Super Ugly Number

Write a program to find the n^th super ugly number.

Super ugly numbers are positive numbers whose all prime factors are in the given prime list primes of size k. For example, [1, 2, 4, 7, 8, 13, 14, 16, 19, 26, 28, 32] is the sequence of the first 12 super ugly numbers given primes = [2, 7, 13, 19] of size 4.

Note:
(1) 1 is a super ugly number for any given primes.
(2) The given numbers in primes are in ascending order.
(3) 0 < k ≤ 100, 0 < n ≤ 10⁶, 0 < primes[i] < 1000.

这题算是Ugly Number II的一个拓展，主要是这里生成丑数的因数不再是[2,3,5]而是一个素数数组．其实做法还可以跟以前一样，只不过用一个和素数数组等长的数组来保存每个素数对应的index.产生一个素数时,更新这个数组．更新的判读方式也和之前一样，这样的复杂为O(nk).主要的消耗在于每次需要从k个素数生成的备选中，选择一个最小值，和更新index这个是Ｏ(k)的复杂度．那么有没有办法改进呢，有，可以看到每次前面的做法每次产生一个丑数，都要用每个素数产生一个备选,这些备选本身极其不可能一次全部更新，所以每次的候选中有相当大的重复部分．我们可以把备选放入heap当中．这样不用每次全部比较得到最小值（heap里面本身已经保存了相对关系）．这样做的时间复杂度为Ｏ（nlogk),具体怎么证明还有待思考．附上一个很好的discuss

    def nthSuperUglyNumber(self, n, primes):
        """
        :type n: int
        :type primes: List[int]
        :rtype: int
        """
        length = len(primes)
        uglys = [1]
        times = [0] * length #prime[i]'s multipy places in uglys
        minHeap = [(primes[i],i) for i in xrange(length)]
        heapq.heapify(minHeap)
        
        while len(uglys) < n:
            (val, index) = minHeap[0]
            uglys.append(val)
            while minHeap[0][0] == uglys[-1]:
                index = heapq.heappop(minHeap)[1]
                times[index] += 1
                heapq.heappush(minHeap,(primes[index]*uglys[times[index]],index))
        return uglys[-1]