結果
| 問題 | No.2936 Sum of Square of Mex |
| ユーザー |
 PNJ
|
| 提出日時 | 2024-10-13 02:22:07 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 11,111 bytes |
| コンパイル時間 | 143 ms |
| コンパイル使用メモリ | 82,396 KB |
| 実行使用メモリ | 409,924 KB |
| 最終ジャッジ日時 | 2024-10-13 02:22:14 |
| 合計ジャッジ時間 | 6,079 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / judge5 |
（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 | 107 ms
96,100 KB |
| testcase_01 | AC | 107 ms
88,048 KB |
| testcase_02 | AC | 1,371 ms
246,720 KB |
| testcase_03 | AC | 196 ms
102,412 KB |
| testcase_04 | AC | 145 ms
99,992 KB |
| testcase_05 | AC | 134 ms
100,268 KB |
| testcase_06 | AC | 108 ms
87,616 KB |
| testcase_07 | AC | 160 ms
100,620 KB |
| testcase_08 | TLE | - |
| testcase_09 | -- | - |
| testcase_10 | -- | - |
| testcase_11 | -- | - |
| testcase_12 | -- | - |
| testcase_13 | -- | - |
| testcase_14 | -- | - |
| testcase_15 | -- | - |
| testcase_16 | -- | - |
| testcase_17 | -- | - |
| testcase_18 | -- | - |
| testcase_19 | -- | - |
| testcase_20 | -- | - |
| testcase_21 | -- | - |
| testcase_22 | -- | - |
| testcase_23 | -- | - |
| testcase_24 | -- | - |
| testcase_25 | -- | - |
| testcase_26 | -- | - |
| testcase_27 | -- | - |
| testcase_28 | -- | - |
ソースコード
mod = 998244353
n = 10**6
inv = [1 for j in range(n+1)]
for a in range(2,n+1):
# ax + py = 1 <=> rx + p(-x-qy) = -q => x = -(inv[r]) * (p//a) (r = p % a)
res = (mod - inv[mod%a]) * (mod // a)
inv[a] = res % mod
fact = [1 for i in range(n+1)]
for i in range(1,n+1):
fact[i] = fact[i-1]*i % mod
fact_inv = [1 for i in range(n+1)]
fact_inv[-1] = pow(fact[-1],mod-2,mod)
for i in range(n,0,-1):
fact_inv[i-1] = fact_inv[i]*i % mod
def binom(n,r):
if n < r or n < 0 or r < 0:
return 0
res = fact_inv[n-r] * fact_inv[r] % mod
res *= fact[n]
res %= mod
return res
NTT_friend = [120586241,167772161,469762049,754974721,880803841,924844033,943718401,998244353,1045430273,1051721729,1053818881]
NTT_dict = {}
for i in range(len(NTT_friend)):
NTT_dict[NTT_friend[i]] = i
NTT_info = [[20,74066978],[25,17],[26,30],[24,362],[23,211],[21,44009197],[22,663003469],[23,31],[20,363],[20,330],[20,2789]]
def popcount(n):
c=(n&0x5555555555555555)+((n>>1)&0x5555555555555555)
c=(c&0x3333333333333333)+((c>>2)&0x3333333333333333)
c=(c&0x0f0f0f0f0f0f0f0f)+((c>>4)&0x0f0f0f0f0f0f0f0f)
c=(c&0x00ff00ff00ff00ff)+((c>>8)&0x00ff00ff00ff00ff)
c=(c&0x0000ffff0000ffff)+((c>>16)&0x0000ffff0000ffff)
c=(c&0x00000000ffffffff)+((c>>32)&0x00000000ffffffff)
return c
def topbit(n):
h = n.bit_length()
h -= 1
return h
def prepared_fft(mod = 998244353):
rank2 = NTT_info[NTT_dict[mod]][0]
root,iroot = [0] * 30,[0] * 30
rate2,irate2= [0] * 30,[0] * 30
rate3,irate3= [0] * 30,[0] * 30
root[rank2] = NTT_info[NTT_dict[mod]][1]
iroot[rank2] = pow(root[rank2],mod - 2,mod)
for i in range(rank2-1,-1,-1):
root[i] = root[i+1] * root[i+1] % mod
iroot[i] = iroot[i+1] * iroot[i+1] % mod
prod,iprod = 1,1
for i in range(rank2-1):
rate2[i] = root[i + 2] * prod % mod
irate2[i] = iroot[i + 2] * iprod % mod
prod = prod * iroot[i + 2] % mod
iprod = iprod * root[i + 2] % mod
prod,iprod = 1,1
for i in range(rank2-2):
rate3[i] = root[i + 3] * prod % mod
irate3[i] = iroot[i + 3] * iprod % mod
prod = prod * iroot[i + 3] % mod
iprod = iprod * root[i + 3] % mod
return root,iroot,rate2,irate2,rate3,irate3
root,iroot,rate2,irate2,rate3,irate3 = prepared_fft()
def ntt(a):
n = len(a)
h = topbit(n)
assert (n == 1 << h)
le = 0
while le < h:
if h - le == 1:
p = 1 << (h - le - 1)
rot = 1
for s in range(1 << le):
offset = s << (h - le)
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
rot = rot * rate2[topbit(~s & -~s)] % mod
le += 1
else:
p = 1 << (h - le - 2)
rot,imag = 1,root[2]
for s in range(1 << le):
rot2 = rot * rot % mod
rot3 = rot2 * rot % mod
offset = s << (h - le)
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
rot = rot * rate3[topbit(~s & -~s)] % mod
le += 2
def intt(a):
n = len(a)
h = topbit(n)
assert (n == 1 << h)
coef = pow(n,mod - 2,mod)
for i in range(n):
a[i] = a[i] * coef % mod
le = h
while le:
if le == 1:
p = 1 << (h - le)
irot = 1
for s in range(1 << (le - 1)):
offset = s << (h - le + 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
irot = irot * irate2[topbit(~s & -~s)] % mod
le -= 1
else:
p = 1 << (h - le)
irot,iimag = 1,iroot[2]
for s in range(1 << (le - 2)):
irot2 = irot * irot % mod
irot3 = irot2 * irot % mod
offset = s << (h - le + 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
irot *= irate3[topbit(~s & -~s)]
irot %= mod
le -= 2
def convolute_naive(a,b):
res = [0] * (len(a) + len(b) - 1)
for i in range(len(a)):
for j in range(len(b)):
res[i+j] = (res[i+j] + a[i] * b[j] % mod) % mod
return res
def convolute(a,b):
s = a[:]
t = b[:]
n = len(s)
m = len(t)
if min(n,m) <= 60:
return convolute_naive(s,t)
le = 1
while le < n + m - 1:
le *= 2
s += [0] * (le - n)
t += [0] * (le - m)
ntt(s)
ntt(t)
for i in range(le):
s[i] = s[i] * t[i] % mod
intt(s)
s = s[:n + m - 1]
return s
def fps_inv(f,deg = -1):
assert (f[0] != 0)
if deg == -1:
deg = len(f)
res = [0] * deg
res[0] = pow(f[0],mod-2,mod)
d = 1
while d < deg:
a = [0] * (d << 1)
tmp = min(len(f),d << 1)
a[:tmp] = f[:tmp]
b = [0] * (d << 1)
b[:d] = res[:d]
ntt(a)
ntt(b)
for i in range(d << 1):
a[i] = a[i] * b[i] % mod
intt(a)
a[:d] = [0] * d
ntt(a)
for i in range(d << 1):
a[i] = a[i] * b[i] % mod
intt(a)
for j in range(d,min(d << 1,deg)):
if a[j]:
res[j] = mod - a[j]
else:
res[j] = 0
d <<= 1
return res
def fps_div(f,g):
n,m = len(f),len(g)
if n < m:
return [],f
rev_f = f[:]
rev_f = rev_f[::-1]
rev_g = g[:]
rev_g = rev_g[::-1]
rev_q = convolute(rev_f,fps_inv(rev_g,n-m+1))[:n-m+1]
q = rev_q[:]
q = q[::-1]
p = convolute(g,q)
r = f[:]
for i in range(min(len(p),len(r))):
r[i] -= p[i]
r[i] %= mod
while len(r):
if r[-1] != 0:
break
r.pop()
return q,r
def fps_add(f,g):
n = max(len(f),len(g))
res = [0] * n
for i in range(len(f)):
res[i] = f[i]
for i in range(len(g)):
res[i] = (res[i] + g[i]) % mod
return res
def fps_diff(f):
if len(f) <= 1:
return [0]
res = []
for i in range(1,len(f)):
res.append(i * f[i] % mod)
return res
def fps_integrate(f):
n = len(f)
res = [0] * (n + 1)
for i in range(n):
res[i+1] = pow(i + 1,mod-2,mod) * f[i] % mod
return res
def fps_log(f,deg = -1):
assert (f[0] == 1)
if deg == -1:
deg = len(f)
res = convolute(fps_diff(f),fps_inv(f,deg))
res = fps_integrate(res)
return res[:deg]
def fps_exp(f,deg = -1):
assert (f[0] == 0)
if deg == -1:
deg = len(f)
res = [1,0]
if len(f) > 1:
res[1] = f[1]
g = [1]
p = []
q = [1,1]
m = 2
while m < deg:
y = res + [0]*m
ntt(y)
p = q[:]
z = [y[i] * p[i] for i in range(len(p))]
intt(z)
z[:m >> 1] = [0] * (m >> 1)
ntt(z)
for i in range(len(p)):
z[i] = z[i] * (-p[i]) % mod
intt(z)
g[m >> 1:] = z[m >> 1:]
q = g + [0] * m
ntt(q)
tmp = min(len(f),m)
x = f[:tmp] + [0] * (m - tmp)
x = fps_diff(x)
x.append(0)
ntt(x)
for i in range(len(x)):
x[i] = x[i] * y[i] % mod
intt(x)
for i in range(len(res)):
if i == 0:
continue
x[i-1] -= res[i] * i % mod
x += [0] * m
for i in range(m-1):
x[m+i],x[i] = x[i],0
ntt(x)
for i in range(len(q)):
x[i] = x[i] * q[i] % mod
intt(x)
x.pop()
x = fps_integrate(x)
x[:m] = [0] * m
for i in range(m,min(len(f),m << 1)):
x[i] += f[i]
ntt(x)
for i in range(len(y)):
x[i] = x[i] * y[i] % mod
intt(x)
res[m:] = x[m:]
m <<= 1
return res[:deg]
def fps_pow(f,k,deg = -1):
if deg == -1:
deg = len(f)
if k == 0:
return [1] + [0] * (deg - 1)
while len(f) < deg:
f.append(0)
p = 0
while p < deg:
if f[p]:
break
p += 1
if p * k >= deg:
return [0] * deg
a = f[p]
g = [0 for _ in range(deg - p)]
a_inv = pow(a,mod-2,mod)
for i in range(deg - p):
g[i] = f[i + p] * a_inv % mod
g = fps_log(g)
for i in range(deg-p):
g[i] = g[i] * k % mod
g = fps_exp(g)
a = pow(a,k,mod)
res = [0] * deg
for i in range(deg):
j = i + p * k
if j >= deg:
break
res[j] = g[i] * a % mod
return res
def composition(f,g):
def rec(n,k,Q,f):
if n == 1:
# part A
p = [0 for _ in range(2 * k)]
ff = f[:]
ff = ff[::-1]
for i in range(k):
p[2 * i] = ff[i]
return p
R = [0 for _ in range(2*n*k)]
for i in range(2*n*k):
R[i] = Q[i]
if i % 2:
R[i] = (mod - R[i]) % mod
QQ = convolute(Q,R)
for i in range(4*n*k - len(QQ)):
QQ.append(0)
for i in range(2*n*k):
QQ[2*n*k + i] = (QQ[2*n*k + i] + Q[i] + R[i]) % mod
nxt_Q = [0 for _ in range(2 * n * k)]
for j in range(2*k):
for i in range(n // 2):
nxt_Q[n*j + i] = QQ[2*n*j + 2*i]
nxt_p = rec(n // 2,k * 2,nxt_Q,f)
# part B
pq = [0 for _ in range(4 * n * k)]
for j in range(2 * k):
for i in range(n // 2):
pq[2*n*j + 2*i+1] = (pq[2*n*j + 2*i+1] + nxt_p[n*j + i]) % mod
p = [0 for _ in range(2 * n * k)]
for i in range(2 * n * k):
p[i] = (p[i] + pq[2 * n * k + i]) % mod
pq.pop()
x = transposed_convolute(pq,R)
for i in range(2*n*k):
p[i] = (p[i] + x[i]) % mod
return p
n = len(g)
assert (popcount(n) == 1)
k = 1
Q = [0 for _ in range(2 * n)]
for i in range(n):
Q[i] = (mod - g[i]) % mod
p = rec(n,k,Q,f)
# part C
output = [0 for i in range(n)]
for i in range(n):
output[i] = p[i]
output = output[::-1]
return output
def transposed_ntt(a):
b = a[:]
intt(b)
b = [b[0]] + b[1:][::-1]
for i in range(len(a)):
a[i] = b[i] * len(a) % mod
return a
def transposed_ntt_inv(a):
b = [a[0]] + a[1:][::-1]
ntt(b)
n = len(b)
n_inv = pow(n,mod - 2,mod)
for i in range(len(b)):
a[i] = b[i] * n_inv % mod
return
def transposed_convolute(aa,bb):
a,b = aa[:],bb[:]
n,m = len(a),len(b)
L = 1
while L < max(len(a),len(b)):
L *= 2
for i in range(L - n):
a.append(0)
for i in range(L - m):
b.append(0)
ntt(b)
transposed_ntt_inv(a)
for i in range(L):
a[i] = a[i] * b[i] % mod
transposed_ntt(a)
a = a[:(n-m+1)]
return a
N,M = map(int,input().split())
f = [0]
g = [1]
for i in range(1,N + 1):
f.append(i * i % mod)
if i > M:
f[-1] = 0
g.append(fact_inv[i])
gg = g[:]
gg[0] = 0
gg = convolute(gg,fps_inv(g))[:N + 1]
while popcount(len(f)) != 1:
f.append(0)
gg.append(0)
f = composition(f,gg)[:N + 1]
f = convolute(f,fps_pow(g,M))[:N + 1]
ans = f[N]
if M == 0:
ans = 0
g[0] = 0
g = fps_pow(g,M + 1)
ans = (ans + g[N] * ((M + 1) * (M + 1) % mod) % mod) % mod
ans = ans * fact[N] % mod
print(ans)