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#include "library/transform/walsh_hadamard.hpp"
シグネチャ
template <typename T, auto add = default_operator::add<T>, auto sub = default_operator::sub<T>> void walsh_hadamard(std::vector<T>&) template <typename T, auto add = default_operator::add<T>, auto sub = default_operator::sub<T>, auto div = default_operator::div<T>> void walsh_hadamard_inv(std::vector<T>&) // std::is_integral_v<T> が true となる型 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>> void walsh_hadamard_inv(std::vector<T>&) // std::is_integral_v<T> が false となる型
概要
長さ $N=2^L$ の列 $(A_0=0,A_1,\ldots,A_{N-1})$ に アダマール変換 を施す関数 walsh_hadamard およびその逆変換を施す関数 walsh_hadamard_inv を提供します.各変換は inplace に行われ,引数として渡した列は書き換えられます.
walsh_hadamard
walsh_hadamard_inv
テンプレート引数
T
add
operator+
sub
operator-
mul
operator*
div
operator/
inv
x
T{1}/x
walsh_hadamard_inv に関して,T が int や long long などの型に対しては除算 div が要求され,double や atcoder::modint などの型に対しては乗法逆元 inv および乗算 mul が要求されます (あとで設計を見直す可能性が高いです).
int
long long
double
atcoder::modint
制約
時間計算量
$\Theta(N\log N)$
#ifndef SUISEN_WALSH_HADAMARD_TRANSFORM #define SUISEN_WALSH_HADAMARD_TRANSFORM #include "library/transform/kronecker_power.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 #endif // SUISEN_WALSH_HADAMARD_TRANSFORM
#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