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This approach uses a min-heap to efficiently find the kth smallest fraction. By iterating through each pair of elements (i, j) in the array where i < j, we can calculate the fraction arr[i]/arr[j] and push it into the heap. The heap is then used to retrieve the k-th smallest element efficiently.
Time Complexity: O(n^2 log(n^2))
Space Complexity: O(n^2)
1var kthSmallestPrimeFraction = function(arr, k) {
2 let n = arr.length;
3 let heap = new MinPriorityQueue({
4 compare: (a, b) => arr[a[0]] * arr[b[1]] - arr[b[0]] * arr[a[1]]
5 });
6
7 for(let i = 0; i < n; i++) {
8 heap.enqueue([i, n - 1]);
9 }
10 let res = null;
11 while(k--) {
12 res = heap.dequeue().element;
13 if(res[1] - 1 > res[0]) heap.enqueue([res[0], res[1] - 1]);
14 }
15 return [arr[res[0]], arr[res[1]]];
16};This JavaScript implementation leverages a MinPriorityQueue for extracting the k-th smallest fraction. The queue uses a custom comparator function to compare based on calculated fraction values.
This approach uses binary search on the value of fractions. By implementing a search over the potential range of fraction values, the algorithm counts how many fractions are less than a given mid-point, narrowing down to the k-th one by manipulating the bounds of the potential fraction values. A two-pointer technique helps count the fractions efficiently.
Time Complexity: O(n log(max_fraction))
Space Complexity: O(1)
1
public class Solution {
public int[] KthSmallestPrimeFraction(int[] arr, int k) {
double lo = 0.0, hi = 1.0;
while (hi - lo > 1e-9) {
double mid = (lo + hi) / 2.0;
int count = 0;
int p = 0, q = 1;
double maxFraction = 0.0;
int i = 0;
for (int j = 1; j < arr.Length; j++) {
while (i < j && arr[i] < arr[j] * mid) i++;
count += i;
if (i > 0) {
double fraction = (double)arr[i - 1] / arr[j];
if (fraction > maxFraction) {
maxFraction = fraction;
p = arr[i - 1];
q = arr[j];
}
}
}
if (count < k) lo = mid;
else {
hi = mid;
result[0] = p;
result[1] = q;
}
}
return result;
}
}The C# implementation uses binary search and a two-pointer method to count and track fractions. Adjusting the fraction value bounds is key to finding the k-th smallest fraction.