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:question: algorithm/DataStructure/SegmentTree/binary_indexed_tree.hpp

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#ifndef ALGORITHM_BINARY_INDEXED_TREE_HPP
#define ALGORITHM_BINARY_INDEXED_TREE_HPP 1

#include <algorithm>
#include <cassert>
#include <initializer_list>
#include <iterator>
#include <type_traits>
#include <utility>
#include <vector>

#include "../../Math/Algebra/algebra.hpp"

namespace algorithm {

namespace binary_indexed_tree {

template <class AbelianGroup>
class BIT {
public:
    using group_type = AbelianGroup;
    using value_type = group_type::value_type;

private:
    int m_sz;  // m_sz:=(要素数).
    std::vector<group_type> m_tree;

    static constexpr int lsb(int bit) { return bit & -bit; }
    group_type sum_internal(int r) const {
        group_type &&res = group_type::one();
        for(; r >= 1; r -= lsb(r)) res = res * m_tree[r - 1];
        return res;
    }
    void build() {
        for(int i = 1; i < m_sz; ++i) {
            int j = i + lsb(i);
            if(j <= m_sz) m_tree[j - 1] = m_tree[j - 1] * m_tree[i - 1];
        }
    }

public:
    // constructor. O(N).
    BIT() : m_sz(0) {};
    explicit BIT(int n) : m_sz(n), m_tree(n, group_type::one()) {
        assert(n >= 0);
    }
    explicit BIT(int n, const value_type &a) : BIT(n, group_type(a)) {}
    explicit BIT(int n, const group_type &a) : m_sz(n), m_tree(n, a) {
        assert(n >= 0);
        build();
    }
    template <std::input_iterator InputIter>
    explicit BIT(InputIter first, InputIter last) : m_tree(first, last) {
        m_sz = m_tree.size();
        build();
    }
    template <typename T>
    explicit BIT(std::initializer_list<T> il) : BIT(il.begin(), il.end()) {}
    explicit BIT(const std::vector<group_type> &v) : m_sz(v.size()), m_tree(v) {
        build();
    }
    explicit BIT(std::vector<group_type> &&v) : m_tree(std::move(v)) {
        m_sz = m_tree.size();
        build();
    }

    // 要素数を取得する．
    int size() const { return m_sz; }
    // k番目の要素をaとの積の結果に置き換える．O(log N).
    void add(int k, const value_type &a) { add(k, group_type(a)); }
    void add(int k, const group_type &a) {
        assert(0 <= k and k < size());
        for(int i = k + 1; i <= m_sz; i += lsb(i)) m_tree[i - 1] = m_tree[i - 1] * a;
    }
    // 区間[0,r)の要素の総積を求める．O(log N).
    value_type sum(int r) const {
        assert(0 <= r and r <= size());
        return sum_internal(r).value();
    }
    // 区間[l,r)の要素の総積を求める．O(log N).
    value_type sum(int l, int r) const {
        assert(0 <= l and l <= r and r <= size());
        return (sum_internal(r) * sum_internal(l).inv()).value();
    }
    // 全要素の総積を求める．O(log N).
    value_type sum_all() const { return sum_internal(m_sz).value(); }
    // pred(sum(r))==true となる区間の最右位値rを二分探索する．
    // ただし，区間[0,n)の要素はpred(S)によって区分化されていること．また，pred(e)==true であること．O(log N).
    template <bool (*pred)(value_type)>
    int most_right() const {
        return most_right([](const value_type &x) -> bool { return pred(x); });
    }
    template <typename Pred>
    int most_right(Pred pred) const {
        static_assert(std::is_invocable_r<bool, Pred, value_type>::value);
        assert(pred(group_type::one().value()));
        int r = 0;
        group_type &&val = group_type::one();
        for(int i = 1; i <= m_sz and pred(m_tree[i - 1].value()); i <<= 1) r = i, val = m_tree[i - 1];
        for(int len = r >> 1; len > 0; len >>= 1) {
            if(r + len <= m_sz and pred((val * m_tree[r + len - 1]).value())) {
                r += len;
                val = val * m_tree[r - 1];
            }
        }
        return r;
    }
    void reset() { std::fill(m_tree.begin(), m_tree.end(), group_type::one()); }
};

template <typename S>
using range_sum_binary_indexed_tree = BIT<algebra::group::addition<S>>;

template <typename S>
using range_xor_binary_indexed_tree = BIT<algebra::group::bit_xor<S>>;

}  // namespace binary_indexed_tree

}  // namespace algorithm

#endif
#line 1 "algorithm/DataStructure/SegmentTree/binary_indexed_tree.hpp"



#include <algorithm>
#include <cassert>
#include <initializer_list>
#include <iterator>
#include <type_traits>
#include <utility>
#include <vector>

#line 1 "algorithm/Math/Algebra/algebra.hpp"



#line 5 "algorithm/Math/Algebra/algebra.hpp"
#include <iostream>
#include <limits>
#include <numeric>
#line 10 "algorithm/Math/Algebra/algebra.hpp"

namespace algorithm {

namespace algebra {

template <typename S>
class Set {
public:
    using value_type = S;

protected:
    value_type val;

public:
    constexpr Set() : val() {}
    constexpr Set(const value_type &val) : val(val) {}
    constexpr Set(value_type &&val) : val(std::move(val)) {}

    friend constexpr bool operator==(const Set &lhs, const Set &rhs) { return lhs.val == rhs.val; }
    friend std::istream &operator>>(std::istream &is, Set &rhs) { return is >> rhs.val; }
    friend std::ostream &operator<<(std::ostream &os, const Set &rhs) { return os << rhs.val; }

    constexpr value_type value() const { return val; }
};

template <typename S, auto op>
class Semigroup : public Set<S> {
    static_assert(std::is_invocable_r<S, decltype(op), S, S>::value);

    using base_type = Set<S>;

public:
    using value_type = typename base_type::value_type;

    constexpr Semigroup() : base_type() {}
    constexpr Semigroup(const value_type &val) : base_type(val) {}
    constexpr Semigroup(value_type &&val) : base_type(std::move(val)) {}

    friend constexpr Semigroup operator*(const Semigroup &lhs, const Semigroup &rhs) { return Semigroup(op(lhs.val, rhs.val)); }

    static constexpr auto get_op() { return op; }
};

template <typename S, auto op, auto e>
class Monoid : public Semigroup<S, op> {
    static_assert(std::is_invocable_r<S, decltype(e)>::value);

    using base_type = Semigroup<S, op>;

public:
    using value_type = typename base_type::value_type;

    constexpr Monoid() : base_type() {}
    constexpr Monoid(const value_type &val) : base_type(val) {}
    constexpr Monoid(value_type &&val) : base_type(std::move(val)) {}

    friend constexpr Monoid operator*(const Monoid &lhs, const Monoid &rhs) { return Monoid(op(lhs.val, rhs.val)); }

    static constexpr auto get_e() { return e; }
    static constexpr Monoid one() { return Monoid(e()); }  // return identity element.
};

template <typename S, auto op, auto e, auto inverse>
class Group : public Monoid<S, op, e> {
    static_assert(std::is_invocable_r<S, decltype(inverse), S>::value);

    using base_type = Monoid<S, op, e>;

public:
    using value_type = typename base_type::value_type;

    constexpr Group() : base_type() {}
    constexpr Group(const value_type &val) : base_type(val) {}
    constexpr Group(value_type &&val) : base_type(std::move(val)) {}

    friend constexpr Group operator*(const Group &lhs, const Group &rhs) { return Group(op(lhs.val, rhs.val)); }

    static constexpr auto get_inverse() { return inverse; }
    static constexpr Group one() { return Group(e()); }                // return identity element.
    constexpr Group inv() const { return Group(inverse(this->val)); }  // return inverse element.
};

template <typename F, auto compose, auto id, typename X, auto mapping>
class OperatorMonoid : public Monoid<F, compose, id> {
    static_assert(std::is_invocable_r<X, decltype(mapping), F, X>::value);

    using base_type = Monoid<F, compose, id>;

public:
    using value_type = typename base_type::value_type;
    using acted_value_type = X;

    constexpr OperatorMonoid() : base_type() {}
    constexpr OperatorMonoid(const value_type &val) : base_type(val) {}
    constexpr OperatorMonoid(value_type &&val) : base_type(std::move(val)) {}

    friend constexpr OperatorMonoid operator*(const OperatorMonoid &lhs, const OperatorMonoid &rhs) { return OperatorMonoid(compose(lhs.val, rhs.val)); }

    static constexpr auto get_mapping() { return mapping; }
    static constexpr OperatorMonoid one() { return OperatorMonoid(id()); }  // return identity mapping.
    constexpr acted_value_type act(const acted_value_type &x) const { return mapping(this->val, x); }
    template <class S>
    constexpr S act(const S &x) const {
        static_assert(std::is_base_of<Set<acted_value_type>, S>::value);
        return S(mapping(this->val, x.value()));
    }
};

namespace element {

template <typename S>
constexpr auto zero = []() -> S { return S(); };

template <typename S>
constexpr auto one = []() -> S { return 1; };

template <typename S>
constexpr auto min = []() -> S { return std::numeric_limits<S>::min(); };

template <typename S>
constexpr auto max = []() -> S { return std::numeric_limits<S>::max(); };

template <typename S>
constexpr auto one_below_max = []() -> S { return std::numeric_limits<S>::max() - 1; };

template <typename S>
constexpr auto lowest = []() -> S { return std::numeric_limits<S>::lowest(); };

template <typename S>
constexpr auto one_above_lowest = []() -> S { return std::numeric_limits<S>::lowest() + 1; };

}  // namespace element

namespace uoperator {

template <typename S>
constexpr auto identity = [](const S &val) -> S { return val; };

template <typename S>
constexpr auto negate = [](const S &val) -> S { return -val; };

}  // namespace uoperator

namespace boperator {

template <typename T, typename S = T>
constexpr auto plus = [](const T &lhs, const S &rhs) -> S { return lhs + rhs; };

template <typename T, typename S = T>
constexpr auto mul = [](const T &lhs, const S &rhs) -> S { return lhs * rhs; };

template <typename T, typename S = T>
constexpr auto bit_and = [](const T &lhs, const S &rhs) -> S { return lhs & rhs; };

template <typename T, typename S = T>
constexpr auto bit_or = [](const T &lhs, const S &rhs) -> S { return lhs | rhs; };

template <typename T, typename S = T>
constexpr auto bit_xor = [](const T &lhs, const S &rhs) -> S { return lhs ^ rhs; };

template <typename T, typename S = T>
constexpr auto min = [](const T &lhs, const S &rhs) -> S { return std::min<S>(lhs, rhs); };

template <typename T, typename S = T>
constexpr auto max = [](const T &lhs, const S &rhs) -> S { return std::max<S>(lhs, rhs); };

template <typename T, typename S = T>
constexpr auto gcd = [](const T &lhs, const S &rhs) -> S { return std::gcd(lhs, rhs); };

template <typename T, typename S = T>
constexpr auto lcm = [](const T &lhs, const S &rhs) -> S { return std::lcm(lhs, rhs); };

template <typename F, auto id, typename X = F>
constexpr auto assign_if_not_id = [](const F &lhs, const X &rhs) -> X {
    static_assert(std::is_invocable_r<F, decltype(id)>::value);
    return (lhs == id() ? rhs : lhs);
};

}  // namespace boperator

namespace monoid {

template <typename S>
using minimum = Monoid<S, boperator::min<S>, element::max<S>>;

template <typename S>
using minimum_safe = Monoid<S, boperator::min<S>, element::one_below_max<S>>;

template <typename S>
using maximum = Monoid<S, boperator::max<S>, element::lowest<S>>;

template <typename S>
using maximum_safe = Monoid<S, boperator::max<S>, element::one_above_lowest<S>>;

template <typename S>
using addition = Monoid<S, boperator::plus<S>, element::zero<S>>;

template <typename S>
using multiplication = Monoid<S, boperator::mul<S>, element::one<S>>;

template <typename S>
using bit_xor = Monoid<S, boperator::bit_xor<S>, element::zero<S>>;

}  // namespace monoid

namespace group {

template <typename S>
using addition = Group<S, boperator::plus<S>, element::zero<S>, uoperator::negate<S>>;

template <typename S>
using bit_xor = Group<S, boperator::bit_xor<S>, element::zero<S>, uoperator::identity<S>>;

}  // namespace group

namespace operator_monoid {

template <typename F, typename X = F>
using assign_for_minimum = OperatorMonoid<
    F, boperator::assign_if_not_id<F, element::max<F>>, element::max<F>,
    X, boperator::assign_if_not_id<F, element::max<F>, X>>;

template <typename F, typename X = F>
using assign_for_maximum = OperatorMonoid<
    F, boperator::assign_if_not_id<F, element::lowest<F>>, element::lowest<F>,
    X, boperator::assign_if_not_id<F, element::lowest<F>, X>>;

template <typename F, typename X = F>
using addition = OperatorMonoid<F, boperator::plus<F>, element::zero<F>, X, boperator::plus<F, X>>;

}  // namespace operator_monoid

}  // namespace algebra

}  // namespace algorithm


#line 13 "algorithm/DataStructure/SegmentTree/binary_indexed_tree.hpp"

namespace algorithm {

namespace binary_indexed_tree {

template <class AbelianGroup>
class BIT {
public:
    using group_type = AbelianGroup;
    using value_type = group_type::value_type;

private:
    int m_sz;  // m_sz:=(要素数).
    std::vector<group_type> m_tree;

    static constexpr int lsb(int bit) { return bit & -bit; }
    group_type sum_internal(int r) const {
        group_type &&res = group_type::one();
        for(; r >= 1; r -= lsb(r)) res = res * m_tree[r - 1];
        return res;
    }
    void build() {
        for(int i = 1; i < m_sz; ++i) {
            int j = i + lsb(i);
            if(j <= m_sz) m_tree[j - 1] = m_tree[j - 1] * m_tree[i - 1];
        }
    }

public:
    // constructor. O(N).
    BIT() : m_sz(0) {};
    explicit BIT(int n) : m_sz(n), m_tree(n, group_type::one()) {
        assert(n >= 0);
    }
    explicit BIT(int n, const value_type &a) : BIT(n, group_type(a)) {}
    explicit BIT(int n, const group_type &a) : m_sz(n), m_tree(n, a) {
        assert(n >= 0);
        build();
    }
    template <std::input_iterator InputIter>
    explicit BIT(InputIter first, InputIter last) : m_tree(first, last) {
        m_sz = m_tree.size();
        build();
    }
    template <typename T>
    explicit BIT(std::initializer_list<T> il) : BIT(il.begin(), il.end()) {}
    explicit BIT(const std::vector<group_type> &v) : m_sz(v.size()), m_tree(v) {
        build();
    }
    explicit BIT(std::vector<group_type> &&v) : m_tree(std::move(v)) {
        m_sz = m_tree.size();
        build();
    }

    // 要素数を取得する．
    int size() const { return m_sz; }
    // k番目の要素をaとの積の結果に置き換える．O(log N).
    void add(int k, const value_type &a) { add(k, group_type(a)); }
    void add(int k, const group_type &a) {
        assert(0 <= k and k < size());
        for(int i = k + 1; i <= m_sz; i += lsb(i)) m_tree[i - 1] = m_tree[i - 1] * a;
    }
    // 区間[0,r)の要素の総積を求める．O(log N).
    value_type sum(int r) const {
        assert(0 <= r and r <= size());
        return sum_internal(r).value();
    }
    // 区間[l,r)の要素の総積を求める．O(log N).
    value_type sum(int l, int r) const {
        assert(0 <= l and l <= r and r <= size());
        return (sum_internal(r) * sum_internal(l).inv()).value();
    }
    // 全要素の総積を求める．O(log N).
    value_type sum_all() const { return sum_internal(m_sz).value(); }
    // pred(sum(r))==true となる区間の最右位値rを二分探索する．
    // ただし，区間[0,n)の要素はpred(S)によって区分化されていること．また，pred(e)==true であること．O(log N).
    template <bool (*pred)(value_type)>
    int most_right() const {
        return most_right([](const value_type &x) -> bool { return pred(x); });
    }
    template <typename Pred>
    int most_right(Pred pred) const {
        static_assert(std::is_invocable_r<bool, Pred, value_type>::value);
        assert(pred(group_type::one().value()));
        int r = 0;
        group_type &&val = group_type::one();
        for(int i = 1; i <= m_sz and pred(m_tree[i - 1].value()); i <<= 1) r = i, val = m_tree[i - 1];
        for(int len = r >> 1; len > 0; len >>= 1) {
            if(r + len <= m_sz and pred((val * m_tree[r + len - 1]).value())) {
                r += len;
                val = val * m_tree[r - 1];
            }
        }
        return r;
    }
    void reset() { std::fill(m_tree.begin(), m_tree.end(), group_type::one()); }
};

template <typename S>
using range_sum_binary_indexed_tree = BIT<algebra::group::addition<S>>;

template <typename S>
using range_xor_binary_indexed_tree = BIT<algebra::group::bit_xor<S>>;

}  // namespace binary_indexed_tree

}  // namespace algorithm
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