[LeetCode] 313. Super Ugly Number 超级丑陋数

Write a program to find the nth super ugly number.

Super ugly numbers are positive numbers whose all prime factors are in the given prime list primes of size k.

Example:

Input: n = 12, primes = [2,7,13,19]
Output: 32 
Explanation: [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 is a super ugly number for any given primes.
• The given numbers in primes are in ascending order.
• 0 < k ≤ 100, 0 < n ≤ 106, 0 < primes[i] < 1000.
• The nth super ugly number is guaranteed to fit in a 32-bit signed integer.

264. Ugly Number II 的拓展，还是找出第n个丑陋数，但质数集合不在只是2，3，5，而是可以任意给定。难度增加了，但本质上和Ugly Number II 没有什么区别，由于不知道质数的个数，可以用一个idx数组来保存当前的位置，然后从每个子链中取出一个数，找出其中最小值，然后更新idx数组对应位置，注意有可能最小值不止一个，要更新所有最小值的位置。

解题思路：
要使得super ugly number不漏掉，那么需要使用每个因子去乘以其对应的“第一个”丑数。那么何为对应的“第一个”丑数？

首先，利用ugly[]数组来保存所有的超级丑数，ugly[i]表示第i+1个超级丑数；

接着利用pointer[]数组来表示每个因子对应的“第一个”丑数的下标。pointer数组长度当然需要和primes长度一致，且初始化为0，代表着每个因子对应的“第一个”丑数都是ugly[0]；

接下来我们以primes[2,7,13,19],pointer[0,0,0,0],ugly[0]=1作为初始条件往下看：

遍历primes数组，用每个因子都乘以其对应的第一个丑数，即ugly[0]=1,可以发现1x2=2是最小值，故ugly[1]=2;但要注意，此时的pointer数组发生了变化：

由于当前产生的丑数2是由2这个因子乘以它的对应“第一个”丑数得到的，因此需要将pointer[0]加一。pointer[0]是2这个因子对应的“第一个”丑数的下标，因为当前已经使用了2x1，如果不更新，则下一轮还是会用2这个因子去乘以第一个丑数(ugly[0]).将其更新后，则意味着2这个因子对应的第一个丑数已经改变了，变成了ugly[1].而其他三个对应的“第一个”丑数还是ugly[0]。

我们接着看下一轮：2x2【即ugly[pointer[1]]x2】,1x7,1x13,1x19，发现还是2这个因子得到的数最小，故更新：ugly[2]=2x2=4,pointer[0]=2；

下一轮：4x2,1x7,1x13,1x19,可以发现当前这一轮最小值是7，且由因子7产生，故更新：ugly[3]=7,pointer[1]=1；

以此类推....
如果更新过程中，出现最小值不止一个的话，则其对应的pointer的值都需要增加1。

Java:

public int nthSuperUglyNumber(int n, int[] primes) {
        int[] ugly = new int[n+1];
        ugly[0]=1;
        int[] pointer = new int[primes.length];
        for(int i=1;i<n;i++) {
            int min=Integer.MAX_VALUE;
            int minIndex = 0;
            for(int j=0;j<primes.length;j++) {
                if(ugly[pointer[j]]*primes[j]<min) {
                    min=ugly[pointer[j]]*primes[j];
                    minIndex = j;
                }else if(ugly[pointer[j]]*primes[j]==min) {
                    pointer[j]++;
                }
            }
            ugly[i]=min;
            pointer[minIndex]++;
        }
        return ugly[n-1];
    }

Java:1

public int nthSuperUglyNumberI(int n, int[] primes) {
    int[] ugly = new int[n];
    int[] idx = new int[primes.length];

    ugly[0] = 1;
    for (int i = 1; i < n; i++) {
        //find next
        ugly[i] = Integer.MAX_VALUE;
        for (int j = 0; j < primes.length; j++)
            ugly[i] = Math.min(ugly[i], primes[j] * ugly[idx[j]]);
        
        //slip duplicate
        for (int j = 0; j < primes.length; j++) {
            while (primes[j] * ugly[idx[j]] <= ugly[i]) idx[j]++;
        }
    }

    return ugly[n - 1];
}

Java:2

public int nthSuperUglyNumber(int n, int[] primes) {
        int[] ugly = new int[n];
        int[] idx = new int[primes.length];
        int[] val = new int[primes.length];
        Arrays.fill(val, 1);

        int next = 1;
        for (int i = 0; i < n; i++) {
            ugly[i] = next;
            
            next = Integer.MAX_VALUE;
            for (int j = 0; j < primes.length; j++) {
                //skip duplicate and avoid extra multiplication
                if (val[j] == ugly[i]) val[j] = ugly[idx[j]++] * primes[j];
                //find next ugly number
                next = Math.min(next, val[j]);
            }
        }

        return ugly[n - 1];
    }

Java: 3 index heap　

public int nthSuperUglyNumberHeap(int n, int[] primes) {
    int[] ugly = new int[n];

    PriorityQueue<Num> pq = new PriorityQueue<>();
    for (int i = 0; i < primes.length; i++) pq.add(new Num(primes[i], 1, primes[i]));
    ugly[0] = 1;

    for (int i = 1; i < n; i++) {
        ugly[i] = pq.peek().val;
        while (pq.peek().val == ugly[i]) {
            Num nxt = pq.poll();
            pq.add(new Num(nxt.p * ugly[nxt.idx], nxt.idx + 1, nxt.p));
        }
    }

    return ugly[n - 1];
}

private class Num implements Comparable<Num> {
    int val;
    int idx;
    int p;

    public Num(int val, int idx, int p) {
        this.val = val;
        this.idx = idx;
        this.p = p;
    }

    @Override
    public int compareTo(Num that) {
        return this.val - that.val;
    }
}　

Python:

def nthSuperUglyNumber(self, n, primes):
        ugly = [1]
        pointers = [0]*len(primes)
        
        for i in range(1,n):
            minu = float("inf")
            minIndex = 0
            for j in range(len(primes)):
                if primes[j] * ugly[pointers[j]] < minu:
                    minu = primes[j] * ugly[pointers[j]]
                    minIndex = j
                elif primes[j] * ugly[pointers[j]] == minu:
                    pointers[j] += 1
            ugly.append(minu)
            pointers[minIndex] += 1
        return ugly[-1]　　

Python:

# Heap solution. (620ms)
class Solution(object):
    def nthSuperUglyNumber(self, n, primes):
        """
        :type n: int
        :type primes: List[int]
        :rtype: int
        """
        heap, uglies, idx, ugly_by_last_prime = [], [0] * n, [0] * len(primes), [0] * n
        uglies[0] = 1

        for k, p in enumerate(primes):
            heapq.heappush(heap, (p, k))

        for i in xrange(1, n):
            uglies[i], k = heapq.heappop(heap)
            ugly_by_last_prime[i] = k
            idx[k] += 1
            while ugly_by_last_prime[idx[k]] > k:
                idx[k] += 1
            heapq.heappush(heap, (primes[k] * uglies[idx[k]], k))

        return uglies[-1]

Python:

# Time:  O(n * k)
# Space: O(n + k)
# Hash solution. (932ms)
class Solution2(object):
    def nthSuperUglyNumber(self, n, primes):
        """
        :type n: int
        :type primes: List[int]
        :rtype: int
        """
        uglies, idx, heap, ugly_set = [0] * n, [0] * len(primes), [], set([1])
        uglies[0] = 1

        for k, p in enumerate(primes):
            heapq.heappush(heap, (p, k))
            ugly_set.add(p)

        for i in xrange(1, n):
            uglies[i], k = heapq.heappop(heap)
            while (primes[k] * uglies[idx[k]]) in ugly_set:
                idx[k] += 1
            heapq.heappush(heap, (primes[k] * uglies[idx[k]], k))
            ugly_set.add(primes[k] * uglies[idx[k]])

        return uglies[-1]

Python:　　

# Time:  O(n * logk) ~ O(n * klogk)
# Space: O(n + k)
class Solution3(object):
    def nthSuperUglyNumber(self, n, primes):
        """
        :type n: int
        :type primes: List[int]
        :rtype: int
        """
        uglies, idx, heap = [1], [0] * len(primes), []
        for k, p in enumerate(primes):
            heapq.heappush(heap, (p, k))

        for i in xrange(1, n):
            min_val, k = heap[0]
            uglies += [min_val]

            while heap[0][0] == min_val:  # worst time: O(klogk)
                min_val, k = heapq.heappop(heap)
                idx[k] += 1
                heapq.heappush(heap, (primes[k] * uglies[idx[k]], k))

        return uglies[-1]　　　　

C++:

class Solution {
public:
    int nthSuperUglyNumber(int n, vector<int>& primes) {
        vector<int> res(1, 1), idx(primes.size(), 0);
        while (res.size() < n) {
            vector<int> tmp;
            int mn = INT_MAX;
            for (int i = 0; i < primes.size(); ++i) {
                tmp.push_back(res[idx[i]] * primes[i]);
            }
            for (int i = 0; i < primes.size(); ++i) {
                mn = min(mn, tmp[i]);
            }
            for (int i = 0; i < primes.size(); ++i) {
                if (mn == tmp[i]) ++idx[i];
            }
            res.push_back(mn);
        }
        return res.back();
    }
};

C++:　　

class Solution {
public:
    int nthSuperUglyNumber(int n, vector<int>& primes) {
        vector<int> dp(n, 1), idx(primes.size(), 0);
        for (int i = 1; i < n; ++i) {
            dp[i] = INT_MAX;
            for (int j = 0; j < primes.size(); ++j) {
                dp[i] = min(dp[i], dp[idx[j]] * primes[j]);
            }
            for (int j = 0; j < primes.size(); ++j) {
                if (dp[i] == dp[idx[j]] * primes[j]) {
                    ++idx[j];
                }
            }
        }
        return dp.back();
    }
};

　　

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