Smallest Divisible Digit Product II — Python

# Time:  O(n + logt)
# Space: O(1)

# freq table, greedy, prefix sum, number theory
class Solution(object):
    def smallestNumber(self, num, t):
        """
        :type num: str
        :type t: int
        :rtype: str
        """
        LOOKUP = [[(0, 0, 0, 0), (0, 1, 0, 0)],
                  [(1, 0, 0, 0), (0, 0, 0, 1)],
                  [(0, 0, 1, 0), (1, 0, 0, 1)]]
        def count(x):
            cnt = [0]*10
            for i in xrange(2, 9+1):
                while x%i == 0:
                    x //= i
                    cnt[i] += 1
                if x == 1:
                    return cnt
            return []
    
        def update(total, cnt, d):
            for i, x in enumerate(cnt):
                total[i] += d*x

        def diff(expect, total):
            return [max(expect[i]-total[i], 0) for i in xrange(len(expect))]

        def min_factors(cnt):
            cnt8, r2 = divmod(cnt[2], 3)
            cnt9, r3 = divmod(cnt[3], 2)
            cnt2, cnt3, cnt4, cnt6 = LOOKUP[r2][r3]
            return (cnt2, cnt3, cnt4, cnt[5], cnt6, cnt[7], cnt8, cnt9)
    
        def format(cnt, l):
            return '1'*(l-sum(cnt))+"".join(str(x)*cnt[x-2] for x in xrange(2, 9+1))
    
        expect = count(t)
        if not expect:
            return "-1"
        nums = map(int, num)
        i = next((i for i in xrange(len(nums)) if not nums[i]), len(nums))
        for j in xrange(i, len(nums)):
            nums[j] = 1
        total = [0]*10
        for x in nums:
            update(total, count(x), +1)
        if all(d == 0 for d in diff(expect, total)):
            return "".join(map(str, nums))
        for i in reversed(xrange(len(nums))):
            update(total, count(nums[i]), -1)
            for x in xrange(nums[i]+1, 9+1):
                update(total, count(x), +1)
                tmp = min_factors(diff(expect, total))
                update(total, count(x), -1)
                if sum(tmp) > len(nums)-1-i:
                    continue
                return "".join(map(str, nums[:i]))+str(x)+format(tmp, len(nums)-1-i)
        return format(min_factors(diff(expect, total)), len(nums)+1)


# Time:  O(nlogt)
# Space: O(logt)
# freq table, greedy, prefix sum, number theory
class Solution2(object):
    def smallestNumber(self, num, t):
        """
        :type num: str
        :type t: int
        :rtype: str
        """
        def gcd(a, b):
            while b:
                a, b = b, a%b
            return a

        def find_candidates(t, l):  # Time: O(logt)
            candidates = []
            for x in reversed(xrange(2, 9+1)):
                while t%x == 0:
                    t //= x
                    candidates.append(x)
                    if len(candidates) > l:
                        return []
                if t == 1:
                    candidates.reverse()
                    return candidates
            return []
    
        def format(candidates, l):
            result = [1]*l
            i = len(result)-len(candidates)
            for x in candidates:
                result[i] = x
                i += 1
            return "".join(map(str, result))

        nums = map(int, num)
        candidates = find_candidates(t, float("inf"))
        if t != 1 and not candidates:
            return "-1"
        i = next((i for i in xrange(len(nums)) if not nums[i]), len(nums))
        for j in xrange(i, len(nums)):
            nums[j] = 1
        prefix = [1]*(len(nums)+1)
        for i in xrange(len(prefix)-1):
            prefix[i+1] = (prefix[i]*nums[i])%t
        if not prefix[-1]:
            return "".join(map(str, nums))
        for i in reversed(xrange(len(nums))):
             target = t//gcd(t, prefix[i])
             for x in xrange(nums[i]+1, 9+1):
                new_target = target//gcd(target, x)
                tmp = find_candidates(new_target, len(nums)-1-i)
                if new_target != 1 and not tmp:
                    continue
                nums[i] = x
                return "".join(map(str, nums[:i+1]))+format(tmp, len(nums)-1-i)
        return format(candidates, max(len(nums)+1, len(candidates)))