結果

問題 No.2062 Sum of Subset mod 999630629
ユーザー zkouzkou
提出日時 2022-08-26 23:33:37
言語 PyPy3
(7.3.15)
結果
AC  
実行時間 2,173 ms / 5,000 ms
コード長 8,660 bytes
コンパイル時間 223 ms
コンパイル使用メモリ 81,664 KB
実行使用メモリ 258,136 KB
最終ジャッジ日時 2024-10-14 01:09:54
合計ジャッジ時間 20,340 ms
ジャッジサーバーID
(参考情報)
judge4 / judge3
このコードへのチャレンジ
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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 44 ms
55,296 KB
testcase_01 AC 44 ms
55,296 KB
testcase_02 AC 47 ms
55,296 KB
testcase_03 AC 45 ms
55,424 KB
testcase_04 AC 44 ms
55,296 KB
testcase_05 AC 44 ms
55,296 KB
testcase_06 AC 45 ms
55,296 KB
testcase_07 AC 45 ms
55,424 KB
testcase_08 AC 69 ms
82,048 KB
testcase_09 AC 65 ms
77,568 KB
testcase_10 AC 64 ms
76,160 KB
testcase_11 AC 588 ms
125,796 KB
testcase_12 AC 582 ms
127,320 KB
testcase_13 AC 360 ms
102,040 KB
testcase_14 AC 586 ms
127,176 KB
testcase_15 AC 190 ms
89,984 KB
testcase_16 AC 617 ms
124,860 KB
testcase_17 AC 584 ms
125,888 KB
testcase_18 AC 356 ms
102,032 KB
testcase_19 AC 188 ms
89,856 KB
testcase_20 AC 237 ms
90,112 KB
testcase_21 AC 357 ms
100,140 KB
testcase_22 AC 242 ms
90,240 KB
testcase_23 AC 66 ms
80,000 KB
testcase_24 AC 67 ms
80,384 KB
testcase_25 AC 2,173 ms
258,020 KB
testcase_26 AC 2,171 ms
258,136 KB
testcase_27 AC 2,129 ms
257,880 KB
testcase_28 AC 2,153 ms
258,016 KB
testcase_29 AC 2,167 ms
257,896 KB
testcase_30 AC 1,246 ms
188,532 KB
testcase_31 AC 1,147 ms
184,396 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_sum_e = (
    911660635,
    509520358,
    369330050,
    332049552,
    983190778,
    123842337,
    238493703,
    975955924,
    603855026,
    856644456,
    131300601,
    842657263,
    730768835,
    942482514,
    806263778,
    151565301,
    510815449,
    503497456,
    743006876,
    741047443,
    56250497,
    867605899,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
)
_fft_sum_ie = (
    86583718,
    372528824,
    373294451,
    645684063,
    112220581,
    692852209,
    155456985,
    797128860,
    90816748,
    860285882,
    927414960,
    354738543,
    109331171,
    293255632,
    535113200,
    308540755,
    121186627,
    608385704,
    438932459,
    359477183,
    824071951,
    103369235,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
)


def _butterfly(a):
    n = len(a)
    h = (n - 1).bit_length()
    for ph in range(1, h + 1):
        w = 1 << (ph - 1)
        p = 1 << (h - ph)
        now = 1
        for s in range(w):
            offset = s << (h - ph + 1)
            for i in range(p):
                l = a[i + offset]
                r = a[i + offset + p] * now % _fft_mod
                a[i + offset] = (l + r) % _fft_mod
                a[i + offset + p] = (l - r) % _fft_mod
            now *= _fft_sum_e[(~s & -~s).bit_length() - 1]
            now %= _fft_mod


def _butterfly_inv(a):
    n = len(a)
    h = (n - 1).bit_length()
    for ph in range(h, 0, -1):
        w = 1 << (ph - 1)
        p = 1 << (h - ph)
        inow = 1
        for s in range(w):
            offset = s << (h - ph + 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) * inow % _fft_mod
            inow *= _fft_sum_ie[(~s & -~s).bit_length() - 1]
            inow %= _fft_mod


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.

    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) <= 100:
        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