# k-th smallest absolute difference of two elements in an array

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.

Examples:

```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
```

## Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

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 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++

 `// C++ program to find k-th absolute difference ` `// between two elements ` `#include ` `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 - a; ` `    ``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; ` ` `  `    ``// 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); ` `    ``cout << kthDiff(a, n, k); ` `    ``return` `0; ` `} `

## Java

 `// Java program to find k-th absolute difference ` `// between two elements ` `import` `java.util.Scanner; ` `import` `java.util.Arrays; ` ` `  `class` `GFG ` `{ ` `    ``// returns number of pairs with absolute ` `    ``// difference less than or equal to mid  ` `    ``static` `int` `countPairs(``int``[] a, ``int` `n, ``int` `mid) ` `    ``{ ` `        ``int` `res = ``0``, value; ` `        ``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 (ub + i + 1) from ` `            ``// this position to count  ` `            ``int` `ub = upperbound(a, n, a[i]+mid); ` `            ``res += (ub- (i-``1``)); ` `        ``} ` `        ``return` `res; ` `    ``} ` ` `  `    ``// returns the upper bound ` `    ``static` `int` `upperbound(``int` `a[], ``int` `n, ``int` `value) ` `    ``{ ` `        ``int` `low = ``0``; ` `        ``int` `high = n; ` `        ``while``(low < high) ` `        ``{ ` `            ``final` `int` `mid = (low + high)/``2``; ` `            ``if``(value >= a[mid]) ` `                ``low = mid + ``1``; ` `            ``else` `                ``high = mid; ` `        ``} ` ` `  `    ``return` `low; ` `    ``} ` ` `  `    ``// Returns k-th absolute difference ` `    ``static` `int` `kthDiff(``int` `a[], ``int` `n, ``int` `k) ` `    ``{ ` `        ``// Sort array ` `        ``Arrays.sort(a); ` ` `  `        ``// Minimum absolute difference ` `        ``int` `low = a[``1``] - a[``0``]; ` `        ``for` `(``int` `i = ``1``; i <= n-``2``; ++i) ` `            ``low = Math.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 function to check the above functions ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``Scanner s = ``new` `Scanner(System.in); ` `        ``int` `k = ``3``; ` `        ``int` `a[] = {``1``,``2``,``3``,``4``}; ` `        ``int` `n = a.length; ` `        ``System.out.println(kthDiff(a, n, k)); ` `    ``} ` ` `  `} ` `// This code is contributed by nishkarsh146 `

## Python3

 `# Python3 program to find  ` `# k-th absolute difference  ` `# between two elements  ` `from` `bisect ``import` `bisect as upper_bound  ` ` `  `# returns number of pairs with  ` `# absolute difference less than  ` `# or equal to mid.  ` `def` `countPairs(a, n, mid):  ` `    ``res ``=` `0` `    ``for` `i ``in` `range``(n):  ` ` `  `        ``# 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, a[i] ``+` `mid)  ` `    ``return` `res  ` ` `  `# Returns k-th absolute difference  ` `def` `kthDiff(a, n, k):  ` `     `  `    ``# Sort array  ` `    ``a ``=` `sorted``(a)  ` ` `  `    ``# Minimum absolute difference  ` `    ``low ``=` `a[``1``] ``-` `a[``0``]  ` `    ``for` `i ``in` `range``(``1``, n ``-` `1``):  ` `        ``low ``=` `min``(low, a[i ``+` `1``] ``-` `a[i])  ` ` `  `    ``# Maximum absolute difference  ` `    ``high ``=` `a[n ``-` `1``] ``-` `a[``0``]  ` ` `  `    ``# Do binary search for k-th absolute difference  ` `    ``while` `(low < high):  ` `        ``mid ``=` `(low ``+` `high) >> ``1` `        ``if` `(countPairs(a, n, mid) < k):  ` `            ``low ``=` `mid ``+` `1` `        ``else``:  ` `            ``high ``=` `mid  ` ` `  `    ``return` `low  ` ` `  `# Driver code  ` `k ``=` `3` `a ``=` `[``1``, ``2``, ``3``, ``4``]  ` `n ``=` `len``(a)  ` `print``(kthDiff(a, n, k))  ` ` `  `# This code is contributed by Mohit Kumar  `

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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