largest-component-size-by-common-factor.py

# Time:  O(f * n), f is the max number of unique prime factors
# Space: O(p + n), p is the total number of unique primes

import collections


class UnionFind(object):
    def __init__(self, n):
        self.set = range(n)
        self.size = [1]*n

    def find_set(self, x):
        if self.set[x] != x:
            self.set[x] = self.find_set(self.set[x])  # path compression.
        return self.set[x]

    def union_set(self, x, y):
        x_root, y_root = map(self.find_set, (x, y))
        if x_root == y_root:
            return False
        self.set[min(x_root, y_root)] = max(x_root, y_root)
        self.size[max(x_root, y_root)] += self.size[min(x_root, y_root)]
        return True


class Solution(object):
    def largestComponentSize(self, A):
        """
        :type A: List[int]
        :rtype: int
        """
        def primeFactors(i):  # prime factor decomposition
            result = []
            d = 2
            if i%d == 0:
                while i%d == 0:
                    i //= d
                result.append(d)
            d = 3
            while d*d <= i:
                if i%d == 0:
                    while i%d == 0:
                        i //= d
                    result.append(d)
                d += 2
            if i > 2:
                result.append(i)
            return result
        
        union_find = UnionFind(len(A))
        nodesWithCommonFactor = collections.defaultdict(int)
        for i in xrange(len(A)):
            for factor in primeFactors(A[i]):
                if factor not in nodesWithCommonFactor:
                    nodesWithCommonFactor[factor] = i
                union_find.union_set(nodesWithCommonFactor[factor], i)
        return max(union_find.size)