Range Xor Range Xor
(acted_monoid/range_xor_range_xor.hpp)

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• Last update: 2026-07-21 20:17:47+09:00
• Include: #include "acted_monoid/range_xor_range_xor.hpp"

Overview

An acted monoid that supports range bitwise XOR operations and range XOR sum queries.

Important Usage Note

When a value $f$ is XORed to all elements in a segment, the total XOR sum of the segment changes by $f$ only if the segment’s length is odd. Therefore, value_type holds the size of the segment. Use the make(val) helper function to initialize leaf nodes.

Example

#include "ds/segtree/lazy_segtree.hpp"
#include "acted_monoid/range_xor_range_xor.hpp"
#include <iostream>
#include <vector>

using AM = m1une::acted_monoid::RangeXorRangeXor<long long>;

int main() {
    std::vector<long long> A = {1, 2, 3, 4, 5};
    int N = A.size();

    std::vector<AM::value_type> init_nodes(N);
    for (int i = 0; i < N; ++i) {
        init_nodes[i] = AM::make(A[i]);
    }

    m1une::ds::LazySegtree<AM> seg(init_nodes);

    seg.apply(1, 4, 7);

    std::cout << seg.prod(1, 4).val << "\n";

    return 0;
}

Interface and Complexity

This is a stateless acted-monoid tag. Lazy data structures use its public value_type, operator_type, id(), op(a, b), op_id(), op_comp(f, g), and mapping(f, x) members. Helpers such as make(...), shifted mappings, or reversal-aware mappings are described above when the header provides them.

The static operations are $O(1)$ for the scalar metadata stored by these range acted monoids, aside from the cost of the underlying arithmetic type.

Verified with

• verify/monoid/commutative_flags.test.cpp

Code

#ifndef M1UNE_ACTED_MONOID_RANGE_XOR_RANGE_XOR_HPP
#define M1UNE_ACTED_MONOID_RANGE_XOR_RANGE_XOR_HPP 1

namespace m1une {
namespace acted_monoid {

template <typename T>
struct RangeXorRangeXorNode {
    T val;
    long long size;
};

template <typename T>
struct RangeXorRangeXor {
    using value_type = RangeXorRangeXorNode<T>;
    using operator_type = T;
    static constexpr bool commutative = true;
    static constexpr bool operator_commutative = true;

    static constexpr value_type id() {
        return {T(0), 0};
    }
    static constexpr value_type op(const value_type& a, const value_type& b) {
        return {a.val ^ b.val, a.size + b.size};
    }
    static constexpr value_type inv(const value_type& x) {
        return {x.val, -x.size};
    }

    static constexpr operator_type op_id() {
        return T(0);
    }
    static constexpr operator_type op_comp(const operator_type& f, const operator_type& g) {
        return f ^ g;
    }

    static constexpr value_type mapping(const operator_type& f, const value_type& x) {
        if (x.size % 2 != 0) {
            return {x.val ^ f, x.size};
        }
        return x;
    }

    static constexpr value_type make(const T& val) {
        return {val, 1};
    }
};

}  // namespace acted_monoid
}  // namespace m1une

#endif  // M1UNE_ACTED_MONOID_RANGE_XOR_RANGE_XOR_HPP

#line 1 "acted_monoid/range_xor_range_xor.hpp"



namespace m1une {
namespace acted_monoid {

template <typename T>
struct RangeXorRangeXorNode {
    T val;
    long long size;
};

template <typename T>
struct RangeXorRangeXor {
    using value_type = RangeXorRangeXorNode<T>;
    using operator_type = T;
    static constexpr bool commutative = true;
    static constexpr bool operator_commutative = true;

    static constexpr value_type id() {
        return {T(0), 0};
    }
    static constexpr value_type op(const value_type& a, const value_type& b) {
        return {a.val ^ b.val, a.size + b.size};
    }
    static constexpr value_type inv(const value_type& x) {
        return {x.val, -x.size};
    }

    static constexpr operator_type op_id() {
        return T(0);
    }
    static constexpr operator_type op_comp(const operator_type& f, const operator_type& g) {
        return f ^ g;
    }

    static constexpr value_type mapping(const operator_type& f, const value_type& x) {
        if (x.size % 2 != 0) {
            return {x.val ^ f, x.size};
        }
        return x;
    }

    static constexpr value_type make(const T& val) {
        return {val, 1};
    }
};

}  // namespace acted_monoid
}  // namespace m1une

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