結果

問題 No.2062 Sum of Subset mod 999630629
ユーザー zkouzkou
提出日時 2022-08-26 23:18:41
言語 PyPy3
(7.3.15)
結果
TLE  
(最新)
AC  
(最初)
実行時間 -
コード長 7,975 bytes
コンパイル時間 166 ms
コンパイル使用メモリ 82,176 KB
実行使用メモリ 275,032 KB
最終ジャッジ日時 2024-04-22 02:27:07
合計ジャッジ時間 26,758 ms
ジャッジサーバーID
（参考情報） 		judge3 / judge5
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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 100 ms
147,200 KB
testcase_01 AC 101 ms
146,816 KB
testcase_02 AC 103 ms
146,816 KB
testcase_03 AC 105 ms
147,072 KB
testcase_04 AC 107 ms
146,816 KB
testcase_05 AC 103 ms
147,328 KB
testcase_06 AC 102 ms
146,944 KB
testcase_07 AC 98 ms
147,328 KB
testcase_08 AC 122 ms
168,704 KB
testcase_09 AC 116 ms
164,608 KB
testcase_10 AC 116 ms
162,816 KB
testcase_11 AC 437 ms
182,144 KB
testcase_12 AC 442 ms
182,912 KB
testcase_13 AC 288 ms
178,688 KB
testcase_14 AC 452 ms
183,296 KB
testcase_15 AC 199 ms
174,848 KB
testcase_16 AC 512 ms
181,248 KB
testcase_17 AC 454 ms
182,144 KB
testcase_18 AC 301 ms
178,432 KB
testcase_19 AC 245 ms
174,464 KB
testcase_20 AC 357 ms
176,256 KB
testcase_21 AC 381 ms
177,792 KB
testcase_22 AC 308 ms
176,384 KB
testcase_23 AC 124 ms
167,040 KB
testcase_24 AC 120 ms
167,168 KB
testcase_25 AC 2,526 ms
269,120 KB
testcase_26 AC 2,440 ms
268,468 KB
testcase_27 AC 2,446 ms
268,520 KB
testcase_28 AC 2,449 ms
268,520 KB
testcase_29 AC 2,001 ms
259,804 KB
testcase_30 AC 1,749 ms
259,880 KB
testcase_31 TLE -
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 # 

# For the sake of speed,
# this convolution is specialized to mod 998244353.


_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
                    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_ - 2)):
                    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] += a[i] * b[j]
                ans[i + j] %= _fft_mod
    else:
        for i in range(n):
            for j in range(m):
                ans[i + j] += a[i] * b[j]
                ans[i + 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] *= b[i]
        a[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] *= iz
        a[i] %= _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] *= iz
        a[i] %= _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) <= 60:
        return _convolution_naive(a, b)
    if a is b:
        return _convolution_square(a)
    return _convolution_fft(a, b)


MOD = 998244353
table_len = 10**6 + 10

fac = [1, 1]
for i in range(2, table_len):
    fac.append(fac[-1] * i % MOD)

finv = [0] * table_len
finv[-1] = pow(fac[-1], MOD - 2, MOD)
for i in range(table_len - 1, 0, -1):
    finv[i - 1] = finv[i] * i % MOD


def comb(n, k):
    if k < 0 or n < 0 or n - k < 0:
        return 0
    return fac[n] * finv[k] % MOD * finv[n - k] % MOD


from collections import Counter

P1 = 998244353
P2 = 999630629

# N = 10**5
# As = [10000] * N

N = int(input())
As = list(map(int, input().split()))

answer = pow(2, N - 1, P1) * sum(As) % P1

# #{ S | P2 <= \sum_{i \in S} A_i }
# = #{ S' | sum(As) - P2 >= \sum_{i \in S'} A_i }

weight_ub = sum(As) - P2 + 1

if weight_ub < 0:
    print(answer)
    exit()

prod = [1]
for A, count in Counter(As).items():
    f = [0] * (weight_ub)
    for i in range(0, weight_ub, A):
        f[i] = comb(count, i // A)
    prod = convolution(prod, f)[:weight_ub]

answer -= P2 * (sum(prod) % P1) % P1
answer %= P1

print(answer)
0