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[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++

// 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; 
}

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:

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).

This article is contributed by Hemang Sarkar. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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

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