LWC 72: 786. K-th Smallest Prime Fraction

Posted 2022-12-08 Demon的黑与白

LWC 72: 786. K-th Smallest Prime Fraction

传送门：786. K-th Smallest Prime Fraction

Problem:

A sorted list A contains 1, plus some number of primes. Then, for every p < q in the list, we consider the fraction p/q.

What is the K-th smallest fraction considered? Return your answer as an array of ints, where answer[0] = p and answer1 = q.

Examples:

Input: A = [1, 2, 3, 5], K = 3
Output: [2, 5]
Explanation:
The fractions to be considered in sorted order are:
1/5, 1/3, 2/5, 1/2, 3/5, 2/3.
The third fraction is 2/5.

Input: A = [1, 7], K = 1
Output: [1, 7]

Note:

• A will have length between 2 and 2000.
• Each A[i] will be between 1 and 30000.
• K will be between 1 and A.length * (A.length + 1) / 2.

思路1：
一种聪明的做法，如果A = [1, 7, 23, 29, 47]，那么有：

1/47   < 1/29    < 1/23 < 1/7
7/47   < 7/29    < 7/23
23/47  < 23/29
29/47

可以采用优先队列的方法，把第一列先送入队列，因为他们一定是每行的最小，选出最小的之后，加入该行的后续元素。直到找到K-1个最小的元素。

代码如下：

    public int[] kthSmallestPrimeFraction(int[] a, int k) 
        int n = a.length;
        PriorityQueue<int[]> pq = new PriorityQueue<>(new Comparator<int[]>() 
            @Override
            public int compare(int[] o1, int[] o2) 
                int s1 = a[o1[0]] * a[o2[1]];
                int s2 = a[o2[0]] * a[o1[1]];
                return s1 - s2;
            
        );
        for (int i = 0; i < n-1; i++) 
            pq.add(new int[]i, n-1);
        
        for (int i = 0; i < k-1; i++) 
            int[] pop = pq.poll();
            if (pop[1] - 1 > pop[0]) 
                pop[1]--;
                pq.offer(pop);
            
        

        int[] peek = pq.peek();
        return new int[]a[peek[0]], a[peek[1]];
    

思路2：
二分，先找到这个第k小的数，接着再小范围暴力搜索，居然能过。

代码如下：

    public int[] kthSmallestPrimeFraction(int[] a, int K) 
        double low = 0, high = 1;
        double eps = 1e-8;
        int n = a.length;
        for(int rep = 0; rep < 50; ++rep)
            double x = low + (high-low) / 2;
            int num = 0;
            for(int i = 0;i < n;i++)
                int ind = Arrays.binarySearch(a, (int)(x * a[i]));
                if(ind < 0) ind = -ind - 2;
                num += ind + 1;
            
            if(num >= K)
                high = x;
            else
                low = x;
            
        

        for (int i = 0; i < n; ++i) 
            double find = a[i] * high;
            int idx = Arrays.binarySearch(a, (int)find);
            if (idx < 0) idx = -idx;
            for (int j = -1; j <= 1; ++j) 
                if (j + idx >= 0 && j + idx < n && Math.abs(a[j + idx] - find) < eps) 
                    return new int[] a[j + idx], a[i];
                
            
        
        return new int[]-1, -1;