結果
| 問題 | No.2582 Random Average^K |
| ユーザー |
 mkawa2
|
| 提出日時 | 2023-12-10 16:47:20 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 13,118 bytes |
| コンパイル時間 | 336 ms |
| コンパイル使用メモリ | 82,220 KB |
| 実行使用メモリ | 470,420 KB |
| 最終ジャッジ日時 | 2024-09-27 04:08:13 |
| 合計ジャッジ時間 | 4,841 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge1 |
（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 | 42 ms
61,184 KB |
| testcase_01 | AC | 41 ms
55,424 KB |
| testcase_02 | AC | 83 ms
76,388 KB |
| testcase_03 | AC | 57 ms
67,200 KB |
| testcase_04 | AC | 49 ms
62,976 KB |
| testcase_05 | AC | 58 ms
67,072 KB |
| testcase_06 | AC | 127 ms
78,196 KB |
| testcase_07 | AC | 116 ms
77,572 KB |
| testcase_08 | AC | 126 ms
78,460 KB |
| testcase_09 | TLE | - |
| testcase_10 | -- | - |
| testcase_11 | -- | - |
| testcase_12 | -- | - |
| testcase_13 | -- | - |
| testcase_14 | -- | - |
| testcase_15 | -- | - |
| testcase_16 | -- | - |
| testcase_17 | -- | - |
ソースコード
import sys
# sys.setrecursionlimit(1000005)
# sys.set_int_max_str_digits(200005)
int1 = lambda x: int(x)-1
pDB = lambda *x: print(*x, end="\n", file=sys.stderr)
p2D = lambda x: print(*x, sep="\n", end="\n\n", file=sys.stderr)
def II(): return int(sys.stdin.readline())
def LI(): return list(map(int, sys.stdin.readline().split()))
def LLI(rows_number): return [LI() for _ in range(rows_number)]
def LI1(): return list(map(int1, sys.stdin.readline().split()))
def LLI1(rows_number): return [LI1() for _ in range(rows_number)]
def SI(): return sys.stdin.readline().rstrip()
dij = [(0, 1), (-1, 0), (0, -1), (1, 0)]
# dij = [(0, 1), (-1, 0), (0, -1), (1, 0), (1, 1), (1, -1), (-1, 1), (-1, -1)]
# inf = -1-(-1 << 31)
inf = -1-(-1 << 63)
# md = 10**9+7
md = 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 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
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%md
a[i+offset] = (l+r)%md
a[i+offset+p] = (l-r)%md
if s+1 != 1 << len_:
rot *= rate2[(~s & -~s).bit_length()-1]
rot %= md
len_ += 1
else:
p = 1 << (h-len_-2)
rot = 1
for s in range(1 << len_):
rot2 = rot*rot%md
rot3 = rot2*rot%md
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)%md*imag
a[i+offset] = (a0+a2+a1+a3)%md
a[i+offset+p] = (a0+a2-a1-a3)%md
a[i+offset+p*2] = (a0-a2+a1na3imag)%md
a[i+offset+p*3] = (a0-a2-a1na3imag)%md
if s+1 != 1 << len_:
rot *= rate3[(~s & -~s).bit_length()-1]
rot %= md
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)%md
a[i+offset+p] = (l-r)*irot%md
if s+1 != (1 << (len_-1)):
irot *= irate2[(~s & -~s).bit_length()-1]
irot %= md
len_ -= 1
else:
p = 1 << (h-len_)
irot = 1
for s in range(1 << (len_-2)):
irot2 = irot*irot%md
irot3 = irot2*irot%md
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%md
a[i+offset] = (a0+a1+a2+a3)%md
a[i+offset+p] = (a0-a1+a2na3iimag)*irot%md
a[i+offset+p*2] = (a0+a1-a2-a3)*irot2%md
a[i+offset+p*3] = (a0-a1-a2na3iimag)*irot3%md
if s+1 != (1 << (len_-2)):
irot *= irate3[(~s & -~s).bit_length()-1]
irot %= md
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[md%i]*(md//i)%md)
a[i] = a[i]*inv[i]%md
return a
def differentiate(a):
n = len(a)
assert n > 0
for i in range(2, n):
a[i] = a[i]*i%md
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])%md
else:
for i in range(n):
for j in range(m):
ans[i+j] = (ans[i+j]+a[i]*b[j])%md
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]%md
butterfly_inv(a)
a = a[:n+m-1]
iz = pow(z, md-2, md)
for i in range(n+m-1):
a[i] = a[i]*iz%md
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]%md
butterfly_inv(a)
a = a[:2*n-1]
iz = pow(z, md-2, md)
for i in range(2*n-1):
a[i] = a[i]*iz%md
return a
def convolution(a, b):
"""It calculates (+, x) convolution in md 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], md-2, md)]
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]%md
butterfly_inv(f)
f = f[m:]+[0]*m
butterfly(f)
for i in range(2*m):
f[i] = f[i]*g[i]%md
butterfly_inv(f)
iz = pow(2*m, md-2, md)
iz = (-iz*iz)%md
for i in range(m):
f[i] = f[i]*iz%md
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]%md 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]%md
butterfly_inv(_f)
_f = _f[:m//2]
iz = pow(m, md-2, md)
iz *= -iz
iz %= md
for i in range(m//2):
_f[i] = _f[i]*iz%md
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]%md
butterfly_inv(r)
im = pow(-m, md-2, md)
for i in range(m):
r[i] = r[i]*im%md
for i in range(m):
t[i] = (t[i]+r[i])%md
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]%md
butterfly_inv(t)
t = t[:m]
i2m = pow(2*m, md-2, md)
for i in range(m):
t[i] = t[i]*i2m%md
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])%md
v += [0]*m
butterfly(v)
for i in range(2*m):
v[i] = v[i]*f_fft[i]%md
butterfly_inv(v)
v = v[:m]
i2m = pow(2*m, md-2, md)
for i in range(m):
v[i] = v[i]*i2m%md
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], md-2, md)
pc = pow(a[l], k, md)
a = log([a[i]*ic%md for i in range(l, len(a))])
for i in range(len(a)):
a[i] = a[i]*k%md
a = exp(a)
for i in range(len(a)):
a[i] = a[i]*pc%md
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], md)
if sqr == None:
return None
T = [0]*(len(a))
T[0] = sqr
res = T.copy()
T[0] = pow(sqr, md-2, md) # T:res^{-1}
m = 1
two_inv = (md+1)//2
F = [sqr]
while m <= len(a)-1:
for i in range(m):
F[i] *= F[i]
F[i] %= md
butterfly_inv(F)
iz = pow(m, md-2, md)
for i in range(m):
F[i] = F[i]*iz%md
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] %= md
butterfly_inv(delta)
iz = pow(2*m, md-2, md)
for i in range(2*m):
delta[i] = delta[i]*iz%md
for i in range(m, min(2*m, len(a))):
res[i] = -delta[i]*two_inv%md
res[i] %= md
if 2*m > len(a)-1:
break
F = res[:2*m]
butterfly(F)
eps = [F[i]*G[i]%md for i in range(2*m)]
butterfly_inv(eps)
for i in range(m):
eps[i] = 0
iz = pow(2*m, md-2, md)
for i in range(m, 2*m):
eps[i] = eps[i]*iz%md
butterfly(eps)
for i in range(2*m):
eps[i] *= G[i]
eps[i] %= md
butterfly_inv(eps)
for i in range(m, 2*m):
T[i] = -eps[i]*iz
T[i] %= md
iz = iz*iz%md
m <<= 1
return res
def inv_arr(n):
if n < 1: return []
res = [-1, 1]
for i in range(2, n+1):
res.append(res[md%i]*(md-md//i)%md)
return res
n,k=LI()
inv=inv_arr(k+2)[1:]
n_max = k+1
fac = [1]
for i in range(1, n_max+1): fac.append(fac[-1]*i%md)
ifac = [1]*(n_max+1)
ifac[n_max] = pow(fac[n_max], md-2, md)
for i in range(n_max-1, 1, -1):
ifac[i] = ifac[i+1]*(i+1)%md
inv[i]*=ifac[i]
inv[i]%=md
inv=power(inv,n)
ans=inv[k]*pow(n,k*(md-2),md)%md*fac[k]%md
print(ans)