結果
| 問題 |
No.1856 Mex Sum 2
|
| コンテスト | |
| ユーザー |
 vwxyz
|
| 提出日時 | 2023-04-30 03:30:44 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
AC
|
| 実行時間 | 1,924 ms / 3,000 ms |
| コード長 | 16,019 bytes |
| コンパイル時間 | 370 ms |
| コンパイル使用メモリ | 82,196 KB |
| 実行使用メモリ | 97,160 KB |
| 最終ジャッジ日時 | 2024-11-18 21:00:07 |
| 合計ジャッジ時間 | 59,600 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge3 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 64 |
ソースコード
import bisect
import copy
import decimal
import fractions
import heapq
import itertools
import math
import random
import sys
import time
from collections import Counter,deque,defaultdict
from functools import lru_cache,reduce
from heapq import heappush,heappop,heapify,heappushpop,_heappop_max,_heapify_max
def _heappush_max(heap,item):
heap.append(item)
heapq._siftdown_max(heap, 0, len(heap)-1)
def _heappushpop_max(heap, item):
if heap and item < heap[0]:
item, heap[0] = heap[0], item
heapq._siftup_max(heap, 0)
return item
from math import gcd as GCD
read=sys.stdin.read
readline=sys.stdin.readline
readlines=sys.stdin.readlines
write=sys.stdout.write
def Tonelli_Shanks(N,p):
if pow(N,p>>1,p)==p-1:
retu=None
elif p%4==3:
retu=pow(N,(p+1)//4,p)
else:
for nonresidue in range(1,p):
if pow(nonresidue,p>>1,p)==p-1:
break
pp=p-1
cnt=0
while pp%2==0:
pp//=2
cnt+=1
s=pow(N,pp,p)
retu=pow(N,(pp+1)//2,p)
for i in range(cnt-2,-1,-1):
if pow(s,1<<i,p)==p-1:
s*=pow(nonresidue,p>>1+i,p)
s%=p
retu*=pow(nonresidue,p>>2+i,p)
retu%=p
return retu
mod = 998244353
imag = 911660635
iimag = 86583718
rate2 = (911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601,
842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497, 867605899)
irate2 = (86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960,
354738543, 109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951, 103369235)
rate3 = (372528824, 337190230, 454590761, 816400692, 578227951, 180142363, 83780245, 6597683, 70046822, 623238099,
183021267, 402682409, 631680428, 344509872, 689220186, 365017329, 774342554, 729444058, 102986190, 128751033, 395565204)
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 % mod
a[i + offset] = (l + r) % mod
a[i + offset + p] = (l - r) % mod
if s + 1 != 1 << len_:
rot *= rate2[(~s & -~s).bit_length() - 1]
rot %= mod
len_ += 1
else:
p = 1 << (h - len_ - 2)
rot = 1
for s in range(1 << len_):
rot2 = rot * rot % mod
rot3 = rot2 * rot % 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) % mod * imag
a[i + offset] = (a0 + a2 + a1 + a3) % mod
a[i + offset + p] = (a0 + a2 - a1 - a3) % mod
a[i + offset + p * 2] = (a0 - a2 + a1na3imag) % mod
a[i + offset + p * 3] = (a0 - a2 - a1na3imag) % mod
if s + 1 != 1 << len_:
rot *= rate3[(~s & -~s).bit_length() - 1]
rot %= 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) % mod
a[i + offset + p] = (l - r) * irot % mod
if s + 1 != (1 << (len_ - 1)):
irot *= irate2[(~s & -~s).bit_length() - 1]
irot %= mod
len_ -= 1
else:
p = 1 << (h - len_)
irot = 1
for s in range(1 << (len_ - 2)):
irot2 = irot * irot % mod
irot3 = irot2 * irot % 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) * iimag % mod
a[i + offset] = (a0 + a1 + a2 + a3) % mod
a[i + offset + p] = (a0 - a1 + a2na3iimag) * irot % mod
a[i + offset + p * 2] = (a0 + a1 - a2 - a3) * irot2 % mod
a[i + offset + p * 3] = (a0 - a1 - a2na3iimag) * irot3 % mod
if s + 1 != (1 << (len_ - 2)):
irot *= irate3[(~s & -~s).bit_length() - 1]
irot %= mod
len_ -= 2
def integrate(a):
a=a.copy()
n = len(a)
assert n > 0
a.pop()
a.insert(0, 0)
inv = [1, 1]
for i in range(2, n):
inv.append(-inv[mod%i] * (mod//i) % mod)
a[i] = a[i] * inv[i] % mod
return a
def differentiate(a):
n = len(a)
assert n > 0
for i in range(2, n):
a[i] = a[i] * i % mod
a.pop(0)
a.append(0)
return a
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]) % mod
else:
for i in range(n):
for j in range(m):
ans[i + j] = (ans[i + j] + a[i] * b[j]) % mod
return ans
def convolution_ntt(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] % mod
butterfly_inv(a)
a = a[:n + m - 1]
iz = pow(z, mod - 2, mod)
for i in range(n + m - 1):
a[i] = a[i] * iz % 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] % mod
butterfly_inv(a)
a = a[:2 * n - 1]
iz = pow(z, mod - 2, mod)
for i in range(2 * n - 1):
a[i] = a[i] * iz % 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.
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) <= 60:
return convolution_naive(a, b)
if a is b:
return convolution_square(a)
return convolution_ntt(a, b)
def inverse(a):
n = len(a)
assert n > 0 and a[0] != 0
res = [pow(a[0], mod - 2, mod)]
m = 1
while m < n:
f = a[:min(n,2*m)] + [0]*(2*m-min(n,2*m))
g = res + [0]*m
butterfly(f)
butterfly(g)
for i in range(2*m):
f[i] = f[i] * g[i] % mod
butterfly_inv(f)
f = f[m:] + [0]*m
butterfly(f)
for i in range(2*m):
f[i] = f[i] * g[i] % mod
butterfly_inv(f)
iz = pow(2*m, mod-2, mod)
iz = (-iz*iz) % mod
for i in range(m):
f[i] = f[i] * iz % mod
res += f[:m]
m <<= 1
return res[:n]
def log(a):
a = a.copy()
n = len(a)
assert n > 0 and a[0] == 1
a_inv = inverse(a)
a=differentiate(a)
a = convolution(a, a_inv)[:n]
a=integrate(a)
return a
def exp(a):
a = a.copy()
n = len(a)
assert n > 0 and a[0] == 0
g = [1]
a[0] = 1
h_drv = a.copy()
h_drv=differentiate(h_drv)
m = 1
while m < n:
f_fft = a[:m] + [0] * m
butterfly(f_fft)
if m > 1:
_f = [f_fft[i] * g_fft[i] % mod for i in range(m)]
butterfly_inv(_f)
_f = _f[m // 2:] + [0] * (m // 2)
butterfly(_f)
for i in range(m):
_f[i] = _f[i] * g_fft[i] % mod
butterfly_inv(_f)
_f = _f[:m//2]
iz = pow(m, mod - 2, mod)
iz *= -iz
iz %= mod
for i in range(m//2):
_f[i] = _f[i] * iz % mod
g.extend(_f)
t = a[:m]
t=differentiate(t)
r = h_drv[:m - 1]
r.append(0)
butterfly(r)
for i in range(m):
r[i] = r[i] * f_fft[i] % mod
butterfly_inv(r)
im = pow(-m, mod - 2, mod)
for i in range(m):
r[i] = r[i] * im % mod
for i in range(m):
t[i] = (t[i] + r[i]) % mod
t = [t[-1]] + t[:-1]
t += [0] * m
butterfly(t)
g_fft = g + [0] * (2 * m - len(g))
butterfly(g_fft)
for i in range(2 * m):
t[i] = t[i] * g_fft[i] % mod
butterfly_inv(t)
t = t[:m]
i2m = pow(2 * m, mod - 2, mod)
for i in range(m):
t[i] = t[i] * i2m % mod
v = a[m:min(n, 2 * m)]
v += [0] * (m - len(v))
t = [0] * (m - 1) + t + [0]
t=integrate(t)
for i in range(m):
v[i] = (v[i] - t[m + i]) % mod
v += [0] * m
butterfly(v)
for i in range(2 * m):
v[i] = v[i] * f_fft[i] % mod
butterfly_inv(v)
v = v[:m]
i2m = pow(2 * m, mod - 2, mod)
for i in range(m):
v[i] = v[i] * i2m % mod
for i in range(min(n - m, m)):
a[m + i] = v[i]
m *= 2
return a
def power(a,k):
n = len(a)
assert n>0
if k==0:
return [1]+[0]*(n-1)
l = 0
while l < len(a) and not a[l]:
l += 1
if l * k >= n:
return [0] * n
ic = pow(a[l], mod - 2, mod)
pc = pow(a[l], k, mod)
a = log([a[i] * ic % mod for i in range(l, len(a))])
for i in range(len(a)):
a[i] = a[i] * k % mod
a = exp(a)
for i in range(len(a)):
a[i] = a[i] * pc % mod
a = [0] * (l * k) + a[:n - l * k]
return a
def sqrt(a):
if len(a) == 0:
return []
if a[0] == 0:
for d in range(1, len(a)):
if a[d]:
if d & 1:
return None
if len(a) - 1 < d // 2:
break
res=sqrt(a[d:]+[0]*(d//2))
if res == None:
return None
res = [0]*(d//2)+res
return res
return [0]*len(a)
sqr = Tonelli_Shanks(a[0],mod)
if sqr == None:
return None
T = [0] * (len(a))
T[0] = sqr
res = T.copy()
T[0] = pow(sqr,mod-2,mod) #T:res^{-1}
m = 1
two_inv = (mod + 1) // 2
F = [sqr]
while m <= len(a) - 1:
for i in range(m):
F[i] *= F[i]
F[i] %= mod
butterfly_inv(F)
iz = pow(m, mod-2, mod)
for i in range(m):
F[i] = F[i] * iz % mod
delta = [0] * (2 * m)
for i in range(m):
delta[i + m] = F[i] - a[i] - (a[i + m] if i+m<len(a) else 0)
butterfly(delta)
G = [0] * (2 * m)
for i in range(m):
G[i] = T[i]
butterfly(G)
for i in range(2 * m):
delta[i] *= G[i]
delta[i] %= mod
butterfly_inv(delta)
iz = pow(2*m, mod-2, mod)
for i in range(2*m):
delta[i] = delta[i] * iz % mod
for i in range(m, min(2 * m, len(a))):
res[i] = -delta[i] * two_inv%mod
res[i]%=mod
if 2 * m > len(a) - 1:
break
F = res[:2 * m]
butterfly(F)
eps = [F[i] * G[i] % mod for i in range(2 * m)]
butterfly_inv(eps)
for i in range(m):
eps[i] = 0
iz = pow(2*m, mod-2, mod)
for i in range(m,2*m):
eps[i] = eps[i] * iz % mod
butterfly(eps)
for i in range(2 * m):
eps[i] *= G[i]
eps[i] %= mod
butterfly_inv(eps)
for i in range(m, 2 * m):
T[i] = -eps[i]*iz
T[i]%=mod
iz = iz*iz % mod
m <<= 1
return res
def Extended_Euclid(n,m):
stack=[]
while m:
stack.append((n,m))
n,m=m,n%m
if n>=0:
x,y=1,0
else:
x,y=-1,0
for i in range(len(stack)-1,-1,-1):
n,m=stack[i]
x,y=y,x-(n//m)*y
return x,y
class MOD:
def __init__(self,p,e=None):
self.p=p
self.e=e
if self.e==None:
self.mod=self.p
else:
self.mod=self.p**self.e
def Pow(self,a,n):
a%=self.mod
if n>=0:
return pow(a,n,self.mod)
else:
assert math.gcd(a,self.mod)==1
x=Extended_Euclid(a,self.mod)[0]
return pow(x,-n,self.mod)
def Build_Fact(self,N):
assert N>=0
self.factorial=[1]
if self.e==None:
for i in range(1,N+1):
self.factorial.append(self.factorial[-1]*i%self.mod)
else:
self.cnt=[0]*(N+1)
for i in range(1,N+1):
self.cnt[i]=self.cnt[i-1]
ii=i
while ii%self.p==0:
ii//=self.p
self.cnt[i]+=1
self.factorial.append(self.factorial[-1]*ii%self.mod)
self.factorial_inve=[None]*(N+1)
self.factorial_inve[-1]=self.Pow(self.factorial[-1],-1)
for i in range(N-1,-1,-1):
ii=i+1
while ii%self.p==0:
ii//=self.p
self.factorial_inve[i]=(self.factorial_inve[i+1]*ii)%self.mod
def Fact(self,N):
if N<0:
return 0
retu=self.factorial[N]
if self.e!=None and self.cnt[N]:
retu*=pow(self.p,self.cnt[N],self.mod)%self.mod
retu%=self.mod
return retu
def Fact_Inve(self,N):
if self.e!=None and self.cnt[N]:
return None
return self.factorial_inve[N]
def Comb(self,N,K,divisible_count=False):
if K<0 or K>N:
return 0
retu=self.factorial[N]*self.factorial_inve[K]%self.mod*self.factorial_inve[N-K]%self.mod
if self.e!=None:
cnt=self.cnt[N]-self.cnt[N-K]-self.cnt[K]
if divisible_count:
return retu,cnt
else:
retu*=pow(self.p,cnt,self.mod)
retu%=self.mod
return retu
N,M=map(int,readline().split())
ans=0
mod=998244353
MD=MOD(mod)
MD.Build_Fact(N+1)
inverse_exp=inverse([MD.Fact_Inve(m) for m in range(N)])
for n in range(1,N+1):
f=[MD.Pow(m+1,n)*MD.Fact_Inve(m+1)%mod for m in range(min(n,M+1))]
dp=convolution(f,inverse_exp[:min(n,M+1)])[:min(n,M+1)]
for m in range(min(n,M+1)):
dp[m]*=MD.Fact(m+1)
dp[m]%=mod
ans+=(m+1)*dp[m]%mod*MD.Comb(N,n)%mod*MD.Pow(M+1+max(M-m-1,0),N-n)%mod
ans%=mod
print(ans)