We are given an array of size n containing positive integers. The absolute difference between values at indices i and j is |a[i] – a[j]|. There are n*(n-1)/2 such pairs and we are asked to print the kth (1 <= k <= n*(n-1)/2) smallest absolute difference among all these pairs.

Input : a[] = {1, 2, 3, 4} k = 3 Output : 1 The possible absolute differences are : {1, 2, 3, 1, 2, 1}. The 3rd smallest value among these is 1. Input : n = 2 a[] = {10, 10} k = 1 Output : 0

**Naive Method** is to find all the n*(n-1)/2 possible absolute differences in O(n^2) and store them in an array. Then sort this array and print the k-th minimum value from this array. This will take time O(n^2 + n^2 * log(n^2)) = O(n^2 + 2*n^2*log(n)).

The naive method won't be efficient for large values of n, say n = 10^5.

An **Efficient Solution** is based on Binary Search.

1) Sort the given array a[]. 2) We can easily find the least possible absolute difference in O(n) after sorting. The largest possible difference will be a[n-1] - a[0] after sorting the array. Let low = minimum_difference and high = maximum_difference. 3) while low < high: 4) mid = (low + high)/2 5) if ((number of pairs with absolute difference <= mid) < k): 6) low = mid + 1 7) else: 8) high = mid 9) return low

We need a function that will tell us number of pairs with difference <= mid efficiently.

Since our array is sorted, this part can be done like this:

1) result = 0 2) for i = 0 to n-1: 3) result = result + (upper_bound(a+i, a+n, a[i] + mid) - (a+i+1)) 4) return result

Here upper_bound is a variant of binary search which returns a pointer to the first element from a[i] to a[n-1] which is greater than a[i] + mid. Let the pointer returned be j. Then a[i] + mid < a[j]. Thus, subtracting (a+i+1) from this will give us the number of values whose difference with a[i] is <= mid. We sum this up for all indices from 0 to n-1 and get the answer for current mid.

// C++ program to find k-th absolute difference // between two elements #include<bits/stdc++.h> using namespace std; // returns number of pairs with absolute difference // less than or equal to mid. int countPairs(int *a, int n, int mid) { int res = 0; for (int i = 0; i < n; ++i) // Upper bound returns pointer to position // of next higher number than a[i]+mid in // a[i..n-1]. We subtract (a + i + 1) from // this position to count res += upper_bound(a+i, a+n, a[i] + mid) - (a + i + 1); return res; } // Returns k-th absolute difference int kthDiff(int a[], int n, int k) { // Sort array sort(a, a+n); // Minimum absolute difference int low = a[1] - a[0]; for (int i = 1; i <= n-2; ++i) low = min(low, a[i+1] - a[i]); // Maximum absolute difference int high = a[n-1] - a[0]; // Do binary search for k-th absolute difference while (low < high) { int mid = (low+high)>>1; if (countPairs(a, n, mid) < k) low = mid + 1; else high = mid; } return low; } // Driver code int main() { int k = 3; int a[] = {1, 2, 3, 4}; int n = sizeof(a)/sizeof(a[0]); cout << kthDiff(a, n, k); return 0; }

Output: 1

The time complexity of the algorithm is O( n*logn + n*logn*logn). Sorting takes O(n*logn). After that the main binary search over low and high takes O(n*logn*logn) time because each call to the function int f(int c, int n, int* a) takes time O(n*logn).

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