結果

問題 No.2084 Mex Subset For All Sequences
ユーザー taiga0629kyoprotaiga0629kyopro
提出日時 2022-09-08 21:29:17
言語 PyPy3
(7.3.15)
結果
AC  
実行時間 225 ms / 2,000 ms
コード長 5,872 bytes
コンパイル時間 1,336 ms
コンパイル使用メモリ 86,660 KB
実行使用メモリ 91,876 KB
最終ジャッジ日時 2023-08-23 17:56:36
合計ジャッジ時間 6,057 ms
ジャッジサーバーID
（参考情報） 		judge14 / judge12
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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 88 ms
80,688 KB
testcase_01 AC 86 ms
80,684 KB
testcase_02 AC 85 ms
80,532 KB
testcase_03 AC 88 ms
80,336 KB
testcase_04 AC 86 ms
80,576 KB
testcase_05 AC 86 ms
80,572 KB
testcase_06 AC 86 ms
80,464 KB
testcase_07 AC 87 ms
80,448 KB
testcase_08 AC 145 ms
85,220 KB
testcase_09 AC 171 ms
86,912 KB
testcase_10 AC 213 ms
91,224 KB
testcase_11 AC 129 ms
83,512 KB
testcase_12 AC 137 ms
84,788 KB
testcase_13 AC 137 ms
84,148 KB
testcase_14 AC 130 ms
83,552 KB
testcase_15 AC 211 ms
91,216 KB
testcase_16 AC 209 ms
91,116 KB
testcase_17 AC 168 ms
87,076 KB
testcase_18 AC 214 ms
91,700 KB
testcase_19 AC 221 ms
91,876 KB
testcase_20 AC 218 ms
91,852 KB
testcase_21 AC 219 ms
91,376 KB
testcase_22 AC 216 ms
91,564 KB
testcase_23 AC 219 ms
91,752 KB
testcase_24 AC 217 ms
91,696 KB
testcase_25 AC 222 ms
91,744 KB
testcase_26 AC 217 ms
91,716 KB
testcase_27 AC 225 ms
91,532 KB
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ソースコード

diff # 




# AtCoder Libary v1.4 を python に移植したもの
# https://github.com/atcoder/ac-library/blob/master/atcoder/convolution.hpp
#https://judge.yosupo.jp/submission/55648

MOD = 998244353
IMAG = 911660635
IIMAG = 86583718
rate2 = (0, 911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601, 842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497, 867605899, 0)
irate2 = (0, 86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960, 354738543, 109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951, 103369235, 0)
rate3 = (0, 372528824, 337190230, 454590761, 816400692, 578227951, 180142363, 83780245, 6597683, 70046822, 623238099, 183021267, 402682409, 631680428, 344509872, 689220186, 365017329, 774342554, 729444058, 102986190, 128751033, 395565204, 0)
irate3 = (0, 509520358, 929031873, 170256584, 839780419, 282974284, 395914482, 444904435, 72135471, 638914820, 66769500, 771127074, 985925487, 262319669, 262341272, 625870173, 768022760, 859816005, 914661783, 430819711, 272774365, 530924681, 0)

def butterfly(a):
  n = len(a)
  h = (n - 1).bit_length()
  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
          a[i + offset] = (l + r) % MOD
          a[i + offset + p] = (l - r) % MOD
        rot *= rate2[(~s & -~s).bit_length()]
        rot %= MOD
      le += 1
    else:
      p = 1 << (h - le - 2)
      rot = 1
      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 *= rate3[(~s & -~s).bit_length()]
        rot %= MOD
      le += 2

def butterfly_inv(a):
  n = len(a)
  h = (n - 1).bit_length()
  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 *= irate2[(~s & -~s).bit_length()]
        irot %= MOD
      le -= 1
    else:
      p = 1 << (h - le)
      irot = 1
      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[(~s & -~s).bit_length()]
        irot %= MOD
      le -= 2

def multiply(s, t):
  n = len(s)
  m = len(t)
  if min(n, m) <= 60:
    a = [0] * (n + m - 1)
    for i in range(n):
      if i % 8 == 0:
        for j in range(m):
          a[i + j] += s[i] * t[j]
          a[i + j] %= MOD
      else:
        for j in range(m):
          a[i + j] += s[i] * t[j]
    return [x % MOD for x in a]
  a = s.copy()
  b = t.copy()
  z = 1 << (n + m - 2).bit_length()
  a += [0] * (z - n)
  b += [0] * (z - m)
  butterfly(a)
  butterfly(b)
  for i in range(z):
    a[i] *= b[i]
    a[i] %= MOD
  butterfly_inv(a)
  a = a[:n + m - 1]
  iz = pow(z, MOD - 2, MOD)
  return [v * iz % MOD for v in a]




#############################
#############
cnb_max=3*10**5
mod=998244353
#############

kai=[1]*(cnb_max+1)
rkai=[1]*(cnb_max+1)
for i in range(cnb_max):
    kai[i+1]=kai[i]*(i+1)%mod

rkai[cnb_max]=pow(kai[cnb_max],mod-2,mod)
for i in range(cnb_max):
    rkai[cnb_max-1-i]=rkai[cnb_max-i]*(cnb_max-i)%mod

def cnb(x,y):
    if y>x:
        return 0
    if x<0:return 0
    if y<0:return 0
    return (kai[x]*rkai[y]%mod)*rkai[x-y]%mod


def inv(n):
    return kai[n-1]*rkai[n]%mod

##################################


def sol(N,M):
    fx=[0]*(M+3)
    gx=[0]*(M+3)
    for j in range(M+2):
        fx[j]=pow(2*M-j,N,mod)*pow(-1,j%2,mod)%mod
        fx[j]*=rkai[j]
        fx[j]%=mod
        gx[j]=rkai[j]
    fg=multiply(fx,gx)
    F=[0]*(M+5)
    F[-1]=(pow(2,N,mod)-1)*pow(M,N,mod)%mod

    for Q in range(M):
        F[Q]=fg[Q+1]*kai[Q+1]%mod
        F[Q]%=mod
    ans=0
    for P in range(M+1):
        ans+=P*(F[P-1]-F[P])
        ans%=mod
    return ans


def naiveMS(N,A):
    #Mex Subset naive
    ans=0
    for bit in range(1,2**N):
        s=set()
        for i in range(N):
            if (bit>>i)&1:s.add(A[i])
        mex=-1
        for x in range(10**9):
            if x not in s:
                mex=x
                break
        ans+=mex
    return ans%mod


def naive(N,M):
    ans = 0
    AS = []

    def dfs(A):
        if len(A) == N:
            AS.append(A[:])
            return
        for x in range(M):
            A.append(x)
            dfs(A)
            A.pop()

    dfs([])
    for A in AS:
        ans += naiveMS(N,A)
    return ans % mod

N,M=map(int,input().split())
print(sol(N,M))
0