Bitwise Xor Convolution
(library/convolution/xor_convolution.hpp)

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• Last update: 2022-03-21 02:24:20+09:00
• Include: #include "library/convolution/xor_convolution.hpp"

xor_convolution

• シグネチャ

  template <
      typename T,
      auto add = default_operator::add<T>,
      auto sub = default_operator::sub<T>,
      auto mul = default_operator::mul<T>,
      auto div = default_operator::div<T>
  >
  std::vector<T> xor_convolution(std::vector<T> a, std::vector<T> b)
  
  template <
      typename T,
      auto add = default_operator::add<T>,
      auto sub = default_operator::sub<T>,
      auto mul = default_operator::mul<T>,
      auto inv = default_operator::inv<T>
  >
  std::vector<T> xor_convolution(std::vector<T> a, std::vector<T> b)
  

• 概要

  長さ $N=2^L$ の列 $(A_0,A_1,\ldots,A_{N-1})$ および $(B_0,B_1,\ldots,B_{N-1})$ を添字 bitwise xor で畳み込みます．即ち，長さ $N$ の列 $(C_0,C_1,\ldots,C_{N-1})$ であって，各 $k=0,\ldots,N-1$ に対して以下を満たす列を計算します．

  \[C _ k = \sum _ { i \oplus j = k } A _ i \cdot B _ j\]

  以下，この畳み込みを $C=A\ast B$ と表記します．

• テンプレート引数

  • T: 列の要素の型．
  • add: 二項演算 (加算)．デフォルトでは operator+ が呼ばれるようになっています．
  • sub: 二項演算 (減算)．デフォルトでは operator- が呼ばれるようになっています．
  • mul: 二項演算 (乗算)．デフォルトでは operator* が呼ばれるようになっています．
  • div: 二項演算 (除算)．デフォルトでは operator/ が呼ばれるようになっています．
  • inv: 単項演算 (乗法逆元)．デフォルトでは x に対して T{1}/x と計算されます．

  内部で用いている Walsh Hadamard 変換が原因で実装が 2 通りに分かれています．詳しくは Walsh Hadamard 変換 のページを参照してください．

• 返り値

  $A\ast B$

• 制約

  • ある非負整数 $L$ が存在して $\vert A \vert=\vert B \vert= 2 ^ L$ を満たす
  • $0\leq L\leq 20$

• 時間計算量

  $\Theta(N\log N)$，あるいは $\Theta(L\cdot 2^L)$

Depends on

• Convolution (library/convolution/convolution.hpp)
• クロネッカー冪による線形変換 (仮称) (library/transform/kronecker_power.hpp)
• Walsh Hadamard 変換 (library/transform/walsh_hadamard.hpp)
• Default Operator (library/util/default_operator.hpp)

Verified with

• test/src/convolution/xor_convolution/xor_convolution.test.cpp

Code

#ifndef SUISEN_XOR_CONVOLUTION
#define SUISEN_XOR_CONVOLUTION

#include "library/transform/walsh_hadamard.hpp"
#include "library/convolution/convolution.hpp"

namespace suisen {
    template <
        typename T,
        auto add = default_operator::add<T>,
        auto sub = default_operator::sub<T>,
        auto mul = default_operator::mul<T>,
        auto div = default_operator::div<T>,
        std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr
    >
    std::vector<T> xor_convolution(std::vector<T> a, std::vector<T> b) {
        return convolution::transform_convolution<
            T,
            walsh_hadamard_transform::walsh_hadamard<T, add, sub>,
            walsh_hadamard_transform::walsh_hadamard_inv<T, add, sub, div>,
            mul
        >(std::move(a), std::move(b));
    }

    template <
        typename T,
        auto add = default_operator::add<T>,
        auto sub = default_operator::sub<T>,
        auto mul = default_operator::mul<T>,
        auto inv = default_operator::inv<T>,
        std::enable_if_t<std::negation_v<std::is_integral<T>>, std::nullptr_t> = nullptr
    >
    std::vector<T> xor_convolution(std::vector<T> a, std::vector<T> b) {
        return convolution::transform_convolution<
            T,
            walsh_hadamard_transform::walsh_hadamard<T, add, sub>,
            walsh_hadamard_transform::walsh_hadamard_inv<T, add, sub, mul, inv>,
            mul
        >(std::move(a), std::move(b));
    }
} // namespace suisen

#endif // SUISEN_XOR_CONVOLUTION

#line 1 "library/convolution/xor_convolution.hpp"



#line 1 "library/transform/walsh_hadamard.hpp"



#line 1 "library/transform/kronecker_power.hpp"



#include <cassert>
#include <utility>
#include <vector>

#line 1 "library/util/default_operator.hpp"



namespace suisen {
    namespace default_operator {
        template <typename T>
        auto zero() -> decltype(T { 0 }) { return T { 0 }; }
        template <typename T>
        auto one()  -> decltype(T { 1 }) { return T { 1 }; }
        template <typename T>
        auto add(const T &x, const T &y) -> decltype(x + y) { return x + y; }
        template <typename T>
        auto sub(const T &x, const T &y) -> decltype(x - y) { return x - y; }
        template <typename T>
        auto mul(const T &x, const T &y) -> decltype(x * y) { return x * y; }
        template <typename T>
        auto div(const T &x, const T &y) -> decltype(x / y) { return x / y; }
        template <typename T>
        auto mod(const T &x, const T &y) -> decltype(x % y) { return x % y; }
        template <typename T>
        auto neg(const T &x) -> decltype(-x) { return -x; }
        template <typename T>
        auto inv(const T &x) -> decltype(one<T>() / x)  { return one<T>() / x; }
    } // default_operator
    namespace default_operator_noref {
        template <typename T>
        auto zero() -> decltype(T { 0 }) { return T { 0 }; }
        template <typename T>
        auto one()  -> decltype(T { 1 }) { return T { 1 }; }
        template <typename T>
        auto add(T x, T y) -> decltype(x + y) { return x + y; }
        template <typename T>
        auto sub(T x, T y) -> decltype(x - y) { return x - y; }
        template <typename T>
        auto mul(T x, T y) -> decltype(x * y) { return x * y; }
        template <typename T>
        auto div(T x, T y) -> decltype(x / y) { return x / y; }
        template <typename T>
        auto mod(T x, T y) -> decltype(x % y) { return x % y; }
        template <typename T>
        auto neg(T x) -> decltype(-x) { return -x; }
        template <typename T>
        auto inv(T x) -> decltype(one<T>() / x)  { return one<T>() / x; }
    } // default_operator
} // namespace suisen


#line 9 "library/transform/kronecker_power.hpp"

namespace suisen {
    namespace kronecker_power_transform {
        namespace internal {
            template <typename UnitTransform, typename ReferenceGetter, std::size_t... Seq>
            void unit_transform(UnitTransform transform, ReferenceGetter ref_getter, std::index_sequence<Seq...>) {
                transform(ref_getter(Seq)...);
            }
        }

        template <typename T, std::size_t D, auto unit_transform>
        void kronecker_power_transform(std::vector<T> &x) {
            const std::size_t n = x.size();
            for (std::size_t block = 1; block < n; block *= D) {
                for (std::size_t l = 0; l < n; l += D * block) {
                    for (std::size_t offset = l; offset < l + block; ++offset) {
                        const auto ref_getter = [&](std::size_t i) -> T& { return x[offset + i * block]; };
                        internal::unit_transform(unit_transform, ref_getter, std::make_index_sequence<D>());
                    }
                }
            }
        }

        template <typename T, typename UnitTransform>
        void kronecker_power_transform(std::vector<T> &x, const std::size_t D, UnitTransform unit_transform) {
            const std::size_t n = x.size();
            std::vector<T> work(D);
            for (std::size_t block = 1; block < n; block *= D) {
                for (std::size_t l = 0; l < n; l += D * block) {
                    for (std::size_t offset = l; offset < l + block; ++offset) {
                        for (std::size_t i = 0; i < D; ++i) work[i] = x[offset + i * block];
                        unit_transform(work);
                        for (std::size_t i = 0; i < D; ++i) x[offset + i * block] = work[i];
                    }
                }
            }
        }

        template <typename T, auto e = default_operator::zero<T>, auto add = default_operator::add<T>, auto mul = default_operator::mul<T>>
        auto kronecker_power_transform(std::vector<T> &x, const std::vector<std::vector<T>> &A) -> decltype(e(), add(std::declval<T>(), std::declval<T>()), mul(std::declval<T>(), std::declval<T>()), void()) {
            const std::size_t D = A.size();
            assert(D == A[0].size());
            auto unit_transform = [&](std::vector<T> &x) {
                std::vector<T> y(D, e());
                for (std::size_t i = 0; i < D; ++i) for (std::size_t j = 0; j < D; ++j) {
                    y[i] = add(y[i], mul(A[i][j], x[j]));
                }
                x.swap(y);
            };
            kronecker_power_transform<T>(x, D, unit_transform);
        }
    }
} // namespace suisen



#line 5 "library/transform/walsh_hadamard.hpp"

namespace suisen::walsh_hadamard_transform {
    namespace internal {
        template <typename T, auto add = default_operator::add<T>, auto sub = default_operator::sub<T>>
        void unit_transform(T& x0, T& x1) {
            T y0 = x0, y1 = x1;
            x0 = add(y0, y1);   // 1,  1
            x1 = sub(y0, y1);   // 1, -1
        }
    } // namespace internal

    using kronecker_power_transform::kronecker_power_transform;

    template <typename T, auto add = default_operator::add<T>, auto sub = default_operator::sub<T>>
    void walsh_hadamard(std::vector<T>& a) {
        kronecker_power_transform<T, 2, internal::unit_transform<T, add, sub>>(a);
    }
    template <typename T, auto add = default_operator::add<T>, auto sub = default_operator::sub<T>, auto div = default_operator::div<T>, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
    void walsh_hadamard_inv(std::vector<T>& a) {
        walsh_hadamard<T, add, sub>(a);
        const T n{ a.size() };
        for (auto& val : a) val = div(val, n);
    }
    template <typename T, auto add = default_operator::add<T>, auto sub = default_operator::sub<T>, auto mul = default_operator::mul<T>, auto inv = default_operator::inv<T>, std::enable_if_t<std::negation_v<std::is_integral<T>>, std::nullptr_t> = nullptr>
    void walsh_hadamard_inv(std::vector<T>& a) {
        walsh_hadamard<T, add, sub>(a);
        const T n{ a.size() };
        const T inv_n = inv(n);
        for (auto& val : a) val = mul(val, inv_n);
    }
} // namespace suisen::walsh_hadamard_transform



#line 1 "library/convolution/convolution.hpp"



#line 6 "library/convolution/convolution.hpp"

#line 8 "library/convolution/convolution.hpp"

namespace suisen {
    namespace convolution {
        template <typename T, auto transform, auto inv_transform, auto mul = default_operator::mul<T>>
        std::vector<T> transform_convolution(std::vector<T> a, std::vector<T> b) {
            const std::size_t n = a.size(), m = b.size();
            assert(n == m);
            transform(a), transform(b);
            for (std::size_t i = 0; i < n; ++i) a[i] = mul(a[i], b[i]);
            inv_transform(a);
            return a;
        }
        template <typename T, auto transform, auto inv_transform, auto mul = default_operator::mul<T>>
        std::vector<T> transform_convolution(std::vector<std::vector<T>> a) {
            const std::size_t num = a.size();
            assert(num);
            const std::size_t n = a[0].size();
            for (auto &v : a) {
                assert(n == int(v.size()));
                transform(v);
            }
            auto &res = a[0];
            for (int i = 1; i < num; ++i) {
                for (int j = 0; j < n; ++j) res[j] = mul(res[j], a[i][j]);
            }
            inv_transform(res);
            return res;
        }
    }
} // namespace suisen



#line 6 "library/convolution/xor_convolution.hpp"

namespace suisen {
    template <
        typename T,
        auto add = default_operator::add<T>,
        auto sub = default_operator::sub<T>,
        auto mul = default_operator::mul<T>,
        auto div = default_operator::div<T>,
        std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr
    >
    std::vector<T> xor_convolution(std::vector<T> a, std::vector<T> b) {
        return convolution::transform_convolution<
            T,
            walsh_hadamard_transform::walsh_hadamard<T, add, sub>,
            walsh_hadamard_transform::walsh_hadamard_inv<T, add, sub, div>,
            mul
        >(std::move(a), std::move(b));
    }

    template <
        typename T,
        auto add = default_operator::add<T>,
        auto sub = default_operator::sub<T>,
        auto mul = default_operator::mul<T>,
        auto inv = default_operator::inv<T>,
        std::enable_if_t<std::negation_v<std::is_integral<T>>, std::nullptr_t> = nullptr
    >
    std::vector<T> xor_convolution(std::vector<T> a, std::vector<T> b) {
        return convolution::transform_convolution<
            T,
            walsh_hadamard_transform::walsh_hadamard<T, add, sub>,
            walsh_hadamard_transform::walsh_hadamard_inv<T, add, sub, mul, inv>,
            mul
        >(std::move(a), std::move(b));
    }
} // namespace suisen

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