結果

問題 No.2062 Sum of Subset mod 999630629
ユーザー chineristACchineristAC
提出日時 2022-08-26 22:42:28
言語 PyPy3
(7.3.13)
結果
AC  
実行時間 3,780 ms / 5,000 ms
コード長 9,164 bytes
コンパイル時間 723 ms
コンパイル使用メモリ 87,220 KB
実行使用メモリ 295,632 KB
最終ジャッジ日時 2023-08-04 04:02:56
合計ジャッジ時間 35,949 ms
ジャッジサーバーID
(参考情報)
judge13 / judge14
このコードへのチャレンジ(β)

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 149 ms
92,632 KB
testcase_01 AC 149 ms
92,804 KB
testcase_02 AC 153 ms
92,648 KB
testcase_03 AC 146 ms
92,640 KB
testcase_04 AC 147 ms
92,516 KB
testcase_05 AC 146 ms
92,776 KB
testcase_06 AC 146 ms
92,636 KB
testcase_07 AC 150 ms
92,696 KB
testcase_08 AC 175 ms
108,448 KB
testcase_09 AC 165 ms
103,220 KB
testcase_10 AC 160 ms
101,776 KB
testcase_11 AC 1,072 ms
201,144 KB
testcase_12 AC 1,066 ms
201,760 KB
testcase_13 AC 636 ms
146,372 KB
testcase_14 AC 1,088 ms
201,148 KB
testcase_15 AC 314 ms
107,476 KB
testcase_16 AC 1,068 ms
201,032 KB
testcase_17 AC 1,082 ms
201,580 KB
testcase_18 AC 647 ms
146,560 KB
testcase_19 AC 321 ms
107,928 KB
testcase_20 AC 429 ms
119,380 KB
testcase_21 AC 638 ms
146,584 KB
testcase_22 AC 425 ms
119,484 KB
testcase_23 AC 169 ms
105,600 KB
testcase_24 AC 165 ms
106,000 KB
testcase_25 AC 3,719 ms
286,500 KB
testcase_26 AC 3,743 ms
286,640 KB
testcase_27 AC 3,751 ms
286,476 KB
testcase_28 AC 3,720 ms
295,632 KB
testcase_29 AC 3,780 ms
286,720 KB
testcase_30 AC 1,982 ms
261,544 KB
testcase_31 AC 1,997 ms
260,804 KB
権限があれば一括ダウンロードができます

ソースコード

diff #

import sys,random,bisect
from collections import deque,defaultdict,Counter
from heapq import heapify,heappop,heappush
from itertools import cycle, permutations
from math import log,gcd

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 = 6*10**5
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 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 integral(f,limit):
    res = [0]+[(f[i] * inv[i+1]) % mod for i in range(len(f)-1)]
    return res[:limit]

def diff(f,limit):
    res = [(f[i+1] * (i+1)) % mod for i in range(len(f)-1)]+[0]
    return res[:limit]

def log(f,limit):
    res = convolution(diff(f,limit),inverse(f,limit))[:limit]
    return integral(res,limit)

def exp(f,limit):
    l = len(f)
    L = 1<<((l-1).bit_length())
    n = L.bit_length()-1
    f = f[:L]
    f+=[0]*(L-len(f))

    res = [1]
    for i in range(1,n+1):
        res += [0]*2**(i-1)
        g = log(res,2**i)
        h = [(f[j]-g[j])%mod for j in range(2**i)]
        h[0] = (h[0]+1) % mod
        res =convolution(res,h)[:2**i]
    return res[:limit]

N = int(input())
A = li()

S = sum(A)

if S < 999630629:
    res = S * pow(2,N-1,mod) % mod
    exit(print(res))

cnt = [0] * (10**4+1)
for a in A:
    cnt[a] += 1

M = S - 999630629
f = [0] * (M+1)
for a in range(1,10**4+1):
    for k in range(a,M+1,a):
        if (k//a) & 1:
            f[k] += cnt[a] * inv[k//a] % mod
        else:
            f[k] -= cnt[a] * inv[k//a] % mod
        f[k] %= mod


f = exp(f,M+1)



res = S * pow(2,N-1,mod) % mod
for i in range(M+1):
    res -= 999630629 * f[i]
    res %= mod

print(res)
0