MOD上での組合せ計算
(Combination/modinv_combination.py)

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• Last update: 2021-09-08 22:15:45+09:00

使い方

Combination(n: int, MOD: int)
MOD 上での $k!$ とその逆元を、n 以下の自然数 $k$ に対して前計算する。計算量 $O(n + \log(\mathrm{MOD}))$

• inv(k: int) -> int
  $k$ の逆元を返す。計算量 $O(1)$

• fact(k: int) -> int
  $k!$ を返す。計算量 $O(1)$

• inv_fact(k: int) -> int
  $k!$ の逆元を返す。計算量 $O(1)$

• perm(k: int, r: int) -> int
  ${}_k\mathrm{P}_r$ を返す。計算量 $O(1)$

• comb(k: int, r: int) -> int
  ${}_k\mathrm{C}_r$ を返す。計算量 $O(1)$

combination(k: int, r: int, MOD: int) -> int
MOD 上での ${}_k\mathrm{C}_r$ をナイーブに計算して返す。計算量 $O(\min(r, k - r))$

Required by

• Lucas の定理 (Combination/LucasTheorem.py)
• カタラン数 (Combination/catalan.py)
• 写像12相 (Combination/twelvefold_way.py)

Verified with

• TestCase/AOJ/DPL_5_E.naive.test.py
• TestCase/yukicoder/yuki0117.test.py

Code

class Combination:
    def __init__(self, n, MOD):
        self.f = [1]
        for i in range(1, n + 1):
            self.f.append(self.f[-1] * i % MOD)
        self.inv_f = [0] * (n + 1)
        self.inv_f[n] = pow(self.f[n], MOD - 2, MOD)
        for i in reversed(range(n)):
            self.inv_f[i] = self.inv_f[i + 1] * (i + 1) % MOD
        self.MOD = MOD

    def inv(self, k):
        """get inverse(k)"""
        return (self.inv_f[k] * self.f[k - 1]) % self.MOD

    def fact(self, k):
        """get k!"""
        return self.f[k]

    def inv_fact(self, k):
        """get inverse(k!)"""
        return self.inv_f[k]

    def perm(self, k, r):
        """get kPr"""
        if k < r:
            return 0
        return (self.f[k] * self.inv_f[k - r]) % self.MOD

    def comb(self, k, r):
        """get kCr"""
        if k < r:
            return 0
        return (self.f[k] * self.inv_f[k - r] % self.MOD
                * self.inv_f[r]) % self.MOD


def combination(k, r, MOD):
    """kCr O(r)"""
    if k < r:
        return 0
    r = min(r, k - r)
    numer, denom = 1, 1
    for l in range(r):
        numer *= (k - l)
        numer %= MOD
        denom *= l + 1
        denom %= MOD
    return numer * pow(denom, MOD - 2, MOD) % MOD

Traceback (most recent call last):
  File "/opt/hostedtoolcache/Python/3.12.4/x64/lib/python3.12/site-packages/onlinejudge_verify/documentation/build.py", line 71, in _render_source_code_stat
    bundled_code = language.bundle(stat.path, basedir=basedir, options={'include_paths': [basedir]}).decode()
                   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/opt/hostedtoolcache/Python/3.12.4/x64/lib/python3.12/site-packages/onlinejudge_verify/languages/python.py", line 96, in bundle
    raise NotImplementedError
NotImplementedError

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