point-update-range-get_nonrecursive.hpp

#pragma once
#include <algorithm>
#include <cassert>
#include <functional>
#include <iostream>
#include <stack>
#include <vector>

// CUT begin
// Nonrecursive Segment Tree (point-update, range-get)
// - Conditions for operations:
//   - merge_data: [TDATA, TDATA] -> TDATA, e(x, y) == e(y, x)
//   - data2ret: [TDATA, TQUERY] -> TRET
//   - merge_ret: [TRET, TRET] -> TRET, g(defaultRET, x) == x, g(x, y) = g(y, x)
//   - commutability f(e(x, y), q) == g(f(x, q), f(y, q))
template <typename TDATA, typename TRET, typename TQUERY> struct NonrecursiveSegmentTree {
    int N;
    TRET defaultRET;
    virtual TDATA merge_data(const TDATA &, const TDATA &) = 0;
    virtual TRET data2ret(const TDATA &, const TQUERY &) = 0;
    virtual TRET merge_ret(const TRET &, const TRET &) = 0;
    std::vector<TDATA> data;
    inline TDATA &at(int i) { return data[i]; }

    inline void _merge(int i) { at(i) = merge_data(at(i << 1), at((i << 1) + 1)); }
    void initialize(const std::vector<TDATA> &seq, TRET RET_ZERO) {
        N = seq.size();
        defaultRET = RET_ZERO;
        data = seq;
        data.insert(data.end(), seq.begin(), seq.end());
        for (int i = N - 1; i; i--) _merge(i);
    }
    NonrecursiveSegmentTree() = default;
    void update(int pos, const TDATA &x) {
        assert(pos >= 0 and pos < N);
        at(pos + N) = x;
        for (int i = pos + N; i > 1;) i >>= 1, _merge(i);
    }

    // [l, r), 0-indexed
    TRET get(int l, int r, TQUERY query = NULL) {
        assert(l >= 0 and r <= N);
        TRET retl = defaultRET, retr = defaultRET;
        l += N, r += N;
        while (l < r) {
            if (l & 1) retl = merge_ret(retl, data2ret(data[l++], query));
            if (r & 1) retr = merge_ret(data2ret(data[--r], query), retr);
            l >>= 1, r >>= 1;
        }
        return merge_ret(retl, retr);
    }

    // Calculate smallest r that satisfies condition(g(f(x_l, q), ..., f(x_{r - 1}, q)) == true
    // Assumption: Monotonicity of g(x_l, ..., x_r) about r (l: fixed)
    // Complexity: O(log N)
    int binary_search(int l, std::function<bool(TRET)> condition, TQUERY query = NULL) {
        std::stack<int> rs;
        l += N;
        int r = N * 2;
        TRET retl = defaultRET;
        if (condition(retl)) return l - N;
        while (l < r) {
            if (l & 1) {
                TRET ret_tmp = merge_ret(retl, data2ret(data[l], query));
                if (condition(ret_tmp)) {
                    while (l * 2 < N * 2) {
                        ret_tmp = merge_ret(retl, data2ret(data[l * 2], query));
                        if (condition(ret_tmp))
                            l *= 2;
                        else
                            retl = ret_tmp, l = l * 2 + 1;
                    }
                    return l - N;
                }
                l++;
                retl = ret_tmp;
            }
            if (r & 1) rs.push(--r);
            l >>= 1, r >>= 1;
        }
        while (!rs.empty()) {
            l = rs.top();
            rs.pop();
            TRET ret_tmp = merge_ret(retl, data2ret(data[l], query));
            if (condition(ret_tmp)) {
                while (l * 2 < N * 2) {
                    ret_tmp = merge_ret(retl, data2ret(data[l * 2], query));
                    if (condition(ret_tmp))
                        l *= 2;
                    else
                        retl = ret_tmp, l = l * 2 + 1;
                }
                return l - N;
            }
            retl = ret_tmp;
        }
        return N;
    }

    template <typename T1, typename T2, typename T3>
    friend std::ostream &operator<<(std::ostream &os, NonrecursiveSegmentTree<T1, T2, T3> s) {
        os << "[SegmentTree (len: " << s.N << ')';
        for (int i = 0; i < s.N; i++) os << s.at(i + s.N) << ',';
        os << "]";
        return os;
    }
};

// Range Minimum Query
// - get: return min(x_l, ..., x_{r - 1})
template <typename T> struct RangeMinimumQuery : public NonrecursiveSegmentTree<T, T, bool> {
    using SegTree = NonrecursiveSegmentTree<T, T, bool>;
    T merge_data(const T &vl, const T &vr) override { return std::min(vl, vr); };
    T data2ret(const T &v, const bool &q) override { return v; }
    T merge_ret(const T &vl, const T &vr) override { return std::min(vl, vr); };
    RangeMinimumQuery(const std::vector<T> &seq, T defaultmin)
        : SegTree::NonrecursiveSegmentTree() {
        SegTree::initialize(seq, defaultmin);
    };
};

template <typename T> struct PointUpdateRangeSum : public NonrecursiveSegmentTree<T, T, bool> {
    using SegTree = NonrecursiveSegmentTree<T, T, bool>;
    T merge_data(const T &vl, const T &vr) override { return vl + vr; };
    T data2ret(const T &v, const bool &q) override { return v; }
    T merge_ret(const T &vl, const T &vr) override { return vl + vr; };
    PointUpdateRangeSum(const std::vector<T> &seq, T zero) : SegTree::NonrecursiveSegmentTree() {
        SegTree::initialize(seq, zero);
    };
};

// Range Counting less than q Query
// - get: return (#{i | l <= i < r, x_i < q}, total sum of them).
template <typename T>
struct CountAndSumLessThan
    : public NonrecursiveSegmentTree<std::vector<std::pair<T, T>>, std::pair<int, T>, T> {
    using TDATA = std::vector<std::pair<T, T>>;
    using TRET = std::pair<int, T>;
    using TQUERY = T;
    TDATA merge_data(const TDATA &vl, const TDATA &vr) override {
        TDATA ret = vl;
        ret.insert(ret.end(), vr.begin(), vr.end());
        std::sort(ret.begin(), ret.end());
        if (ret.size()) {
            ret[0].second = ret[0].first;
            for (size_t i = 1; i < ret.size(); i++)
                ret[i].second = ret[i - 1].second + ret[i].first;
        }
        return ret;
    }
    TRET data2ret(const TDATA &vec, const TQUERY &q) override {
        int i = std::lower_bound(vec.begin(), vec.end(), std::make_pair(q, q)) - vec.begin();
        if (!i)
            return std::make_pair(0, 0);
        else
            return std::make_pair(i, vec[i - 1].second);
    }
    TRET merge_ret(const TRET &l, const TRET &r) override {
        return std::make_pair(l.first + r.first, l.second + r.second);
    }
    using SegTree = NonrecursiveSegmentTree<TDATA, TRET, TQUERY>;
    CountAndSumLessThan(const std::vector<T> &seq) : SegTree::NonrecursiveSegmentTree() {
        std::vector<TDATA> init;
        for (auto x : seq) init.emplace_back(TDATA{std::pair<T, T>(x, x)});
        SegTree::initialize(init, TRET(0, 0));
    }
};