頻出関数

整数問題

約数

def make_divisors(n):
    divisors = []
    for i in range(1, int(n**0.5)+1):
        if n % i == 0:
            divisors.append(i)
            if i != n // i:
                divisors.append(n//i)
    
    # divisors.sort()
    return divisors

最大公約数

def gcd(x, y):
    if y == 0:
        return x
    else:
        return gcd(y, x % y)

最小公倍数

def lcm(x, y):
    d = gcd(x, y)
    return a // x * y

素因数分解

def prime_factorize(n):
    res = []
    for i in range(2, int(n**(1/2))+1):
        if n % i != 0: continue
        num = 0
        while n % i == 0:
            num += 1
            n /= i
        res.append((i, num))
    if n != 1:
        res.append((n, 1))
    
    return res

べき乗 (10**9+7で割った余り)

mod = 10**9 + 7

def pow(x, y):
    ret = 1
    while y:
        if y & 1:
            ret = ret * x % mod
        x = x * x % mod
        y >>= 1
    
    return ret

二項係数 (10**9+7で割った余り)

def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 #出力の制限
N = 10**4
g1 = [1, 1] # 元テーブル
g2 = [1, 1] #逆元テーブル
inverse = [0, 1] #逆元テーブル計算用テーブル

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )

a = cmb(n,r,mod)

素数判定

def is_prime(n):
    if n == 1: return False

    for k in range(2, int(n**(1/2)) + 1):
        if n % k == 0:
            return False

    return True

素数列挙

def primes(n):
    is_prime = [True] * (n + 1)
    is_prime[0] = False
    is_prime[1] = False
    for i in range(2, int(n**0.5) + 1):
        if not is_prime[i]:
            continue
        for j in range(i * 2, n + 1, i):
            is_prime[j] = False
    return [i for i in range(n + 1) if is_prime[i]]

木

UnionFind木

class UnionFind():
    def __init__(self, n):
        self.n = n
        self.parents = [-1] * n

    def find(self, x):
        if self.parents[x] < 0:
            return x
        else:
            self.parents[x] = self.find(self.parents[x])
            return self.parents[x]

    def union(self, x, y):
        x = self.find(x)
        y = self.find(y)

        if x == y:
            return

        if self.parents[x] > self.parents[y]:
            x, y = y, x

        self.parents[x] += self.parents[y]
        self.parents[y] = x

    def size(self, x):
        return -self.parents[self.find(x)]

    def same(self, x, y):
        return self.find(x) == self.find(y)

    def members(self, x):
        root = self.find(x)
        return [i for i in range(self.n) if self.find(i) == root]

    def roots(self):
        return [i for i, x in enumerate(self.parents) if x < 0]

    def group_count(self):
        return len(self.roots())

    def all_group_members(self):
        return {r: self.members(r) for r in self.roots()}

    def __str__(self):
        return '\n'.join('{}: {}'.format(r, self.members(r)) for r in self.roots())

uf = UnionFind(n) # n: 頂点数