結果
| 問題 | No.2459 Stampaholic (Hard) |
| ユーザー |
 chineristAC
|
| 提出日時 | 2023-09-01 23:27:46 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
AC
|
| 実行時間 | 3,015 ms / 4,000 ms |
| コード長 | 11,910 bytes |
| コンパイル時間 | 384 ms |
| コンパイル使用メモリ | 82,560 KB |
| 実行使用メモリ | 267,312 KB |
| 最終ジャッジ日時 | 2025-01-03 12:34:27 |
| 合計ジャッジ時間 | 36,350 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge5 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 19 |
ソースコード
import sys,random
from itertools import permutations
from collections import deque
input = lambda :sys.stdin.readline().rstrip()
mi = lambda :map(int,input().split())
li = lambda :list(mi())
mod = 998244353
omega = pow(3,119,mod)
rev_omega = pow(omega,mod-2,mod)
N = 10**6
g1 = [1]*(N+1) # 元テーブル
g2 = [1]*(N+1) #逆元テーブル
inv = [1]*(N+1) #逆元テーブル計算用テーブル
for i in range( 2, N + 1 ):
g1[i]=( ( g1[i-1] * i ) % mod )
inv[i]=( ( -inv[mod % i] * (mod//i) ) % mod )
g2[i]=( (g2[i-1] * inv[i]) % mod )
inv[0]=0
_fft_mod = 998244353
_fft_imag = 911660635
_fft_iimag = 86583718
_fft_rate2 = (911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601,
842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497, 867605899)
_fft_irate2 = (86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960,
354738543, 109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951, 103369235)
_fft_rate3 = (372528824, 337190230, 454590761, 816400692, 578227951, 180142363, 83780245, 6597683, 70046822, 623238099,
183021267, 402682409, 631680428, 344509872, 689220186, 365017329, 774342554, 729444058, 102986190, 128751033, 395565204)
_fft_irate3 = (509520358, 929031873, 170256584, 839780419, 282974284, 395914482, 444904435, 72135471, 638914820, 66769500,
771127074, 985925487, 262319669, 262341272, 625870173, 768022760, 859816005, 914661783, 430819711, 272774365, 530924681)
def _butterfly(a):
n = len(a)
h = (n - 1).bit_length()
len_ = 0
while len_ < h:
if h - len_ == 1:
p = 1 << (h - len_ - 1)
rot = 1
for s in range(1 << len_):
offset = s << (h - len_)
for i in range(p):
l = a[i + offset]
r = a[i + offset + p] * rot % _fft_mod
a[i + offset] = (l + r) % _fft_mod
a[i + offset + p] = (l - r) % _fft_mod
if s + 1 != (1 << len_):
rot *= _fft_rate2[(~s & -~s).bit_length() - 1]
rot %= _fft_mod
len_ += 1
else:
p = 1 << (h - len_ - 2)
rot = 1
for s in range(1 << len_):
rot2 = rot * rot % _fft_mod
rot3 = rot2 * rot % _fft_mod
offset = s << (h - len_)
for i in range(p):
a0 = a[i + offset]
a1 = a[i + offset + p] * rot
a2 = a[i + offset + p * 2] * rot2
a3 = a[i + offset + p * 3] * rot3
a1na3imag = (a1 - a3) % _fft_mod * _fft_imag
a[i + offset] = (a0 + a2 + a1 + a3) % _fft_mod
a[i + offset + p] = (a0 + a2 - a1 - a3) % _fft_mod
a[i + offset + p * 2] = (a0 - a2 + a1na3imag) % _fft_mod
a[i + offset + p * 3] = (a0 - a2 - a1na3imag) % _fft_mod
if s + 1 != (1 << len_):
rot *= _fft_rate3[(~s & -~s).bit_length() - 1]
rot %= _fft_mod
len_ += 2
def _butterfly_inv(a):
n = len(a)
h = (n - 1).bit_length()
len_ = h
while len_:
if len_ == 1:
p = 1 << (h - len_)
irot = 1
for s in range(1 << (len_ - 1)):
offset = s << (h - len_ + 1)
for i in range(p):
l = a[i + offset]
r = a[i + offset + p]
a[i + offset] = (l + r) % _fft_mod
a[i + offset + p] = (l - r) * irot % _fft_mod
if s + 1 != (1 << (len_ - 1)):
irot *= _fft_irate2[(~s & -~s).bit_length() - 1]
irot %= _fft_mod
len_ -= 1
else:
p = 1 << (h - len_)
irot = 1
for s in range(1 << (len_ - 2)):
irot2 = irot * irot % _fft_mod
irot3 = irot2 * irot % _fft_mod
offset = s << (h - len_ + 2)
for i in range(p):
a0 = a[i + offset]
a1 = a[i + offset + p]
a2 = a[i + offset + p * 2]
a3 = a[i + offset + p * 3]
a2na3iimag = (a2 - a3) * _fft_iimag % _fft_mod
a[i + offset] = (a0 + a1 + a2 + a3) % _fft_mod
a[i + offset + p] = (a0 - a1 +
a2na3iimag) * irot % _fft_mod
a[i + offset + p * 2] = (a0 + a1 -
a2 - a3) * irot2 % _fft_mod
a[i + offset + p * 3] = (a0 - a1 -
a2na3iimag) * irot3 % _fft_mod
if s + 1 != (1 << (len_ - 1)):
irot *= _fft_irate3[(~s & -~s).bit_length() - 1]
irot %= _fft_mod
len_ -= 2
def _convolution_naive(a, b):
n = len(a)
m = len(b)
ans = [0] * (n + m - 1)
if n < m:
for j in range(m):
for i in range(n):
ans[i + j] = (ans[i + j] + a[i] * b[j]) % _fft_mod
else:
for i in range(n):
for j in range(m):
ans[i + j] = (ans[i + j] + a[i] * b[j]) % _fft_mod
return ans
def _convolution_fft(a, b):
a = a.copy()
b = b.copy()
n = len(a)
m = len(b)
z = 1 << (n + m - 2).bit_length()
a += [0] * (z - n)
_butterfly(a)
b += [0] * (z - m)
_butterfly(b)
for i in range(z):
a[i] = a[i] * b[i] % _fft_mod
_butterfly_inv(a)
a = a[:n + m - 1]
iz = pow(z, _fft_mod - 2, _fft_mod)
for i in range(n + m - 1):
a[i] = a[i] * iz % _fft_mod
return a
def _convolution_square(a):
a = a.copy()
n = len(a)
z = 1 << (2 * n - 2).bit_length()
a += [0] * (z - n)
_butterfly(a)
for i in range(z):
a[i] = a[i] * a[i] % _fft_mod
_butterfly_inv(a)
a = a[:2 * n - 1]
iz = pow(z, _fft_mod - 2, _fft_mod)
for i in range(2 * n - 1):
a[i] = a[i] * iz % _fft_mod
return a
def convolution(a, b):
"""It calculates (+, x) convolution in mod 998244353.
Given two arrays a[0], a[1], ..., a[n - 1] and b[0], b[1], ..., b[m - 1],
it calculates the array c of length n + m - 1, defined by
> c[i] = sum(a[j] * b[i - j] for j in range(i + 1)) % 998244353.
It returns an empty list if at least one of a and b are empty.
Constraints
-----------
> len(a) + len(b) <= 8388609
Complexity
----------
> O(n log n), where n = len(a) + len(b).
"""
n = len(a)
m = len(b)
if n == 0 or m == 0:
return []
if min(n, m) <= 0:
return _convolution_naive(a, b)
if a is b:
return _convolution_square(a)
return _convolution_fft(a, b)
def bostan_mori(P,Q,N):
"""
[x^N]P(x)/Q(x)を求める
"""
d = len(Q) - 1
z = 1 << (2*d).bit_length()
iz = pow(z, _fft_mod - 2, _fft_mod)
while N:
"""
P(x)/Q(x) = P(x)Q(-x)/Q(x)Q(-x)
"""
P += [0] * (z-len(P))
Q += [0] * (z-len(Q))
_butterfly(P)
_butterfly(Q)
dft_t = Q.copy()
for i in range(0,z,2):
dft_t[i],dft_t[i^1] = dft_t[i^1],dft_t[i]
P = [a*b % mod for a,b in zip(P,dft_t)]
_butterfly_inv(P)
Q = [a*b % mod for a,b in zip(Q,dft_t)]
_butterfly_inv(Q)
P = [a * iz % mod for a in P][N&1::2]
Q = [a * iz % mod for a in Q][0::2]
N >>= 1
res = P[0] * pow(Q[0],mod-2,mod) % mod
return res
def taylor_shift(f,a):
g = [f[i]*g1[i]%mod for i in range(len(f))][::-1]
e = [g2[i] for i in range(len(f))]
t = 1
for i in range(1,len(f)):
t = t * a % mod
e[i] = e[i] * t % mod
res = convolution(g,e)[:len(f)]
return [res[len(f)-1-i]*g2[i]%mod for i in range(len(f))]
def inverse(f,limit):
assert(f[0]!=0)
l = len(f)
L = 1<<((l-1).bit_length())
n = L.bit_length()-1
f = f[:L]
f+=[0]*(L-len(f))
res = [pow(f[0],mod-2,mod)]
for i in range(1,n+1):
h = convolution(res,f[:2**i])[:2**i]
h = [(-h[i]) % mod for i in range(2**i)]
h[0] = (h[0]+2) % mod
res = convolution(res,h)[:2**i]
return res[:limit]
def cmb(n,r):
if r < 0 or n < r:
return 0
return g1[n] * (g2[r] * g2[n-r] % mod) % mod
def brute(H,W,N,K):
inv = pow((H-K+1)*(W-K+1),mod-2,mod)
ans = 0
for i in range(H):
for j in range(W):
"""
h <= i < h+K and 0 <= h < H-K+1
max(i-K+1,0) <= h <= min(i,H-K)
"""
h = min(i,H-K) - max(i-K+1,0) + 1
w = min(j,W-K) - max(j-K+1,0) + 1
ans += 1 - pow(1-h*w*inv % mod,N,mod)
ans %= mod
return ans
def calc_const_h(H,W,N,K,h):
inv = pow((H-K+1)*(W-K+1),mod-2,mod)
ans = W
cosnt_W = W
for j in range(K-1):
w = min(j,W-K) - max(j-K+1,0) + 1
cosnt_W -= 1
ans -= pow(1-h*w*inv,N,mod)
ans %= mod
if (W-1-j) < K-1:
continue
w = min(W-1-j,W-K) - max(W-1-j-K+1,0) + 1
cosnt_W -= 1
ans -= pow(1-h*w*inv,N,mod)
ans %= mod
w = K
ans -= pow(1-h*w*inv,N,mod) * cosnt_W % mod
ans %= mod
return ans
def solve_easy(H,W,N,K):
inv = pow((H-K+1)*(W-K+1),mod-2,mod)
ans = 0
const_H = H
for i in range(K-1):
h = min(i,H-K) - max(i-K+1,0) + 1
const_H -= 1
ans += calc_const_h(H,W,N,K,h)
ans %= mod
if H-1-i < K-1:
continue
h = min(H-1-i,H-K) - max(H-1-i-K+1,0) + 1
const_H -= 1
ans += calc_const_h(H,W,N,K,h)
ans %= mod
h = K
ans += const_H * calc_const_h(H,W,N,K,h) % mod
ans %= mod
return ans
def calc_pow_sum(N,K):
"""
k^i for k in range(1,K+1)
を i = 1,2,...,N まで計算
"""
f = [1] * (N+2)
f[0] = 1
for n in range(1,N+2):
f[n] = f[n-1] * ((K+1) * inv[n] % mod) % mod
f[0] = 0
f = f[1:]
g = [g2[i] for i in range(N+2)]
g[0] = 0
g = g[1:]
ig = inverse(g,N+1)
h = convolution(f,ig)
h = [h[i]*g1[i]%mod for i in range(N+1)]
return [K] + h[1:]
def solve_hard(H,W,N,K):
all_inv = pow((H-K+1)*(W-K+1),mod-2,mod)
ph = min(K-1,H-K)
tmp_h = calc_pow_sum(N,ph)
tmp_h = [2*x % mod for x in tmp_h]
need = H - 2 * ph
maxi = ph + 1
for i in range(N+1):
tmp_h[i] += pow(maxi,i,mod) * need % mod
tmp_h[i] %= mod
pw = min(K-1,W-K)
tmp_w = calc_pow_sum(N,pw)
tmp_w = [2*x % mod for x in tmp_w]
need = W - 2 * pw
maxi = pw + 1
for i in range(N+1):
tmp_w[i] += pow(maxi,i,mod) * (need) % mod
tmp_w[i] %= mod
#print(tmp_h,tmp_w)
ans = H*W
for i in range(N+1):
if i & 1 == 0:
ans -= (tmp_h[i] * tmp_w[i] % mod) * (cmb(N,i) * pow(all_inv,i,mod) % mod) % mod
else:
ans += (tmp_h[i] * tmp_w[i] % mod) * (cmb(N,i) * pow(all_inv,i,mod) % mod) % mod
ans %= mod
return ans
while False:
H,W = random.randint(2,100),random.randint(2,100)
K = random.randint(1,min(H,W))
N = random.randint(1,100)
if solve_easy(H,W,N,K)!=solve_hard(H,W,N,K):
print(H,W,N,K)
check = [min(i,H-K) - max(i-K+1,0) + 1 for i in range(H)]
print(check,H-K+1)
assert False
H,W,N,K = mi()
print(solve_hard(H,W,N,K))