test/generalized_floor_sum.test.cpp

test/generalized_floor_sum.test.cpp · GitHub で見る

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#define PROBLEM "https://yukicoder.me/problems/no/2362"
#include "../math/generalized_floor_sum.hpp"
#include "../components/fast_io.hpp"

int main() {
	FastInput in;
	FastOutput out;
	int t;
	in.read(t);
	while (t--) {
		std::uint64_t n, m, a, b;
		in.read(n);
		in.read(m);
		in.read(a);
		in.read(b);
		auto x = generalized_floor_sum_moments<2>(n, m, a, b);
		auto y = generalized_floor_sum_moments<2>(n, m, a, 0);
		out.write(static_cast<long long>(2 * x[1][1] - (n - 1) * x[0][1] + y[1][1] - n * y[0][1]));
		out.write('\n');
	}
}

#line 1 "test/generalized_floor_sum.test.cpp"
#define PROBLEM "https://yukicoder.me/problems/no/2362"
#line 1 "math/generalized_floor_sum.hpp"


#include <array>
#include <cassert>
#include <cstdint>
#include <type_traits>
#include <utility>

namespace generalized_floor_sum_detail {
using U = std::uint64_t;
template <class T>
constexpr bool integer = std::is_integral_v<T> || std::is_same_v<T, __int128_t> || std::is_same_v<T, __uint128_t>;
template <int N, class T>
T divide(T value) {
	if constexpr (integer<T>) return value / T(N);
	else {
		static const T inverse = T(1) / T(N);
		return value * inverse;
	}
}
template <int D, class T>
using Table = std::array<std::array<T, D + 1>, D + 1>;

template <int D, class T>
struct Coefficients {
	std::array<std::array<T, D + 2>, D + 1> sums{};
	std::array<std::array<T, D + 2>, D + 2> choose{};
	std::array<T, D + 1> denominator{}, inverse{};
	Coefficients() {
		for (int i = 0; i <= D + 1; ++i) {
			choose[i][0] = choose[i][i] = T(1);
			for (int j = 1; j < i; ++j) choose[i][j] = choose[i - 1][j - 1] + choose[i - 1][j];
		}
		T factorial = 1;
		for (int p = 0; p <= D; ++p) {
			denominator[p] = factorial * T(p + 1);
			for (int k = 0; k <= p + 1; ++k) {
				T value = factorial * choose[p + 1][k];
				for (int j = 0; j < p; ++j)
					if (k <= j + 1) value -= factorial / denominator[j] * choose[p + 1][j] * sums[j][k];
				sums[p][k] = value;
			}
			factorial = denominator[p];
			if constexpr (!integer<T>) inverse[p] = T(1) / denominator[p];
		}
	}
	T divide(T value, int p) const {
		if constexpr (integer<T>) return value / denominator[p];
		else return value * inverse[p];
	}
};

template <int D, class T>
const Coefficients<D, T> &coefficients() {
	static const Coefficients<D, T> value;
	return value;
}

template <int D, class T, class A, class B>
[[gnu::always_inline]] inline void translate(Table<D, T> &result, const Coefficients<D, T> &c, A a, B b,
	std::array<T, D + 1> &ak, std::array<T, D + 1> &bk) {
	if (a != A(0)) {
		ak[0] = T(1);
		for (int i = 1; i <= D; ++i) ak[i] = ak[i - 1] * T(a);
		for (int q = D; q >= 1; --q)
			for (int p = 0; p + q <= D; ++p)
				for (int j = 0; j < q; ++j) result[p][q] += c.choose[q][j] * ak[q - j] * result[p + q - j][j];
	}
	if (b != B(0)) {
		bk[0] = T(1);
		for (int i = 1; i <= D; ++i) bk[i] = bk[i - 1] * T(b);
		for (int q = D; q >= 1; --q)
			for (int p = 0; p + q <= D; ++p)
				for (int j = 0; j < q; ++j) result[p][q] += c.choose[q][j] * bk[q - j] * result[p][j];
	}
}

template <int D, class T>
Table<D, T> euclidean(U n, U m, U a, U b) {
	Table<D, T> result{};
	if (!n) return result;
	if (n == 1) {
		result[0][0] = T(1);
		T value = T(b / m);
		for (int q = 1; q <= D; ++q) result[0][q] = result[0][q - 1] * value;
		return result;
	}
	const auto &c = coefficients<D, T>();
	std::array<T, D + 1> sums{}, ak{}, bk{}, kp{};
	T x = T(n - 1);
	for (int p = 0; p <= D; ++p) {
		T value = c.sums[p][p + 1];
		for (int j = p; j >= 0; --j) value = value * x + c.sums[p][j];
		sums[p] = c.divide(value, p);
	}
	U u = a / m, v = b / m;
	a %= m;
	b %= m;
	if (a) {
		__uint128_t z = __uint128_t(a) * (n - 1) + b;
		U k = z >> 64 ? U(z / m) : U(z) / m;
		if (k) {
			auto child = euclidean<D, T>(k, a, m, m - b - 1);
			kp[0] = T(1);
			for (int j = 1; j <= D; ++j) kp[j] = kp[j - 1] * T(k);
			for (int p = 0; p < D; ++p) {
				std::array<T, D + 1> inner{};
				for (int i = 0; i < D - p; ++i) {
					T value = 0;
					for (int j = 0; j <= p + 1; ++j) value += c.sums[p][j] * child[i][j];
					inner[i] = c.divide(value, p);
				}
				for (int q = 1; p + q <= D; ++q) {
					T value = sums[p] * kp[q];
					for (int i = 0; i < q; ++i) value -= c.choose[q][i] * inner[i];
					result[p][q] = value;
				}
			}
		}
	}
	for (int p = 0; p <= D; ++p) result[p][0] = sums[p];
	translate<D>(result, c, u, v, ak, bk);
	return result;
}

template <class A, class B>
using Signed = std::enable_if_t<std::is_integral_v<A> && std::is_integral_v<B> &&
	(std::is_signed_v<A> || std::is_signed_v<B>), int>;

template <class A>
bool negative(A value) {
	if constexpr (std::is_signed_v<A>) return value < 0;
	else return false;
}

template <class T, class A>
std::pair<U, T> normalize(A value, U m) {
	U x = static_cast<U>(value);
	if (!negative(value)) return {x % m, T(x / m)};
	x = U(0) - x;
	U r = x % m, q = x / m + (r != 0);
	if constexpr (integer<T>) return {r ? m - r : 0, T(-__int128_t(q))};
	else return {r ? m - r : 0, -T(q)};
}

template <class T, bool Full = false>
auto quadratic(U n, U m, U a, U b) {
	std::conditional_t<Full, Table<2, T>, std::array<T, 3>> result{};
	if (!n) return result;
	T count = T(n), s1 = T(n >> 1) * T(n - 1 + (n & 1)), s2 = divide<3>(s1 * (T(2) * count - T(1)));
	U u = a / m, v = b / m;
	a %= m;
	b %= m;
	T f = 0, g = 0, h = 0;
	if (a) {
		__uint128_t z = __uint128_t(a) * (n - 1) + b;
		U k = z >> 64 ? U(z / m) : U(z) / m;
		if (k) {
			auto child = quadratic<T>(k, a, m, m - b - 1);
			T height = T(k);
			f = T(n - 1) * height - child[0];
			g = s1 * height - divide<2>(child[2] + child[0]);
			h = T(n - 1) * height * height - child[0] - T(2) * child[1];
		}
	}
	T at = T(u), bt = T(v);
	T f0 = f + at * s1 + bt * count;
	T f1 = g + at * s2 + bt * s1;
	T f2 = h + T(2) * (at * g + bt * f) + at * at * s2 + bt * (T(2) * at * s1 + bt * count);
	if constexpr (Full) {
		result[0][0] = count;
		result[1][0] = s1;
		result[2][0] = s2;
		result[0][1] = f0;
		result[1][1] = f1;
		result[0][2] = f2;
	} else result = {f0, f1, f2};
	return result;
}

template <class T>
T linear(U n, U m, U a, U b) {
	T result = 0;
	for (;;) {
		if (!n) return result;
		if (n == 1) return result + T(b / m);
		if (a >= m) result += divide<2>(T(n) * T(n - 1)) * T(a / m), a %= m;
		if (b >= m) result += T(n) * T(b / m), b %= m;
		__uint128_t z = __uint128_t(a) * n + b;
		if (z < m) return result;
		if (z >> 64) n = z / m, b = z % m;
		else {
			U value = z;
			n = value / m;
			b = value % m;
		}
		std::swap(a, m);
	}
}
}

template <int D, class T = __int128_t>
auto generalized_floor_sum_moments(std::uint64_t n, std::uint64_t m, std::uint64_t a, std::uint64_t b) {
	static_assert(D >= 0);
	assert(m > 0);
	if constexpr (D == 0) return generalized_floor_sum_detail::Table<0, T>{std::array<T, 1>{T(n)}};
	else if constexpr (D == 1) {
		generalized_floor_sum_detail::Table<1, T> result{};
		result[0][0] = T(n);
		result[1][0] = n ? generalized_floor_sum_detail::divide<2>(T(n) * T(n - 1)) : T(0);
		result[0][1] = generalized_floor_sum_detail::linear<T>(n, m, a, b);
		return result;
	} else if constexpr (D == 2) return generalized_floor_sum_detail::quadratic<T, true>(n, m, a, b);
	else return generalized_floor_sum_detail::euclidean<D, T>(n, m, a, b);
}
template <int P, int Q, class T = __int128_t>
T generalized_floor_sum(std::uint64_t n, std::uint64_t m, std::uint64_t a, std::uint64_t b) {
	static_assert(P >= 0 && Q >= 0);
	assert(m > 0);
	if constexpr (Q == 0) {
		if (!n) return T(0);
		if constexpr (P == 0) return T(n);
		else if constexpr (P == 1) return generalized_floor_sum_detail::divide<2>(T(n) * T(n - 1));
		else if constexpr (P == 2) return generalized_floor_sum_detail::divide<6>(T(n) * T(n - 1) * (T(2) * T(n) - T(1)));
		else {
			const auto &c = generalized_floor_sum_detail::coefficients<P, T>();
			T value = c.sums[P][P + 1], x = T(n - 1);
			for (int j = P; j >= 0; --j) value = value * x + c.sums[P][j];
			return c.divide(value, P);
		}
	} else if constexpr (P == 0 && Q == 1) return generalized_floor_sum_detail::linear<T>(n, m, a, b);
	else return generalized_floor_sum_moments<P + Q, T>(n, m, a, b)[P][Q];
}

template <int D, class T = __int128_t, class A, class B, generalized_floor_sum_detail::Signed<A, B> = 0>
auto generalized_floor_sum_moments(std::uint64_t n, std::uint64_t m, A a, B b) {
	using namespace generalized_floor_sum_detail;
	static_assert(D >= 0);
	assert(m > 0);
	if (!negative(a) && !negative(b)) return generalized_floor_sum_moments<D, T>(n, m, U(a), U(b));
	if constexpr (D == 0) return generalized_floor_sum_moments<0, T>(n, m, U(0), U(0));
	else {
		if (!n) return Table<D, T>{};
		auto [rb, v] = normalize<T>(b, m);
		if (n == 1) {
			Table<D, T> result{};
			result[0][0] = T(1);
			for (int q = 1; q <= D; ++q) result[0][q] = result[0][q - 1] * v;
			return result;
		}
		auto [ra, u] = normalize<T>(a, m);
		auto result = generalized_floor_sum_moments<D, T>(n, m, ra, rb);
		std::array<T, D + 1> ak{}, bk{};
		translate<D>(result, coefficients<D, T>(), u, v, ak, bk);
		return result;
	}
}

template <int P, int Q, class T = __int128_t, class A, class B, generalized_floor_sum_detail::Signed<A, B> = 0>
T generalized_floor_sum(std::uint64_t n, std::uint64_t m, A a, B b) {
	using namespace generalized_floor_sum_detail;
	static_assert(P >= 0 && Q >= 0);
	assert(m > 0);
	if constexpr (Q == 0) return generalized_floor_sum<P, Q, T>(n, m, U(0), U(0));
	else {
		if (!negative(a) && !negative(b)) return generalized_floor_sum<P, Q, T>(n, m, U(a), U(b));
		return generalized_floor_sum_moments<P + Q, T>(n, m, a, b)[P][Q];
	}
}

#line 1 "components/fast_io.hpp"



#include <algorithm>
#line 6 "components/fast_io.hpp"
#include <bit>
#line 8 "components/fast_io.hpp"
#include <charconv>
#include <cstddef>
#line 11 "components/fast_io.hpp"
#include <cstdio>
#include <cstring>
#include <memory>
#include <stdexcept>
#include <string>
#include <string_view>
#line 18 "components/fast_io.hpp"
#if defined(__unix__) || defined(__APPLE__)
#include <sys/mman.h>
#include <sys/stat.h>
#include <unistd.h>
#endif

namespace fast_io_detail { struct BulkInput; struct BulkOutput; }

class FastInput {
	friend struct fast_io_detail::BulkInput;
	static constexpr std::size_t capacity = 1 << 16;
	FILE *file = nullptr;
	std::unique_ptr<char[]> buffer;
	const char *current = "", *end = current;
	std::string scratch;
	void *mapping = nullptr;
	std::size_t mapped_size = 0;
	bool refill() {
		if (!file) return false;
		std::size_t n = std::fread(buffer.get(), 1, capacity, file);
		current = buffer.get();
		end = current + n;
		return n;
	}
	static const char *scan(const char *p, const char *end) {
		while (end - p >= 8) {
			std::uint64_t x;
			std::memcpy(&x, p, 8);
			if ((x - 0x2121212121212121ULL) & ~x & 0x8080808080808080ULL) break;
			p += 8;
		}
		while (p != end && static_cast<unsigned char>(*p) > ' ') ++p;
		return p;
	}

public:
	explicit FastInput(FILE *input = stdin) {
#if defined(__unix__) || defined(__APPLE__)
		struct stat status;
		int fd = fileno(input);
		auto offset = ftello(input);
		if (fd >= 0 && offset >= 0 && !fstat(fd, &status) && S_ISREG(status.st_mode) && status.st_size > offset) {
			mapped_size = status.st_size;
			mapping = mmap(nullptr, mapped_size, PROT_READ, MAP_PRIVATE, fd, 0);
			if (mapping != MAP_FAILED) {
				current = static_cast<const char *>(mapping) + offset;
				end = static_cast<const char *>(mapping) + mapped_size;
				return;
			}
			mapping = nullptr;
		}
#endif
		file = input;
		buffer = std::make_unique_for_overwrite<char[]>(capacity);
	}
	explicit FastInput(std::string_view input) : current(input.empty() ? "" : input.data()), end(current + input.size()) {}
	FastInput(const FastInput &) = delete;
	FastInput &operator=(const FastInput &) = delete;
	~FastInput() {
#if defined(__unix__) || defined(__APPLE__)
		if (mapping) munmap(mapping, mapped_size);
#endif
	}
	std::string_view token() {
		while (true) {
			while (current != end && static_cast<unsigned char>(*current) <= ' ') ++current;
			if (current != end) break;
			if (!refill()) return {};
		}
		const char *begin = current;
		current = scan(current, end);
		if (current != end || !file) return {begin, std::size_t(current - begin)};
		scratch.assign(begin, end);
		while (refill()) {
			current = scan(current, end);
			scratch.append(buffer.get(), std::size_t(current - buffer.get()));
			if (current != end) break;
		}
		return scratch;
	}
	template <class T>
	bool read(T &value) {
		if constexpr (std::is_integral_v<T> && !std::is_same_v<T, bool>) {
			while (true) {
				while (current != end && static_cast<unsigned char>(*current) <= ' ') ++current;
				if (current != end) break;
				if (!refill()) return false;
			}
			if constexpr (std::is_unsigned_v<T> && sizeof(T) == 4 && std::endian::native == std::endian::little) {
				if (end - current >= 11) {
					std::uint64_t x;
					std::memcpy(&x, current, 8);
					if (!(((x + 0x4646464646464646ULL) | (x - 0x3030303030303030ULL)) & 0x8080808080808080ULL)) {
						x &= 0x0f0f0f0f0f0f0f0fULL;
						x = (x * 10 + (x >> 8)) & 0x00ff00ff00ff00ffULL;
						x = (x * 100 + (x >> 16)) & 0x0000ffff0000ffffULL;
						std::uint64_t v = std::uint32_t(x * 10000 + (x >> 32));
						int n = 8;
						for (; n < 10 && static_cast<unsigned char>(current[n]) - unsigned('0') < 10; ++n)
							v = v * 10 + (current[n] - '0');
						if (static_cast<unsigned char>(current[n]) <= ' ' && v <= 0xffffffffULL) {
							current += n;
							value = T(v);
							return true;
						}
					}
				}
			}
			T parsed{};
			auto result = std::from_chars(current, end, parsed);
			if (result.ec == std::errc{} && (result.ptr != end ? static_cast<unsigned char>(*result.ptr) <= ' ' : !file)) {
				current = result.ptr;
				value = parsed;
				return true;
			}
		}
		auto s = token();
		if (s.empty()) return false;
		if constexpr (std::is_same_v<T, bool>) {
			assert(s == "0" || s == "1");
			value = s[0] == '1';
		} else if constexpr (std::is_integral_v<T>) {
			auto result = std::from_chars(s.data(), s.data() + s.size(), value);
			assert(result.ec == std::errc{} && result.ptr == s.data() + s.size());
			(void)result;
		} else value = T(s);
		return true;
	}
};

class FastOutput {
	friend struct fast_io_detail::BulkOutput;
	inline static constexpr auto quads = [] {
		std::array<std::array<char, 4>, 10000> table{};
		for (int i = 0; i < 10000; ++i)
			table[i] = {char('0' + i / 1000), char('0' + i / 100 % 10), char('0' + i / 10 % 10), char('0' + i % 10)};
		return table;
	}();
	FILE *file;
	std::unique_ptr<char[]> buffer = std::make_unique_for_overwrite<char[]>(1 << 16);
	std::size_t capacity = 1 << 16, length = 0;
	char *reserve(std::size_t n) {
		if (n > capacity - length) {
			if (!flush()) throw std::runtime_error("output failed");
			if (n > capacity) {
				std::size_t next = std::max(n, 2 * capacity);
				buffer = std::make_unique_for_overwrite<char[]>(next);
				capacity = next;
			}
		}
		return buffer.get() + length;
	}

public:
	explicit FastOutput(FILE *output = stdout) : file(output) {}
	FastOutput(const FastOutput &) = delete;
	FastOutput &operator=(const FastOutput &) = delete;
	~FastOutput() { flush(); }
	bool flush() {
		std::size_t offset = 0;
		while (offset < length) {
			auto n = std::fwrite(buffer.get() + offset, 1, length - offset, file);
			if (!n) {
				std::memmove(buffer.get(), buffer.get() + offset, length - offset);
				length -= offset;
				return false;
			}
			offset += n;
		}
		length = 0;
		return std::fflush(file) == 0;
	}
	void write(char c) { *reserve(1) = c; ++length; }
	void write(__uint128_t value) {
		char digits[40];
		char *end = digits + sizeof(digits), *begin = end;
		do { *--begin = char('0' + value % 10); value /= 10; } while (value);
		write(std::string_view(begin, end - begin));
	}
	void write(__int128_t value) {
		__uint128_t magnitude = __uint128_t(value);
		if (value < 0) write('-'), magnitude = 0 - magnitude;
		write(magnitude);
	}
	void write(std::string_view s) {
		char *out = reserve(s.size());
		if (!s.empty()) std::memcpy(out, s.data(), s.size());
		length += s.size();
	}
	template <class T> requires std::is_integral_v<T>
	void write(T value) {
		char *out = reserve(32);
		if constexpr (std::is_same_v<T, bool>) *out = '0' + value, ++length;
		else if constexpr (std::is_unsigned_v<T> && sizeof(T) == 4) {
			char *p = out;
			if (value >= 100000000) {
				auto high = value / 100000000, low = value % 100000000;
				p = std::to_chars(p, p + 2, high).ptr;
				std::memcpy(p, quads[low / 10000].data(), 4);
				std::memcpy(p + 4, quads[low % 10000].data(), 4);
				p += 8;
			} else if (value >= 10000) {
				p = std::to_chars(p, p + 4, value / 10000).ptr;
				std::memcpy(p, quads[value % 10000].data(), 4);
				p += 4;
			} else p = std::to_chars(p, p + 4, value).ptr;
			length += p - out;
		} else length += std::to_chars(out, out + 32, value).ptr - out;
	}
	template <class T> requires requires(const T &value, char *out) { value.decimal_capacity(); value.to_chars(out); }
	void write(const T &value) {
		char *out = reserve(value.decimal_capacity());
		length += value.to_chars(out) - out;
	}
	template <class T>
	void println(const T &value) { write(value); write('\n'); }
};


#line 4 "test/generalized_floor_sum.test.cpp"

int main() {
	FastInput in;
	FastOutput out;
	int t;
	in.read(t);
	while (t--) {
		std::uint64_t n, m, a, b;
		in.read(n);
		in.read(m);
		in.read(a);
		in.read(b);
		auto x = generalized_floor_sum_moments<2>(n, m, a, b);
		auto y = generalized_floor_sum_moments<2>(n, m, a, 0);
		out.write(static_cast<long long>(2 * x[1][1] - (n - 1) * x[0][1] + y[1][1] - n * y[0][1]));
		out.write('\n');
	}
}