結果

問題 No.2062 Sum of Subset mod 999630629
ユーザー zkouzkou
提出日時 2022-08-26 23:34:09
言語 PyPy3
(7.3.15)
結果
AC  
実行時間 1,668 ms / 5,000 ms
コード長 11,789 bytes
コンパイル時間 322 ms
コンパイル使用メモリ 82,268 KB
実行使用メモリ 257,676 KB
最終ジャッジ日時 2024-10-14 01:10:12
合計ジャッジ時間 16,824 ms
ジャッジサーバーID
(参考情報)
judge2 / judge3
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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 51 ms
55,808 KB
testcase_01 AC 50 ms
56,064 KB
testcase_02 AC 50 ms
55,936 KB
testcase_03 AC 50 ms
56,320 KB
testcase_04 AC 49 ms
55,936 KB
testcase_05 AC 50 ms
55,680 KB
testcase_06 AC 49 ms
56,064 KB
testcase_07 AC 50 ms
55,680 KB
testcase_08 AC 77 ms
82,304 KB
testcase_09 AC 71 ms
78,464 KB
testcase_10 AC 69 ms
76,672 KB
testcase_11 AC 524 ms
124,976 KB
testcase_12 AC 527 ms
126,512 KB
testcase_13 AC 331 ms
101,048 KB
testcase_14 AC 528 ms
126,240 KB
testcase_15 AC 209 ms
89,344 KB
testcase_16 AC 525 ms
122,784 KB
testcase_17 AC 535 ms
125,072 KB
testcase_18 AC 338 ms
101,168 KB
testcase_19 AC 206 ms
89,472 KB
testcase_20 AC 250 ms
89,984 KB
testcase_21 AC 342 ms
99,268 KB
testcase_22 AC 252 ms
89,344 KB
testcase_23 AC 74 ms
80,768 KB
testcase_24 AC 74 ms
81,024 KB
testcase_25 AC 1,668 ms
257,676 KB
testcase_26 AC 1,641 ms
257,396 KB
testcase_27 AC 1,642 ms
257,276 KB
testcase_28 AC 1,641 ms
257,468 KB
testcase_29 AC 1,638 ms
257,548 KB
testcase_30 AC 904 ms
182,284 KB
testcase_31 AC 905 ms
182,244 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 #

# 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)


# Reference: https://opt-cp.com/fps-fast-algorithms/
def inv(a):
    """It calculates the inverse of formal power series in O(n log n) time, where n = len(a)."""
    n = len(a)
    assert n > 0 and a[0] != 0
    res = [pow(a[0], _fft_mod - 2, _fft_mod)]
    m = 1
    while m < n:
        f = a[: min(n, 2 * m)]
        g = res.copy()
        f += [0] * (2 * m - len(f))
        _butterfly(f)
        g += [0] * (2 * m - len(g))
        _butterfly(g)
        for i in range(2 * m):
            f[i] = f[i] * g[i] % _fft_mod
        _butterfly_inv(f)
        f = f[m:] + [0] * m
        _butterfly(f)
        for i in range(2 * m):
            f[i] = f[i] * g[i] % _fft_mod
        _butterfly_inv(f)
        f = f[:m]
        iz = pow(2 * m, _fft_mod - 2, _fft_mod)
        iz *= -iz
        iz %= _fft_mod
        for i in range(m):
            f[i] = f[i] * iz % _fft_mod
        res.extend(f)
        m *= 2
    res = res[:n]
    return res


def integ_inplace(a):
    n = len(a)
    assert n > 0
    if n == 1:
        return []
    a.pop()
    a.insert(0, 0)
    inv = [1, 1]
    for i in range(2, n):
        inv.append(-inv[_fft_mod % i] * (_fft_mod // i) % _fft_mod)
        a[i] = a[i] * inv[i] % _fft_mod


def deriv_inplace(a):
    n = len(a)
    assert n > 0
    for i in range(2, n):
        a[i] = a[i] * i % _fft_mod
    a.pop(0)
    a.append(0)


def log(a):
    a = a.copy()
    n = len(a)
    assert n > 0 and a[0] == 1
    a_inv = inv(a)
    deriv_inplace(a)
    a = convolution(a, a_inv)[:n]
    integ_inplace(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()
    deriv_inplace(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] % _fft_mod 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] % _fft_mod
            _butterfly_inv(_f)
            _f = _f[: m // 2]
            iz = pow(m, _fft_mod - 2, _fft_mod)
            iz *= -iz
            iz %= _fft_mod
            for i in range(m // 2):
                _f[i] = _f[i] * iz % _fft_mod
            g.extend(_f)

        t = a[:m]
        deriv_inplace(t)
        r = h_drv[: m - 1]
        r.append(0)
        _butterfly(r)
        for i in range(m):
            r[i] = r[i] * f_fft[i] % _fft_mod
        _butterfly_inv(r)
        im = pow(-m, _fft_mod - 2, _fft_mod)
        for i in range(m):
            r[i] = r[i] * im % _fft_mod
        for i in range(m):
            t[i] = (t[i] + r[i]) % _fft_mod
        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] % _fft_mod
        _butterfly_inv(t)
        t = t[:m]
        i2m = pow(2 * m, _fft_mod - 2, _fft_mod)
        for i in range(m):
            t[i] = t[i] * i2m % _fft_mod

        v = a[m : min(n, 2 * m)]
        v += [0] * (m - len(v))
        t = [0] * (m - 1) + t + [0]
        integ_inplace(t)
        for i in range(m):
            v[i] = (v[i] - t[m + i]) % _fft_mod

        v += [0] * m
        _butterfly(v)
        for i in range(2 * m):
            v[i] = v[i] * f_fft[i] % _fft_mod
        _butterfly_inv(v)
        v = v[:m]
        i2m = pow(2 * m, _fft_mod - 2, _fft_mod)
        for i in range(m):
            v[i] = v[i] * i2m % _fft_mod

        for i in range(min(n - m, m)):
            a[m + i] = v[i]

        m *= 2
    return a


def pow_fps(a, k):
    a = a.copy()
    n = len(a)
    l = 0
    while l < len(a) and not a[l]:
        l += 1
    if l * k >= n:
        return [0] * n
    ic = pow(a[l], _fft_mod - 2, _fft_mod)
    pc = pow(a[l], k, _fft_mod)
    a = log([a[i] * ic % _fft_mod for i in range(l, len(a))])
    for i in range(len(a)):
        a[i] = a[i] * k % _fft_mod
    a = exp(a)
    for i in range(len(a)):
        a[i] = a[i] * pc % _fft_mod
    a = [0] * (l * k) + a[: n - l * k]
    return a


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()

arg = [0] * weight_ub
arg[0] = arg[1] = 1
log_1_x = log(arg)
s = [0] * weight_ub
for A, count in Counter(As).items():
    for i in range(0, weight_ub, A):
        s[i] += log_1_x[i // A] * count
        s[i] %= P1
e = exp(s)

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

print(answer)
0