature
        for (int i = 0; i < _n; i++) {
            if constexpr (requires(U x) { Monoid::make(x); }) {
                _st[0][i] = Monoid::make(v[i]);
            } else if constexpr (requires(U x, int idx) { Monoid::make(x, idx); }) {
                _st[0][i] = Monoid::make(v[i], i);
            } else {
                _st[0][i] = static_cast<T>(v[i]);
            }
        }
        for (int k = 1; k < max_log; k++) {
            for (int i = 0; i + (1 << k) <= _n; i++) {
                _st[k][i] = Monoid::op(_st[k - 1][i], _st[k - 1][i + (1 << (k - 1))]);
            }
        }
    }

    // Returns the product (result of the monoid operation) in the range [l, r) in O(1) time.
    // Requires the monoid operation to be idempotent.
    T prod(int l, int r) const {
        assert(0 <= l && l <= r && r <= _n);
        if (l == r) return Monoid::id();

        // Calculate the largest power of 2 less than or equal to the interval length
        int k = std::bit_width((unsigned int)(r - l)) - 1;
        return Monoid::op(_st[k][l], _st[k][r - (1 << k)]);
    }
};

}  // namespace ds
}  // namespace m1une


#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 12 "graph/tree/sparse_table_lca.hpp"

namespace m1une {
namespace tree {

template <class T = int>
struct SparseTableLca {
    using cost_type = T;
    using edge_type = m1une::graph::Edge<T>;

    int root;
    std::vector<int> parent;
    std::vector<int> parent_edge;
    std::vector<int> depth;
    std::vector<T> dist;
    std::vector<int> subtree_size;
    std::vector<int> tin;
    std::vector<int> tout;
    std::vector<int> order;
    std::vector<int> first;
    std::vector<int> euler;

   private:
    struct RmqNode {
        int depth;
        int vertex;
    };

    struct RmqMonoid {
        using value_type = RmqNode;

        static value_type id() {
            return {std::numeric_limits<int>::max(), -1};
        }

        static value_type op(const value_type& a, const value_type& b) {
            if (a.depth != b.depth) return a.depth < b.depth ? a : b;
            return a.vertex < b.vertex ? a : b;
        }
    };

    int _n;
    m1une::ds::SparseTable<RmqMonoid> _st;

    void check_vertex(int v) const {
        assert(0 <= v && v < _n);
        assert(first[v] != -1);
    }

   public:
    SparseTableLca() : root(-1), _n(0) {}
    explicit SparseTableLca(const m1une::graph::Graph<T>& g, int root_ = 0) {
        build(g, root_);
    }

    void build(const m1une::graph::Graph<T>& g, int root_ = 0) {
        _n = g.size();
        root = _n == 0 ? -1 : root_;
        parent.assign(_n, -2);
        parent_edge.assign(_n, -1);
        depth.assign(_n, 0);
        dist.assign(_n, T(0));
        subtree_size.assign(_n, 0);
        tin.assign(_n, -1);
        tout.assign(_n, -1);
        order.clear();
        order.reserve(_n);
        first.assign(_n, -1);
        euler.clear();
        euler.reserve(std::max(0, 2 * _n - 1));
        _st = m1une::ds::SparseTable<RmqMonoid>();

        if (_n == 0) return;
        assert(0 <= root && root < _n);

        std::vector<int> it(_n, 0);
        std::vector<char> visited(_n, false);
        std::vector<int> stack = {root};
        visited[root] = true;
        parent[root] = -1;

        int timer = 0;
        tin[root] = timer++;
        order.push_back(root);
        first[root] = 0;
        euler.push_back(root);

        while (!stack.empty()) {
            int v = stack.back();
            if (it[v] < int(g[v].size())) {
                const auto& e = g[v][it[v]++];
                if (!e.alive) continue;
                if (visited[e.to]) continue;
                visited[e.to] = true;
                parent[e.to] = v;
                parent_edge[e.to] = e.id;
                depth[e.to] = depth[v] + 1;
                dist[e.to] = dist[v] + e.cost;
                tin[e.to] = timer++;
                order.push_back(e.to);
                first[e.to] = int(euler.size());
                euler.push_back(e.to);
                stack.push_back(e.to);
            } else {
                subtree_size[v]++;
                if (parent[v] != -1) subtree_size[parent[v]] += subtree_size[v];
                tout[v] = timer;
                stack.pop_back();
                if (!stack.empty()) euler.push_back(stack.back());
            }
        }

        std::vector<RmqNode> rmq;
        rmq.reserve(euler.size());
        for (int v : euler) rmq.push_back({depth[v], v});
        _st = m1une::ds::SparseTable<RmqMonoid>(std::move(rmq));
    }

    int size() const {
        return _n;
    }

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

    bool is_ancestor(int u, int v) const {
        check_vertex(u);
        check_vertex(v);
        return tin[u] <= tin[v] && tout[v] <= tout[u];
    }

    bool in_subtree(int v, int u) const {
        return is_ancestor(u, v);
    }

    int lca(int u, int v) const {
        check_vertex(u);
        check_vertex(v);
        int l = first[u], r = first[v];
        if (l > r) std::swap(l, r);
        return _st.prod(l, r + 1).vertex;
    }

    int dist_edges(int u, int v) const {
        int w = lca(u, v);
        return depth[u] + depth[v] - 2 * depth[w];
    }

    T dist_cost(int u, int v) const {
        int w = lca(u, v);
        return dist[u] + dist[v] - dist[w] - dist[w];
    }

    std::pair<int, int> subtree_range(int v) const {
        check_vertex(v);
        return {tin[v], tout[v]};
    }
};

}  // namespace tree
}  // namespace m1une

m1une's library

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

Sparse Table LCA
(graph/tree/sparse_table_lca.hpp)

Overview

Public Members

Methods

Example

Depends on

Required by

Verified with

Code

Sparse Table LCA (graph/tree/sparse_table_lca.hpp)

Overview

Public Members

Methods

Example

Depends on

Required by

Verified with

Code

Sparse Table LCA
(graph/tree/sparse_table_lca.hpp)