結果
| 問題 | No.2503 Typical Path Counting Problem on a Grid |
| ユーザー |
 rniya
|
| 提出日時 | 2023-09-10 13:57:46 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 187 ms / 2,000 ms |
| コード長 | 22,476 bytes |
| コンパイル時間 | 2,319 ms |
| コンパイル使用メモリ | 205,528 KB |
| 実行使用メモリ | 42,496 KB |
| 最終ジャッジ日時 | 2024-09-15 13:58:27 |
| 合計ジャッジ時間 | 5,016 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge3 |
（要ログイン）
テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | AC | 142 ms
42,368 KB |
| testcase_01 | AC | 156 ms
42,496 KB |
| testcase_02 | AC | 147 ms
42,368 KB |
| testcase_03 | AC | 167 ms
42,368 KB |
| testcase_04 | AC | 183 ms
42,496 KB |
| testcase_05 | AC | 163 ms
42,368 KB |
| testcase_06 | AC | 187 ms
42,368 KB |
| testcase_07 | AC | 184 ms
42,368 KB |
| testcase_08 | AC | 169 ms
42,496 KB |
| testcase_09 | AC | 182 ms
42,368 KB |
ソースコード
#include <bits/stdc++.h>
#ifdef LOCAL
#include <debug.hpp>
#else
#define debug(...) void(0)
#endif
#include <type_traits>
#ifdef _MSC_VER
#include <intrin.h>
#endif
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
return x;
}
// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
unsigned int _m;
unsigned long long im;
// @param m `1 <= m < 2^31`
explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
// @return m
unsigned int umod() const { return _m; }
// @param a `0 <= a < m`
// @param b `0 <= b < m`
// @return `a * b % m`
unsigned int mul(unsigned int a, unsigned int b) const {
// [1] m = 1
// a = b = im = 0, so okay
// [2] m >= 2
// im = ceil(2^64 / m)
// -> im * m = 2^64 + r (0 <= r < m)
// let z = a*b = c*m + d (0 <= c, d < m)
// a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
// c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
// ((ab * im) >> 64) == c or c + 1
unsigned long long z = a;
z *= b;
#ifdef _MSC_VER
unsigned long long x;
_umul128(z, im, &x);
#else
unsigned long long x =
(unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
unsigned int v = (unsigned int)(z - x * _m);
if (_m <= v) v += _m;
return v;
}
};
// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1;
unsigned long long y = safe_mod(x, m);
while (n) {
if (n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
if (n <= 1) return false;
if (n == 2 || n == 7 || n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = {2, 7, 61};
for (long long a : bases) {
long long t = d;
long long y = pow_mod_constexpr(a, t, n);
while (t != n - 1 && y != 1 && y != n - 1) {
y = y * y % n;
t <<= 1;
}
if (y != n - 1 && t % 2 == 0) {
return false;
}
}
return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);
// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
a = safe_mod(a, b);
if (a == 0) return {b, 0};
// Contracts:
// [1] s - m0 * a = 0 (mod b)
// [2] t - m1 * a = 0 (mod b)
// [3] s * |m1| + t * |m0| <= b
long long s = b, t = a;
long long m0 = 0, m1 = 1;
while (t) {
long long u = s / t;
s -= t * u;
m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b
// [3]:
// (s - t * u) * |m1| + t * |m0 - m1 * u|
// <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
// = s * |m1| + t * |m0| <= b
auto tmp = s;
s = t;
t = tmp;
tmp = m0;
m0 = m1;
m1 = tmp;
}
// by [3]: |m0| <= b/g
// by g != b: |m0| < b/g
if (m0 < 0) m0 += b / s;
return {s, m0};
}
// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
if (m == 2) return 1;
if (m == 167772161) return 3;
if (m == 469762049) return 3;
if (m == 754974721) return 11;
if (m == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (m - 1) / 2;
while (x % 2 == 0) x /= 2;
for (int i = 3; (long long)(i)*i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0) {
x /= i;
}
}
}
if (x > 1) {
divs[cnt++] = x;
}
for (int g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
unsigned long long m,
unsigned long long a,
unsigned long long b) {
unsigned long long ans = 0;
while (true) {
if (a >= m) {
ans += n * (n - 1) / 2 * (a / m);
a %= m;
}
if (b >= m) {
ans += n * (b / m);
b %= m;
}
unsigned long long y_max = a * n + b;
if (y_max < m) break;
// y_max < m * (n + 1)
// floor(y_max / m) <= n
n = (unsigned long long)(y_max / m);
b = (unsigned long long)(y_max % m);
std::swap(m, a);
}
return ans;
}
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value ||
std::is_same<T, __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int128 =
typename std::conditional<std::is_same<T, __uint128_t>::value ||
std::is_same<T, unsigned __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using make_unsigned_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value,
__uint128_t,
unsigned __int128>;
template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
is_signed_int128<T>::value ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
std::is_signed<T>::value) ||
is_signed_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<(is_integral<T>::value &&
std::is_unsigned<T>::value) ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<
is_signed_int128<T>::value,
make_unsigned_int128<T>,
typename std::conditional<std::is_signed<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type>::type;
#else
template <class T> using is_integral = typename std::is_integral<T>;
template <class T>
using is_signed_int =
typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<is_integral<T>::value &&
std::is_unsigned<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type;
#endif
template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;
template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;
template <class T> using to_unsigned_t = typename to_unsigned<T>::type;
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
struct modint_base {};
struct static_modint_base : modint_base {};
template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;
} // namespace internal
template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
using mint = static_modint;
public:
static constexpr int mod() { return m; }
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
static_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
static_modint(T v) {
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
static_modint(T v) {
_v = (unsigned int)(v % umod());
}
unsigned int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint& operator*=(const mint& rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (unsigned int)(z % umod());
return *this;
}
mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
if (prime) {
assert(_v);
return pow(umod() - 2);
} else {
auto eg = internal::inv_gcd(_v, m);
assert(eg.first == 1);
return eg.second;
}
}
friend mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
private:
unsigned int _v;
static constexpr unsigned int umod() { return m; }
static constexpr bool prime = internal::is_prime<m>;
};
template <int id> struct dynamic_modint : internal::modint_base {
using mint = dynamic_modint;
public:
static int mod() { return (int)(bt.umod()); }
static void set_mod(int m) {
assert(1 <= m);
bt = internal::barrett(m);
}
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
dynamic_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
dynamic_modint(T v) {
long long x = (long long)(v % (long long)(mod()));
if (x < 0) x += mod();
_v = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
dynamic_modint(T v) {
_v = (unsigned int)(v % mod());
}
unsigned int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v += mod() - rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator*=(const mint& rhs) {
_v = bt.mul(_v, rhs._v);
return *this;
}
mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
auto eg = internal::inv_gcd(_v, mod());
assert(eg.first == 1);
return eg.second;
}
friend mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
private:
unsigned int _v;
static internal::barrett bt;
static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
namespace internal {
template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;
template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;
template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};
template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;
} // namespace internal
} // namespace atcoder
template <typename T, size_t N> struct SquareMatrix {
std::array<std::array<T, N>, N> A;
SquareMatrix() : A{{}} {}
size_t size() const { return N; }
inline const std::array<T, N>& operator[](int k) const { return A[k]; }
inline std::array<T, N>& operator[](int k) { return A[k]; }
static SquareMatrix I() {
SquareMatrix res;
for (size_t i = 0; i < N; i++) res[i][i] = 1;
return res;
}
SquareMatrix& operator+=(const SquareMatrix& B) {
for (size_t i = 0; i < N; i++) {
for (size_t j = 0; j < N; j++) {
(*this)[i][j] += B[i][j];
}
}
return *this;
}
SquareMatrix& operator-=(const SquareMatrix& B) {
for (size_t i = 0; i < N; i++) {
for (size_t j = 0; j < N; j++) {
(*this)[i][j] -= B[i][j];
}
}
return *this;
}
SquareMatrix& operator*=(const SquareMatrix& B) {
std::array<std::array<T, N>, N> C = {};
for (size_t i = 0; i < N; i++) {
for (size_t k = 0; k < N; k++) {
for (size_t j = 0; j < N; j++) {
C[i][j] += (*this)[i][k] * B[k][j];
}
}
}
A.swap(C);
return *this;
}
SquareMatrix& operator*=(const T& v) {
for (size_t i = 0; i < N; i++) {
for (size_t j = 0; j < N; j++) {
(*this)[i][j] *= v;
}
}
return *this;
}
SquareMatrix& operator/=(const T& v) {
T inv = T(1) / v;
for (size_t i = 0; i < N; i++) {
for (size_t j = 0; j < N; j++) {
(*this)[i][j] *= inv;
}
}
return *this;
}
SquareMatrix& operator^=(long long k) {
assert(0 <= k);
SquareMatrix B = SquareMatrix::I();
while (k > 0) {
if (k & 1) B *= *this;
*this *= *this;
k >>= 1;
}
A.swap(B.A);
return *this;
}
SquareMatrix operator-() const {
SquareMatrix res;
for (size_t i = 0; i < N; i++) {
for (size_t j = 0; j < N; j++) {
res[i][j] = -(*this)[i][j];
}
}
return res;
}
SquareMatrix operator+(const SquareMatrix& B) const { return SquareMatrix(*this) += B; }
SquareMatrix operator-(const SquareMatrix& B) const { return SquareMatrix(*this) -= B; }
SquareMatrix operator*(const SquareMatrix& B) const { return SquareMatrix(*this) *= B; }
SquareMatrix operator*(const T& v) const { return SquareMatrix(*this) *= v; }
SquareMatrix operator/(const T& v) const { return SquareMatrix(*this) /= v; }
SquareMatrix operator^(const long long k) const { return SquareMatrix(*this) ^= k; }
bool operator==(const SquareMatrix& B) const { return A == B.A; }
bool operator!=(const SquareMatrix& B) const { return A != B.A; }
SquareMatrix transpose() const {
SquareMatrix res;
for (size_t i = 0; i < N; i++) {
for (size_t j = 0; j < N; j++) {
res[j][i] = (*this)[i][j];
}
}
return res;
}
T determinant(size_t n = N) const {
SquareMatrix B(*this);
T res = 1;
for (size_t i = 0; i < n; i++) {
int pivot = -1;
for (size_t j = i; j < n; j++) {
if (B[j][i] != 0) {
pivot = j;
break;
}
}
if (pivot == -1) return 0;
if (pivot != (int)i) {
res *= -1;
std::swap(B[i], B[pivot]);
}
res *= B[i][i];
T inv = T(1) / B[i][i];
for (size_t j = 0; j < n; j++) B[i][j] *= inv;
for (size_t j = i + 1; j < n; j++) {
T a = B[j][i];
for (size_t k = 0; k < n; k++) {
B[j][k] -= B[i][k] * a;
}
}
}
}
friend std::ostream& operator<<(std::ostream& os, const SquareMatrix& p) {
os << "[(" << N << " * " << N << " Matrix)";
os << "\n[columun sums: ";
for (size_t j = 0; j < N; j++) {
T sum = 0;
for (size_t i = 0; i < N; i++) sum += p[i][j];
;
os << sum << (j + 1 < N ? "," : "");
}
os << "]";
for (size_t i = 0; i < N; i++) {
os << "\n[";
for (size_t j = 0; j < N; j++) os << p[i][j] << (j + 1 < N ? "," : "");
os << "]";
}
os << "]\n";
return os;
}
};
using namespace std;
typedef long long ll;
#define all(x) begin(x), end(x)
constexpr int INF = (1 << 30) - 1;
constexpr long long IINF = (1LL << 60) - 1;
constexpr int dx[4] = {1, 0, -1, 0}, dy[4] = {0, 1, 0, -1};
template <class T> istream& operator>>(istream& is, vector<T>& v) {
for (auto& x : v) is >> x;
return is;
}
template <class T> ostream& operator<<(ostream& os, const vector<T>& v) {
auto sep = "";
for (const auto& x : v) os << exchange(sep, " ") << x;
return os;
}
template <class T, class U = T> bool chmin(T& x, U&& y) { return y < x and (x = forward<U>(y), true); }
template <class T, class U = T> bool chmax(T& x, U&& y) { return x < y and (x = forward<U>(y), true); }
template <class T> void mkuni(vector<T>& v) {
sort(begin(v), end(v));
v.erase(unique(begin(v), end(v)), end(v));
}
template <class T> int lwb(const vector<T>& v, const T& x) { return lower_bound(begin(v), end(v), x) - begin(v); }
using mint = atcoder::modint998244353;
using SM = SquareMatrix<mint, 2>;
const int MAX_N = 10000010;
mint dp[MAX_N];
void precalc() {
dp[0] = 1;
for (int i = 0; i + 2 < MAX_N; i++) {
mint val = dp[i];
val *= i + 1;
dp[i + 2] += val;
dp[i + 1] += val * 2;
}
}
void solve() {
ll n, m;
cin >> n >> m;
if (n > m) swap(n, m);
if (n == 0) {
cout << 1 << '\n';
return;
}
mint ans = 0;
if (n == m) {
ans += dp[n - 1] * dp[n - 1] * n;
ans += dp[n] * dp[n];
} else {
SM M;
M[0][1] = 1;
M[1][0] = n;
M[1][1] = 2 * n + 1;
M ^= m - n;
ans += (dp[n - 1] * M[0][0] + dp[n] * M[0][1]) * dp[n - 1] * n;
ans += (dp[n - 1] * M[1][0] + dp[n] * M[1][1]) * dp[n];
}
cout << ans.val() << '\n';
}
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
precalc();
int T;
cin >> T;
for (; T--;) solve();
return 0;
}