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:heavy_check_mark: Euler Tour (subtree queries, with commutative monoids) (data_structure/euler_tour_subtree_query.hpp)

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Operations

根付き木 $T$ であってそれぞれの頂点 $x$ に可換 monoid 重み $a_x \in M$ が乗ったものがあるとする。頂点数を $N$ として均し $O(\log N)$ で次が処理できる。

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Code

#pragma once
#include <cassert>
#include <vector>
#include "utils/macros.hpp"
#include "graph/euler_tour_preorder.hpp"
#include "data_structure/segment_tree.hpp"

/**
 * @brief Euler Tour (subtree queries, with commutative monoids)
 * @docs data_structure/euler_tour_subtree_query.md
 */
template <class CommutativeMonoid>
class euler_tour_subtree_query {
    typedef typename CommutativeMonoid::value_type value_type;
    segment_tree<CommutativeMonoid> data;
    std::vector<int> left, right;

public:
    euler_tour_subtree_query(const std::vector<std::vector<int> > & g, int root, const CommutativeMonoid & mon_ = CommutativeMonoid())
            : data(g.size(), mon_) {
        std::vector<int> tour;
        do_euler_tour_preorder(g, root, tour, left, right);
    }
    template <class InputIterator>
    euler_tour_subtree_query(const std::vector<std::vector<int> > & g, int root, InputIterator first, InputIterator last, const CommutativeMonoid & mon_ = CommutativeMonoid())
            : data(std::distance(first, last), mon_) {
        assert ((int)g.size() == std::distance(first, last));
        std::vector<int> tour;
        do_euler_tour_preorder(g, root, tour, left, right);
        REP (x, g.size()) {
            data.unsafe_point_set(left[x], *(first ++));
        }
        data.unsafe_rebuild();
    }

    void vertex_set(int x, value_type a) {
        assert (0 <= x and x < (int)left.size());
        return data.point_set(left[x], a);
    }
    value_type vertex_get(int x) {
        assert (0 <= x and x < (int)left.size());
        return data.point_get(left[x]);
    }
    value_type subtree_get(int x) {
        assert (0 <= x and x < (int)left.size());
        return data.range_get(left[x], right[x]);
    }
};

#line 2 "data_structure/euler_tour_subtree_query.hpp"
#include <cassert>
#include <vector>
#line 2 "utils/macros.hpp"
#define REP(i, n) for (int i = 0; (i) < (int)(n); ++ (i))
#define REP3(i, m, n) for (int i = (m); (i) < (int)(n); ++ (i))
#define REP_R(i, n) for (int i = (int)(n) - 1; (i) >= 0; -- (i))
#define REP3R(i, m, n) for (int i = (int)(n) - 1; (i) >= (int)(m); -- (i))
#define ALL(x) std::begin(x), std::end(x)
#line 2 "graph/euler_tour_preorder.hpp"
#include <functional>
#line 4 "graph/euler_tour_preorder.hpp"

/**
 * @brief Euler Tour (preorder)
 * @arg g must be a rooted tree, directed or undirected
 */
void do_euler_tour_preorder(std::vector<std::vector<int> > const & g, int root, std::vector<int> & tour, std::vector<int> & left, std::vector<int> & right) {
    int n = g.size();
    tour.clear();
    left.assign(n, -1);
    right.assign(n, -1);
    std::function<void (int, int)> go = [&](int x, int parent) {
        left[x] = tour.size();
        tour.push_back(x);
        for (int y : g[x]) if (y != parent) {
            go(y, x);
        }
        right[x] = tour.size();
    };
    go(root, -1);
}
#line 2 "data_structure/segment_tree.hpp"
#include <algorithm>
#line 6 "data_structure/segment_tree.hpp"

/**
 * @brief Segment Tree / セグメント木 (monoids, 完全二分木)
 * @docs data_structure/segment_tree.md
 * @tparam Monoid (commutativity is not required)
 */
template <class Monoid>
struct segment_tree {
    typedef typename Monoid::value_type value_type;
    const Monoid mon;
    int n;
    std::vector<value_type> a;
    segment_tree() = default;
    segment_tree(int n_, const Monoid & mon_ = Monoid()) : mon(mon_) {
        n = 1; while (n < n_) n *= 2;
        a.resize(2 * n - 1, mon.unit());
    }
    void point_set(int i, value_type b) {  // 0-based
        assert (0 <= i and i < n);
        a[i + n - 1] = b;
        for (i = (i + n) / 2; i > 0; i /= 2) {  // 1-based
            a[i - 1] = mon.mult(a[2 * i - 1], a[2 * i]);
        }
    }
    value_type range_get(int l, int r) {  // 0-based, [l, r)
        assert (0 <= l and l <= r and r <= n);
        value_type lacc = mon.unit(), racc = mon.unit();
        for (l += n, r += n; l < r; l /= 2, r /= 2) {  // 1-based loop, 2x faster than recursion
            if (l % 2 == 1) lacc = mon.mult(lacc, a[(l ++) - 1]);
            if (r % 2 == 1) racc = mon.mult(a[(-- r) - 1], racc);
        }
        return mon.mult(lacc, racc);
    }

    value_type point_get(int i) {  // 0-based
        assert (0 <= i and i < n);
        return a[i + n - 1];
    }

    /**
     * @brief a fast & semigroup-friendly version constructor
     * @note $O(n)$
     */
    template <class InputIterator>
    segment_tree(InputIterator first, InputIterator last, const Monoid & mon_ = Monoid()) : mon(mon_) {
        int size = std::distance(first, last);
        n = 1; while (n < size) n *= 2;
        a.resize(2 * n - 1, mon.unit());
        std::copy(first, last, a.begin() + (n - 1));
        unsafe_rebuild();
    }
    /**
     * @brief update a leaf node without updating ancestors
     * @note $O(1)$
     */
    void unsafe_point_set(int i, value_type b) {  // 0-based
        assert (0 <= i and i < n);
        a[i + n - 1] = b;
    }
    /**
     * @brief re-build non-leaf nodes from leaf nodes
     * @note $O(n)$
     */
    void unsafe_rebuild() {
        REP_R (i, n - 1) {
            a[i] = mon.mult(a[2 * i + 1], a[2 * i + 2]);
        }
    }
};
#line 7 "data_structure/euler_tour_subtree_query.hpp"

/**
 * @brief Euler Tour (subtree queries, with commutative monoids)
 * @docs data_structure/euler_tour_subtree_query.md
 */
template <class CommutativeMonoid>
class euler_tour_subtree_query {
    typedef typename CommutativeMonoid::value_type value_type;
    segment_tree<CommutativeMonoid> data;
    std::vector<int> left, right;

public:
    euler_tour_subtree_query(const std::vector<std::vector<int> > & g, int root, const CommutativeMonoid & mon_ = CommutativeMonoid())
            : data(g.size(), mon_) {
        std::vector<int> tour;
        do_euler_tour_preorder(g, root, tour, left, right);
    }
    template <class InputIterator>
    euler_tour_subtree_query(const std::vector<std::vector<int> > & g, int root, InputIterator first, InputIterator last, const CommutativeMonoid & mon_ = CommutativeMonoid())
            : data(std::distance(first, last), mon_) {
        assert ((int)g.size() == std::distance(first, last));
        std::vector<int> tour;
        do_euler_tour_preorder(g, root, tour, left, right);
        REP (x, g.size()) {
            data.unsafe_point_set(left[x], *(first ++));
        }
        data.unsafe_rebuild();
    }

    void vertex_set(int x, value_type a) {
        assert (0 <= x and x < (int)left.size());
        return data.point_set(left[x], a);
    }
    value_type vertex_get(int x) {
        assert (0 <= x and x < (int)left.size());
        return data.point_get(left[x]);
    }
    value_type subtree_get(int x) {
        assert (0 <= x and x < (int)left.size());
        return data.range_get(left[x], right[x]);
    }
};

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