結果
| 問題 | No.1939 Numbered Colorful Balls |
| ユーザー |
 noimi
|
| 提出日時 | 2022-05-13 22:53:51 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 64 ms / 2,000 ms |
| コード長 | 43,992 bytes |
| コンパイル時間 | 5,459 ms |
| コンパイル使用メモリ | 327,916 KB |
| 実行使用メモリ | 14,376 KB |
| 最終ジャッジ日時 | 2023-09-29 08:32:49 |
| 合計ジャッジ時間 | 7,212 ms |
|
ジャッジサーバーID （参考情報） |
judge11 / judge14 |
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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 | 63 ms
14,092 KB |
| testcase_01 | AC | 2 ms
5,420 KB |
| testcase_02 | AC | 62 ms
13,936 KB |
| testcase_03 | AC | 63 ms
13,944 KB |
| testcase_04 | AC | 2 ms
5,624 KB |
| testcase_05 | AC | 64 ms
14,116 KB |
| testcase_06 | AC | 64 ms
14,376 KB |
| testcase_07 | AC | 16 ms
7,252 KB |
| testcase_08 | AC | 9 ms
6,060 KB |
| testcase_09 | AC | 62 ms
13,356 KB |
| testcase_10 | AC | 3 ms
5,520 KB |
| testcase_11 | AC | 32 ms
9,580 KB |
| testcase_12 | AC | 6 ms
5,820 KB |
| testcase_13 | AC | 3 ms
5,612 KB |
| testcase_14 | AC | 16 ms
7,420 KB |
| testcase_15 | AC | 9 ms
6,384 KB |
| testcase_16 | AC | 32 ms
9,988 KB |
| testcase_17 | AC | 32 ms
9,516 KB |
| testcase_18 | AC | 17 ms
7,340 KB |
| testcase_19 | AC | 60 ms
13,180 KB |
| testcase_20 | AC | 17 ms
7,408 KB |
| testcase_21 | AC | 61 ms
13,488 KB |
| testcase_22 | AC | 6 ms
5,752 KB |
| testcase_23 | AC | 31 ms
9,728 KB |
| testcase_24 | AC | 61 ms
13,496 KB |
| testcase_25 | AC | 32 ms
9,700 KB |
| testcase_26 | AC | 63 ms
14,372 KB |
| testcase_27 | AC | 2 ms
5,360 KB |
| testcase_28 | AC | 2 ms
5,384 KB |
| testcase_29 | AC | 32 ms
9,656 KB |
ソースコード
#include <bits/stdc++.h>
using namespace std;
namespace Modular998 {
#line 2 "library/modulo/binomial.hpp"
template <typename T> struct Binomial {
vector<T> f, g, h;
Binomial(int MAX = 0) : f(1, T(1)), g(1, T(1)), h(1, T(1)) {
while(MAX >= (int)f.size()) extend();
}
void extend() {
int n = f.size();
int m = n * 2;
f.resize(m);
g.resize(m);
h.resize(m);
for(int i = n; i < m; i++) f[i] = f[i - 1] * T(i);
g[m - 1] = f[m - 1].inverse();
h[m - 1] = g[m - 1] * f[m - 2];
for(int i = m - 2; i >= n; i--) {
g[i] = g[i + 1] * T(i + 1);
h[i] = g[i] * f[i - 1];
}
}
T fac(int i) {
if(i < 0) return T(0);
while(i >= (int)f.size()) extend();
return f[i];
}
T finv(int i) {
if(i < 0) return T(0);
while(i >= (int)g.size()) extend();
return g[i];
}
T inv(int i) {
if(i < 0) return -inv(-i);
while(i >= (int)h.size()) extend();
return h[i];
}
T C(int n, int r) {
if(n < 0 || n < r || r < 0) return T(0);
return fac(n) * finv(n - r) * finv(r);
}
inline T operator()(int n, int r) { return C(n, r); }
template <typename I> T multinomial(const vector<I> &r) {
static_assert(is_integral<I>::value == true);
int n = 0;
for(auto &x : r) {
if(x < 0) return T(0);
n += x;
}
T res = fac(n);
for(auto &x : r) res *= finv(x);
return res;
}
template <typename I> T operator()(const vector<I> &r) { return multinomial(r); }
T C_naive(int n, int r) {
if(n < 0 || n < r || r < 0) return T(0);
T ret = T(1);
r = min(r, n - r);
for(int i = 1; i <= r; ++i) ret *= inv(i) * (n--);
return ret;
}
T P(int n, int r) {
if(n < 0 || n < r || r < 0) return T(0);
return fac(n) * finv(n - r);
}
T H(int n, int r) {
if(n < 0 || r < 0) return T(0);
return r == 0 ? 1 : C(n + r - 1, r);
}
};
#line 2 "library/modint/montgomery-modint.hpp"
template <uint32_t mod> struct LazyMontgomeryModInt {
using mint = LazyMontgomeryModInt;
using i32 = int32_t;
using u32 = uint32_t;
using u64 = uint64_t;
static constexpr u32 get_r() {
u32 ret = mod;
for(i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
return ret;
}
static constexpr u32 r = get_r();
static constexpr u32 n2 = -u64(mod) % mod;
static_assert(r * mod == 1, "invalid, r * mod != 1");
static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
u32 a;
constexpr LazyMontgomeryModInt() : a(0) {}
constexpr LazyMontgomeryModInt(const int64_t &b) : a(reduce(u64(b % mod + mod) * n2)){};
static constexpr u32 reduce(const u64 &b) { return (b + u64(u32(b) * u32(-r)) * mod) >> 32; }
constexpr mint &operator+=(const mint &b) {
if(i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
return *this;
}
constexpr mint &operator-=(const mint &b) {
if(i32(a -= b.a) < 0) a += 2 * mod;
return *this;
}
constexpr mint &operator*=(const mint &b) {
a = reduce(u64(a) * b.a);
return *this;
}
constexpr mint &operator/=(const mint &b) {
*this *= b.inverse();
return *this;
}
constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
constexpr bool operator==(const mint &b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); }
constexpr bool operator!=(const mint &b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); }
constexpr mint operator-() const { return mint() - mint(*this); }
constexpr mint pow(u64 n) const {
mint ret(1), mul(*this);
while(n > 0) {
if(n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
constexpr mint inverse() const { return pow(mod - 2); }
friend ostream &operator<<(ostream &os, const mint &b) { return os << b.get(); }
friend istream &operator>>(istream &is, mint &b) {
int64_t t;
is >> t;
b = LazyMontgomeryModInt<mod>(t);
return (is);
}
constexpr u32 get() const {
u32 ret = reduce(a);
return ret >= mod ? ret - mod : ret;
}
static constexpr u32 get_mod() { return mod; }
};
#line 2 "library/fps/ntt-friendly-fps.hpp"
#line 2 "library/ntt/ntt-avx2.hpp"
#line 2 "library/modint/simd-montgomery.hpp"
#include <immintrin.h>
__attribute__((target("sse4.2"))) inline __m128i my128_mullo_epu32(const __m128i &a, const __m128i &b) { return _mm_mullo_epi32(a, b); }
__attribute__((target("sse4.2"))) inline __m128i my128_mulhi_epu32(const __m128i &a, const __m128i &b) {
__m128i a13 = _mm_shuffle_epi32(a, 0xF5);
__m128i b13 = _mm_shuffle_epi32(b, 0xF5);
__m128i prod02 = _mm_mul_epu32(a, b);
__m128i prod13 = _mm_mul_epu32(a13, b13);
__m128i prod = _mm_unpackhi_epi64(_mm_unpacklo_epi32(prod02, prod13), _mm_unpackhi_epi32(prod02, prod13));
return prod;
}
__attribute__((target("sse4.2"))) inline __m128i montgomery_mul_128(const __m128i &a, const __m128i &b, const __m128i &r, const __m128i &m1) {
return _mm_sub_epi32(_mm_add_epi32(my128_mulhi_epu32(a, b), m1), my128_mulhi_epu32(my128_mullo_epu32(my128_mullo_epu32(a, b), r), m1));
}
__attribute__((target("sse4.2"))) inline __m128i montgomery_add_128(const __m128i &a, const __m128i &b, const __m128i &m2, const __m128i &m0) {
__m128i ret = _mm_sub_epi32(_mm_add_epi32(a, b), m2);
return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret);
}
__attribute__((target("sse4.2"))) inline __m128i montgomery_sub_128(const __m128i &a, const __m128i &b, const __m128i &m2, const __m128i &m0) {
__m128i ret = _mm_sub_epi32(a, b);
return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret);
}
__attribute__((target("avx2"))) inline __m256i my256_mullo_epu32(const __m256i &a, const __m256i &b) { return _mm256_mullo_epi32(a, b); }
__attribute__((target("avx2"))) inline __m256i my256_mulhi_epu32(const __m256i &a, const __m256i &b) {
__m256i a13 = _mm256_shuffle_epi32(a, 0xF5);
__m256i b13 = _mm256_shuffle_epi32(b, 0xF5);
__m256i prod02 = _mm256_mul_epu32(a, b);
__m256i prod13 = _mm256_mul_epu32(a13, b13);
__m256i prod = _mm256_unpackhi_epi64(_mm256_unpacklo_epi32(prod02, prod13), _mm256_unpackhi_epi32(prod02, prod13));
return prod;
}
__attribute__((target("avx2"))) inline __m256i montgomery_mul_256(const __m256i &a, const __m256i &b, const __m256i &r, const __m256i &m1) {
return _mm256_sub_epi32(_mm256_add_epi32(my256_mulhi_epu32(a, b), m1), my256_mulhi_epu32(my256_mullo_epu32(my256_mullo_epu32(a, b), r), m1));
}
__attribute__((target("avx2"))) inline __m256i montgomery_add_256(const __m256i &a, const __m256i &b, const __m256i &m2, const __m256i &m0) {
__m256i ret = _mm256_sub_epi32(_mm256_add_epi32(a, b), m2);
return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2), ret);
}
__attribute__((target("avx2"))) inline __m256i montgomery_sub_256(const __m256i &a, const __m256i &b, const __m256i &m2, const __m256i &m0) {
__m256i ret = _mm256_sub_epi32(a, b);
return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2), ret);
}
#line 4 "library/ntt/ntt-avx2.hpp"
namespace ntt_inner {
using u64 = uint64_t;
constexpr uint32_t get_pr(uint32_t mod) {
if(mod == 2) return 1;
u64 ds[32] = {};
int idx = 0;
u64 m = mod - 1;
for(u64 i = 2; i * i <= m; ++i) {
if(m % i == 0) {
ds[idx++] = i;
while(m % i == 0) m /= i;
}
}
if(m != 1) ds[idx++] = m;
uint32_t pr = 2;
while(1) {
int flg = 1;
for(int i = 0; i < idx; ++i) {
u64 a = pr, b = (mod - 1) / ds[i], r = 1;
while(b) {
if(b & 1) r = r * a % mod;
a = a * a % mod;
b >>= 1;
}
if(r == 1) {
flg = 0;
break;
}
}
if(flg == 1) break;
++pr;
}
return pr;
}
constexpr int SZ_FFT_BUF = 1 << 23;
uint32_t _buf1[SZ_FFT_BUF] __attribute__((aligned(64)));
uint32_t _buf2[SZ_FFT_BUF] __attribute__((aligned(64)));
} // namespace ntt_inner
template <typename mint> struct NTT {
static constexpr uint32_t mod = mint::get_mod();
static constexpr uint32_t pr = ntt_inner::get_pr(mint::get_mod());
static constexpr int level = __builtin_ctzll(mod - 1);
mint dw[level], dy[level];
mint *buf1, *buf2;
constexpr NTT() {
setwy(level);
union raw_cast {
mint dat;
uint32_t _;
};
buf1 = &(((raw_cast *)(ntt_inner::_buf1))->dat);
buf2 = &(((raw_cast *)(ntt_inner::_buf2))->dat);
}
constexpr void setwy(int k) {
mint w[level], y[level];
w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
y[k - 1] = w[k - 1].inverse();
for(int i = k - 2; i > 0; --i) w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
dw[0] = dy[0] = w[1] * w[1];
dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
for(int i = 3; i < k; ++i) {
dw[i] = dw[i - 1] * y[i - 2] * w[i];
dy[i] = dy[i - 1] * w[i - 2] * y[i];
}
}
__attribute__((target("avx2"))) void ntt(mint *a, int n) {
int k = n ? __builtin_ctz(n) : 0;
if(k == 0) return;
if(k == 1) {
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
if(k & 1) {
int v = 1 << (k - 1);
if(v < 8) {
for(int j = 0; j < v; ++j) {
mint ajv = a[j + v];
a[j + v] = a[j] - ajv;
a[j] += ajv;
}
} else {
const __m256i m0 = _mm256_set1_epi32(0);
const __m256i m2 = _mm256_set1_epi32(mod + mod);
int j0 = 0;
int j1 = v;
for(; j0 < v; j0 += 8, j1 += 8) {
__m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
__m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
__m256i naj = montgomery_add_256(T0, T1, m2, m0);
__m256i najv = montgomery_sub_256(T0, T1, m2, m0);
_mm256_storeu_si256((__m256i *)(a + j0), naj);
_mm256_storeu_si256((__m256i *)(a + j1), najv);
}
}
}
int u = 1 << (2 + (k & 1));
int v = 1 << (k - 2 - (k & 1));
mint one = mint(1);
mint imag = dw[1];
while(v) {
if(v == 1) {
mint ww = one, xx = one, wx = one;
for(int jh = 0; jh < u;) {
ww = xx * xx, wx = ww * xx;
mint t0 = a[jh + 0], t1 = a[jh + 1] * xx;
mint t2 = a[jh + 2] * ww, t3 = a[jh + 3] * wx;
mint t0p2 = t0 + t2, t1p3 = t1 + t3;
mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
a[jh + 0] = t0p2 + t1p3, a[jh + 1] = t0p2 - t1p3;
a[jh + 2] = t0m2 + t1m3, a[jh + 3] = t0m2 - t1m3;
xx *= dw[__builtin_ctz((jh += 4))];
}
} else if(v == 4) {
const __m128i m0 = _mm_set1_epi32(0);
const __m128i m1 = _mm_set1_epi32(mod);
const __m128i m2 = _mm_set1_epi32(mod + mod);
const __m128i r = _mm_set1_epi32(mint::r);
const __m128i Imag = _mm_set1_epi32(imag.a);
mint ww = one, xx = one, wx = one;
for(int jh = 0; jh < u;) {
if(jh == 0) {
int j0 = 0;
int j1 = v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = v;
for(; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
const __m128i T0P2 = montgomery_add_128(T0, T2, m2, m0);
const __m128i T1P3 = montgomery_add_128(T1, T3, m2, m0);
const __m128i T0M2 = montgomery_sub_128(T0, T2, m2, m0);
const __m128i T1M3 = montgomery_mul_128(montgomery_sub_128(T1, T3, m2, m0), Imag, r, m1);
_mm_storeu_si128((__m128i *)(a + j0), montgomery_add_128(T0P2, T1P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j1), montgomery_sub_128(T0P2, T1P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j2), montgomery_add_128(T0M2, T1M3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j3), montgomery_sub_128(T0M2, T1M3, m2, m0));
}
} else {
ww = xx * xx, wx = ww * xx;
const __m128i WW = _mm_set1_epi32(ww.a);
const __m128i WX = _mm_set1_epi32(wx.a);
const __m128i XX = _mm_set1_epi32(xx.a);
int j0 = jh * v;
int j1 = j0 + v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = j1;
for(; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
const __m128i MT1 = montgomery_mul_128(T1, XX, r, m1);
const __m128i MT2 = montgomery_mul_128(T2, WW, r, m1);
const __m128i MT3 = montgomery_mul_128(T3, WX, r, m1);
const __m128i T0P2 = montgomery_add_128(T0, MT2, m2, m0);
const __m128i T1P3 = montgomery_add_128(MT1, MT3, m2, m0);
const __m128i T0M2 = montgomery_sub_128(T0, MT2, m2, m0);
const __m128i T1M3 = montgomery_mul_128(montgomery_sub_128(MT1, MT3, m2, m0), Imag, r, m1);
_mm_storeu_si128((__m128i *)(a + j0), montgomery_add_128(T0P2, T1P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j1), montgomery_sub_128(T0P2, T1P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j2), montgomery_add_128(T0M2, T1M3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j3), montgomery_sub_128(T0M2, T1M3, m2, m0));
}
}
xx *= dw[__builtin_ctz((jh += 4))];
}
} else {
const __m256i m0 = _mm256_set1_epi32(0);
const __m256i m1 = _mm256_set1_epi32(mod);
const __m256i m2 = _mm256_set1_epi32(mod + mod);
const __m256i r = _mm256_set1_epi32(mint::r);
const __m256i Imag = _mm256_set1_epi32(imag.a);
mint ww = one, xx = one, wx = one;
for(int jh = 0; jh < u;) {
if(jh == 0) {
int j0 = 0;
int j1 = v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = v;
for(; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
const __m256i T0P2 = montgomery_add_256(T0, T2, m2, m0);
const __m256i T1P3 = montgomery_add_256(T1, T3, m2, m0);
const __m256i T0M2 = montgomery_sub_256(T0, T2, m2, m0);
const __m256i T1M3 = montgomery_mul_256(montgomery_sub_256(T1, T3, m2, m0), Imag, r, m1);
_mm256_storeu_si256((__m256i *)(a + j0), montgomery_add_256(T0P2, T1P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j1), montgomery_sub_256(T0P2, T1P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j2), montgomery_add_256(T0M2, T1M3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j3), montgomery_sub_256(T0M2, T1M3, m2, m0));
}
} else {
ww = xx * xx, wx = ww * xx;
const __m256i WW = _mm256_set1_epi32(ww.a);
const __m256i WX = _mm256_set1_epi32(wx.a);
const __m256i XX = _mm256_set1_epi32(xx.a);
int j0 = jh * v;
int j1 = j0 + v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = j1;
for(; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
const __m256i MT1 = montgomery_mul_256(T1, XX, r, m1);
const __m256i MT2 = montgomery_mul_256(T2, WW, r, m1);
const __m256i MT3 = montgomery_mul_256(T3, WX, r, m1);
const __m256i T0P2 = montgomery_add_256(T0, MT2, m2, m0);
const __m256i T1P3 = montgomery_add_256(MT1, MT3, m2, m0);
const __m256i T0M2 = montgomery_sub_256(T0, MT2, m2, m0);
const __m256i T1M3 = montgomery_mul_256(montgomery_sub_256(MT1, MT3, m2, m0), Imag, r, m1);
_mm256_storeu_si256((__m256i *)(a + j0), montgomery_add_256(T0P2, T1P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j1), montgomery_sub_256(T0P2, T1P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j2), montgomery_add_256(T0M2, T1M3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j3), montgomery_sub_256(T0M2, T1M3, m2, m0));
}
}
xx *= dw[__builtin_ctz((jh += 4))];
}
}
u <<= 2;
v >>= 2;
}
}
__attribute__((target("avx2"))) void intt(mint *a, int n, int normalize = true) {
int k = n ? __builtin_ctz(n) : 0;
if(k == 0) return;
if(k == 1) {
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
if(normalize) {
a[0] *= mint(2).inverse();
a[1] *= mint(2).inverse();
}
return;
}
int u = 1 << (k - 2);
int v = 1;
mint one = mint(1);
mint imag = dy[1];
while(u) {
if(v == 1) {
mint ww = one, xx = one, yy = one;
u <<= 2;
for(int jh = 0; jh < u;) {
ww = xx * xx, yy = xx * imag;
mint t0 = a[jh + 0], t1 = a[jh + 1];
mint t2 = a[jh + 2], t3 = a[jh + 3];
mint t0p1 = t0 + t1, t2p3 = t2 + t3;
mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
a[jh + 0] = t0p1 + t2p3, a[jh + 2] = (t0p1 - t2p3) * ww;
a[jh + 1] = t0m1 + t2m3, a[jh + 3] = (t0m1 - t2m3) * ww;
xx *= dy[__builtin_ctz(jh += 4)];
}
} else if(v == 4) {
const __m128i m0 = _mm_set1_epi32(0);
const __m128i m1 = _mm_set1_epi32(mod);
const __m128i m2 = _mm_set1_epi32(mod + mod);
const __m128i r = _mm_set1_epi32(mint::r);
const __m128i Imag = _mm_set1_epi32(imag.a);
mint ww = one, xx = one, yy = one;
u <<= 2;
for(int jh = 0; jh < u;) {
if(jh == 0) {
int j0 = 0;
int j1 = v;
int j2 = v + v;
int j3 = j2 + v;
for(; j0 < v; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
const __m128i T0P1 = montgomery_add_128(T0, T1, m2, m0);
const __m128i T2P3 = montgomery_add_128(T2, T3, m2, m0);
const __m128i T0M1 = montgomery_sub_128(T0, T1, m2, m0);
const __m128i T2M3 = montgomery_mul_128(montgomery_sub_128(T2, T3, m2, m0), Imag, r, m1);
_mm_storeu_si128((__m128i *)(a + j0), montgomery_add_128(T0P1, T2P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j2), montgomery_sub_128(T0P1, T2P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j1), montgomery_add_128(T0M1, T2M3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j3), montgomery_sub_128(T0M1, T2M3, m2, m0));
}
} else {
ww = xx * xx, yy = xx * imag;
const __m128i WW = _mm_set1_epi32(ww.a);
const __m128i XX = _mm_set1_epi32(xx.a);
const __m128i YY = _mm_set1_epi32(yy.a);
int j0 = jh * v;
int j1 = j0 + v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = j1;
for(; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
const __m128i T0P1 = montgomery_add_128(T0, T1, m2, m0);
const __m128i T2P3 = montgomery_add_128(T2, T3, m2, m0);
const __m128i T0M1 = montgomery_mul_128(montgomery_sub_128(T0, T1, m2, m0), XX, r, m1);
__m128i T2M3 = montgomery_mul_128(montgomery_sub_128(T2, T3, m2, m0), YY, r, m1);
_mm_storeu_si128((__m128i *)(a + j0), montgomery_add_128(T0P1, T2P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j2), montgomery_mul_128(montgomery_sub_128(T0P1, T2P3, m2, m0), WW, r, m1));
_mm_storeu_si128((__m128i *)(a + j1), montgomery_add_128(T0M1, T2M3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j3), montgomery_mul_128(montgomery_sub_128(T0M1, T2M3, m2, m0), WW, r, m1));
}
}
xx *= dy[__builtin_ctz(jh += 4)];
}
} else {
const __m256i m0 = _mm256_set1_epi32(0);
const __m256i m1 = _mm256_set1_epi32(mod);
const __m256i m2 = _mm256_set1_epi32(mod + mod);
const __m256i r = _mm256_set1_epi32(mint::r);
const __m256i Imag = _mm256_set1_epi32(imag.a);
mint ww = one, xx = one, yy = one;
u <<= 2;
for(int jh = 0; jh < u;) {
if(jh == 0) {
int j0 = 0;
int j1 = v;
int j2 = v + v;
int j3 = j2 + v;
for(; j0 < v; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
const __m256i T0P1 = montgomery_add_256(T0, T1, m2, m0);
const __m256i T2P3 = montgomery_add_256(T2, T3, m2, m0);
const __m256i T0M1 = montgomery_sub_256(T0, T1, m2, m0);
const __m256i T2M3 = montgomery_mul_256(montgomery_sub_256(T2, T3, m2, m0), Imag, r, m1);
_mm256_storeu_si256((__m256i *)(a + j0), montgomery_add_256(T0P1, T2P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j2), montgomery_sub_256(T0P1, T2P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j1), montgomery_add_256(T0M1, T2M3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j3), montgomery_sub_256(T0M1, T2M3, m2, m0));
}
} else {
ww = xx * xx, yy = xx * imag;
const __m256i WW = _mm256_set1_epi32(ww.a);
const __m256i XX = _mm256_set1_epi32(xx.a);
const __m256i YY = _mm256_set1_epi32(yy.a);
int j0 = jh * v;
int j1 = j0 + v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = j1;
for(; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
const __m256i T0P1 = montgomery_add_256(T0, T1, m2, m0);
const __m256i T2P3 = montgomery_add_256(T2, T3, m2, m0);
const __m256i T0M1 = montgomery_mul_256(montgomery_sub_256(T0, T1, m2, m0), XX, r, m1);
const __m256i T2M3 = montgomery_mul_256(montgomery_sub_256(T2, T3, m2, m0), YY, r, m1);
_mm256_storeu_si256((__m256i *)(a + j0), montgomery_add_256(T0P1, T2P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j2), montgomery_mul_256(montgomery_sub_256(T0P1, T2P3, m2, m0), WW, r, m1));
_mm256_storeu_si256((__m256i *)(a + j1), montgomery_add_256(T0M1, T2M3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j3), montgomery_mul_256(montgomery_sub_256(T0M1, T2M3, m2, m0), WW, r, m1));
}
}
xx *= dy[__builtin_ctz(jh += 4)];
}
}
u >>= 4;
v <<= 2;
}
if(k & 1) {
v = 1 << (k - 1);
if(v < 8) {
for(int j = 0; j < v; ++j) {
mint ajv = a[j] - a[j + v];
a[j] += a[j + v];
a[j + v] = ajv;
}
} else {
const __m256i m0 = _mm256_set1_epi32(0);
const __m256i m2 = _mm256_set1_epi32(mod + mod);
int j0 = 0;
int j1 = v;
for(; j0 < v; j0 += 8, j1 += 8) {
const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
__m256i naj = montgomery_add_256(T0, T1, m2, m0);
__m256i najv = montgomery_sub_256(T0, T1, m2, m0);
_mm256_storeu_si256((__m256i *)(a + j0), naj);
_mm256_storeu_si256((__m256i *)(a + j1), najv);
}
}
}
if(normalize) {
mint invn = mint(n).inverse();
for(int i = 0; i < n; i++) a[i] *= invn;
}
}
__attribute__((target("avx2"))) void inplace_multiply(int l1, int l2, int zero_padding = true) {
int l = l1 + l2 - 1;
int M = 4;
while(M < l) M <<= 1;
if(zero_padding) {
for(int i = l1; i < M; i++) ntt_inner::_buf1[i] = 0;
for(int i = l2; i < M; i++) ntt_inner::_buf2[i] = 0;
}
const __m256i m0 = _mm256_set1_epi32(0);
const __m256i m1 = _mm256_set1_epi32(mod);
const __m256i r = _mm256_set1_epi32(mint::r);
const __m256i N2 = _mm256_set1_epi32(mint::n2);
for(int i = 0; i < l1; i += 8) {
__m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
__m256i b = montgomery_mul_256(a, N2, r, m1);
_mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), b);
}
for(int i = 0; i < l2; i += 8) {
__m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf2 + i));
__m256i b = montgomery_mul_256(a, N2, r, m1);
_mm256_storeu_si256((__m256i *)(ntt_inner::_buf2 + i), b);
}
ntt(buf1, M);
ntt(buf2, M);
for(int i = 0; i < M; i += 8) {
__m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
__m256i b = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf2 + i));
__m256i c = montgomery_mul_256(a, b, r, m1);
_mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), c);
}
intt(buf1, M, false);
const __m256i INVM = _mm256_set1_epi32((mint(M).inverse()).a);
for(int i = 0; i < l; i += 8) {
__m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
__m256i b = montgomery_mul_256(a, INVM, r, m1);
__m256i c = my256_mulhi_epu32(my256_mullo_epu32(b, r), m1);
__m256i d = _mm256_and_si256(_mm256_cmpgt_epi32(c, m0), m1);
__m256i e = _mm256_sub_epi32(d, c);
_mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), e);
}
}
void ntt(vector<mint> &a) {
int M = (int)a.size();
for(int i = 0; i < M; i++) buf1[i].a = a[i].a;
ntt(buf1, M);
for(int i = 0; i < M; i++) a[i].a = buf1[i].a;
}
void intt(vector<mint> &a) {
int M = (int)a.size();
for(int i = 0; i < M; i++) buf1[i].a = a[i].a;
intt(buf1, M, true);
for(int i = 0; i < M; i++) a[i].a = buf1[i].a;
}
vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
if(a.size() == 0 && b.size() == 0) return vector<mint>{};
int l = a.size() + b.size() - 1;
if(min<int>(a.size(), b.size()) <= 40) {
vector<mint> s(l);
for(int i = 0; i < (int)a.size(); ++i)
for(int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j];
return s;
}
assert(l <= ntt_inner::SZ_FFT_BUF);
int M = 4;
while(M < l) M <<= 1;
for(int i = 0; i < (int)a.size(); ++i) buf1[i].a = a[i].a;
for(int i = (int)a.size(); i < M; ++i) buf1[i].a = 0;
for(int i = 0; i < (int)b.size(); ++i) buf2[i].a = b[i].a;
for(int i = (int)b.size(); i < M; ++i) buf2[i].a = 0;
ntt(buf1, M);
ntt(buf2, M);
for(int i = 0; i < M; ++i) buf1[i].a = mint::reduce(uint64_t(buf1[i].a) * buf2[i].a);
intt(buf1, M, false);
vector<mint> s(l);
mint invm = mint(M).inverse();
for(int i = 0; i < l; ++i) s[i] = buf1[i] * invm;
return s;
}
void ntt_doubling(vector<mint> &a) {
int M = (int)a.size();
for(int i = 0; i < M; i++) buf1[i].a = a[i].a;
intt(buf1, M);
mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
for(int i = 0; i < M; i++) buf1[i] *= r, r *= zeta;
ntt(buf1, M);
a.resize(2 * M);
for(int i = 0; i < M; i++) a[M + i].a = buf1[i].a;
}
};
#line 2 "library/fps/formal-power-series.hpp"
template <typename mint> struct FormalPowerSeries : vector<mint> {
using vector<mint>::vector;
using FPS = FormalPowerSeries;
FPS &operator+=(const FPS &r) {
if(r.size() > this->size()) this->resize(r.size());
for(int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
return *this;
}
FPS &operator+=(const mint &r) {
if(this->empty()) this->resize(1);
(*this)[0] += r;
return *this;
}
FPS &operator-=(const FPS &r) {
if(r.size() > this->size()) this->resize(r.size());
for(int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
return *this;
}
FPS &operator-=(const mint &r) {
if(this->empty()) this->resize(1);
(*this)[0] -= r;
return *this;
}
FPS &operator*=(const mint &v) {
for(int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
return *this;
}
FPS &operator/=(const FPS &r) {
if(this->size() < r.size()) {
this->clear();
return *this;
}
int n = this->size() - r.size() + 1;
if((int)r.size() <= 64) {
FPS f(*this), g(r);
g.shrink();
mint coeff = g.back().inverse();
for(auto &x : g) x *= coeff;
int deg = (int)f.size() - (int)g.size() + 1;
int gs = g.size();
FPS quo(deg);
for(int i = deg - 1; i >= 0; i--) {
quo[i] = f[i + gs - 1];
for(int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
}
*this = quo * coeff;
this->resize(n, mint(0));
return *this;
}
return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
}
FPS &operator%=(const FPS &r) {
*this -= *this / r * r;
shrink();
return *this;
}
FPS operator+(const FPS &r) const { return FPS(*this) += r; }
FPS operator+(const mint &v) const { return FPS(*this) += v; }
FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
FPS operator-(const mint &v) const { return FPS(*this) -= v; }
FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
FPS operator*(const mint &v) const { return FPS(*this) *= v; }
FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
FPS operator-() const {
FPS ret(this->size());
for(int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
return ret;
}
void shrink() {
while(this->size() && this->back() == mint(0)) this->pop_back();
}
FPS rev() const {
FPS ret(*this);
reverse(begin(ret), end(ret));
return ret;
}
FPS dot(FPS r) const {
FPS ret(min(this->size(), r.size()));
for(int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
return ret;
}
FPS pre(int sz) const { return FPS(begin(*this), begin(*this) + min((int)this->size(), sz)); }
FPS operator>>(int sz) const {
if((int)this->size() <= sz) return {};
FPS ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
FPS operator<<(int sz) const {
FPS ret(*this);
ret.insert(ret.begin(), sz, mint(0));
return ret;
}
FPS diff() const {
const int n = (int)this->size();
FPS ret(max(0, n - 1));
mint one(1), coeff(1);
for(int i = 1; i < n; i++) {
ret[i - 1] = (*this)[i] * coeff;
coeff += one;
}
return ret;
}
FPS integral() const {
const int n = (int)this->size();
FPS ret(n + 1);
ret[0] = mint(0);
if(n > 0) ret[1] = mint(1);
auto mod = mint::get_mod();
for(int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
for(int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
return ret;
}
mint eval(mint x) const {
mint r = 0, w = 1;
for(auto &v : *this) r += w * v, w *= x;
return r;
}
FPS log(int deg = -1) const {
assert((*this)[0] == mint(1));
if(deg == -1) deg = (int)this->size();
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
FPS pow(int64_t k, int deg = -1) const {
const int n = (int)this->size();
if(deg == -1) deg = n;
for(int i = 0; i < n; i++) {
if((*this)[i] != mint(0)) {
if(i * k > deg) return FPS(deg, mint(0));
mint rev = mint(1) / (*this)[i];
FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg) * ((*this)[i].pow(k));
ret = (ret << (i * k)).pre(deg);
if((int)ret.size() < deg) ret.resize(deg, mint(0));
return ret;
}
}
return FPS(deg, mint(0));
}
static void *ntt_ptr;
static void set_fft();
FPS &operator*=(const FPS &r);
void ntt();
void intt();
void ntt_doubling();
static int ntt_pr();
FPS inv(int deg = -1) const;
FPS exp(int deg = -1) const;
};
template <typename mint> void *FormalPowerSeries<mint>::ntt_ptr = nullptr;
/**
* @brief 多項式/形式的冪級数ライブラリ
* @docs docs/fps/formal-power-series.md
*/
#line 5 "library/fps/ntt-friendly-fps.hpp"
template <typename mint> void FormalPowerSeries<mint>::set_fft() {
if(!ntt_ptr) ntt_ptr = new NTT<mint>;
}
template <typename mint> FormalPowerSeries<mint> &FormalPowerSeries<mint>::operator*=(const FormalPowerSeries<mint> &r) {
if(this->empty() || r.empty()) {
this->clear();
return *this;
}
set_fft();
auto ret = static_cast<NTT<mint> *>(ntt_ptr)->multiply(*this, r);
return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <typename mint> void FormalPowerSeries<mint>::ntt() {
set_fft();
static_cast<NTT<mint> *>(ntt_ptr)->ntt(*this);
}
template <typename mint> void FormalPowerSeries<mint>::intt() {
set_fft();
static_cast<NTT<mint> *>(ntt_ptr)->intt(*this);
}
template <typename mint> void FormalPowerSeries<mint>::ntt_doubling() {
set_fft();
static_cast<NTT<mint> *>(ntt_ptr)->ntt_doubling(*this);
}
template <typename mint> int FormalPowerSeries<mint>::ntt_pr() {
set_fft();
return static_cast<NTT<mint> *>(ntt_ptr)->pr;
}
template <typename mint> FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
assert((*this)[0] != mint(0));
if(deg == -1) deg = (int)this->size();
FormalPowerSeries<mint> res(deg);
res[0] = {mint(1) / (*this)[0]};
for(int d = 1; d < deg; d <<= 1) {
FormalPowerSeries<mint> f(2 * d), g(2 * d);
for(int j = 0; j < min((int)this->size(), 2 * d); j++) f[j] = (*this)[j];
for(int j = 0; j < d; j++) g[j] = res[j];
f.ntt();
g.ntt();
for(int j = 0; j < 2 * d; j++) f[j] *= g[j];
f.intt();
for(int j = 0; j < d; j++) f[j] = 0;
f.ntt();
for(int j = 0; j < 2 * d; j++) f[j] *= g[j];
f.intt();
for(int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
}
return res.pre(deg);
}
template <typename mint> FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
using fps = FormalPowerSeries<mint>;
assert((*this).size() == 0 || (*this)[0] == mint(0));
if(deg == -1) deg = this->size();
fps inv;
inv.reserve(deg + 1);
inv.push_back(mint(0));
inv.push_back(mint(1));
auto inplace_integral = [&](fps &F) -> void {
const int n = (int)F.size();
auto mod = mint::get_mod();
while((int)inv.size() <= n) {
int i = inv.size();
inv.push_back((-inv[mod % i]) * (mod / i));
}
F.insert(begin(F), mint(0));
for(int i = 1; i <= n; i++) F[i] *= inv[i];
};
auto inplace_diff = [](fps &F) -> void {
if(F.empty()) return;
F.erase(begin(F));
mint coeff = 1, one = 1;
for(int i = 0; i < (int)F.size(); i++) {
F[i] *= coeff;
coeff += one;
}
};
fps b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
for(int m = 2; m < deg; m *= 2) {
auto y = b;
y.resize(2 * m);
y.ntt();
z1 = z2;
fps z(m);
for(int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
z.intt();
fill(begin(z), begin(z) + m / 2, mint(0));
z.ntt();
for(int i = 0; i < m; ++i) z[i] *= -z1[i];
z.intt();
c.insert(end(c), begin(z) + m / 2, end(z));
z2 = c;
z2.resize(2 * m);
z2.ntt();
fps x(begin(*this), begin(*this) + min<int>(this->size(), m));
x.resize(m);
inplace_diff(x);
x.push_back(mint(0));
x.ntt();
for(int i = 0; i < m; ++i) x[i] *= y[i];
x.intt();
x -= b.diff();
x.resize(2 * m);
for(int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = mint(0);
x.ntt();
for(int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
x.intt();
x.pop_back();
inplace_integral(x);
for(int i = m; i < min<int>(this->size(), 2 * m); ++i) x[i] += (*this)[i];
fill(begin(x), begin(x) + m, mint(0));
x.ntt();
for(int i = 0; i < 2 * m; ++i) x[i] *= y[i];
x.intt();
b.insert(end(b), begin(x) + m, end(x));
}
return fps{begin(b), begin(b) + deg};
}
using mint = LazyMontgomeryModInt<998244353>;
using fps = FormalPowerSeries<mint>;
using vmint = vector<mint>;
Binomial<mint> binomial;
mint inv(int i) { return binomial.inv(i); }
mint C(int r, int c) { return binomial.C(r, c); }
mint P(int r, int c) { return binomial.P(r, c); }
mint fact(int r) { return binomial.fac(r); }
mint ifact(int r) { return binomial.finv(r); }
} // namespace Modular998
using namespace Modular998;
int main() {
int n, m;
cin >> n >> m;
fps f(n + 1);
f[0] = 1;
while(m--) {
int x;
cin >> x;
f[x] = 1;
}
n++;
mint r = f.pow(n)[n - 1] * inv(n);
cout << r << endl;
}