Minimum Possible value of |ai + aj - k| for given array and k.

Last Updated : 19 Sep, 2023

You are given an array of n integer and an integer K. Find the number of total unordered pairs {i, j} such that absolute value of (ai + aj - K), i.e., |ai + aj - k| is minimal possible, where i != j.
Examples:

Input: arr[] = {0, 4, 6, 2, 4}, K = 7
Output: Minimal Value = 1, Total Pairs = 5
Explanation: Pairs resulting minimal value are : {a1, a3}, {a2, a4}, {a2, a5}, {a3, a4}, {a4, a5}
Input: arr[] = {4, 6, 2, 4} , K = 9
Output: Minimal Value = 1, Total Pairs = 4
Explanation: Pairs resulting minimal value are : {a1, a2}, {a1, a4}, {a2, a3}, {a2, a4}

A simple solution is iterate over all possible pairs and for each pair we will check whether the value of (ai + aj - K) is smaller than our current smallest value of not. So as per result of above condition we have total of three cases :

abs( ai + aj - K) > smallest : do nothing as this pair will not count in minimal possible value.
abs(ai + aj - K) = smallest : increment the count of pair resulting minimal possible value.
abs( ai + aj - K) < smallest : update the smallest value and set count to 1.

Below is the implementation of the above approach:

C++

// CPP program to find number of pairs  and minimal
// possible value
#include <bits/stdc++.h>
using namespace std;

// function for finding pairs and min value
void pairs(int arr[], int n, int k)
{
    // initialize smallest and count
    int smallest = INT_MAX;
    int count = 0;

    // iterate over all pairs
    for (int i = 0; i < n; i++)
        for (int j = i + 1; j < n; j++) {
            // is abs value is smaller than smallest
            // update smallest and reset count to 1
            if (abs(arr[i] + arr[j] - k) < smallest) {
                smallest = abs(arr[i] + arr[j] - k);
                count = 1;
            }

            // if abs value is equal to smallest
            // increment count value
            else if (abs(arr[i] + arr[j] - k) == smallest)
                count++;
        }

    // print result
    cout << "Minimal Value = " << smallest << "\n";
    cout << "Total Pairs = " << count << "\n";
}

// driver program
int main()
{
    int arr[] = { 3, 5, 7, 5, 1, 9, 9 };
    int k = 12;
    int n = sizeof(arr) / sizeof(arr[0]);
    pairs(arr, n, k);
    return 0;
}

Java

// Java program to find number of pairs
// and minimal possible value
import java.util.*;

class GFG {

    // function for finding pairs and min value
    static void pairs(int arr[], int n, int k)
    {
        // initialize smallest and count
        int smallest = Integer.MAX_VALUE;
        int count = 0;

        // iterate over all pairs
        for (int i = 0; i < n; i++)
            for (int j = i + 1; j < n; j++) {
                // is abs value is smaller than
                // smallest update smallest and
                // reset count to 1
                if (Math.abs(arr[i] + arr[j] - k)
                    < smallest) {
                    smallest
                        = Math.abs(arr[i] + arr[j] - k);
                    count = 1;
                }

                // if abs value is equal to smallest
                // increment count value
                else if (Math.abs(arr[i] + arr[j] - k)
                         == smallest)
                    count++;
            }

        // print result
        System.out.println("Minimal Value = " + smallest);
        System.out.println("Total Pairs = " + count);
    }

    /* Driver program to test above function */
    public static void main(String[] args)
    {
        int arr[] = { 3, 5, 7, 5, 1, 9, 9 };
        int k = 12;
        int n = arr.length;
        pairs(arr, n, k);
    }
}
// This code is contributed by Arnav Kr. Mandal.

Python3

# Python3 program to find number of pairs
# and minimal possible value

# function for finding pairs and min value


def pairs(arr, n, k):

    # initialize smallest and count
    smallest = 999999999999
    count = 0

    # iterate over all pairs
    for i in range(n):
        for j in range(i + 1, n):

            # is abs value is smaller than smallest
            # update smallest and reset count to 1
            if abs(arr[i] + arr[j] - k) < smallest:
                smallest = abs(arr[i] + arr[j] - k)
                count = 1

            # if abs value is equal to smallest
            # increment count value
            elif abs(arr[i] + arr[j] - k) == smallest:
                count += 1

    # print result
    print("Minimal Value = ", smallest)
    print("Total Pairs = ", count)


# Driver Code
if __name__ == '__main__':
    arr = [3, 5, 7, 5, 1, 9, 9]
    k = 12
    n = len(arr)
    pairs(arr, n, k)

# This code is contributed by PranchalK

// C# program to find number
// of pairs and minimal
// possible value
using System;

class GFG {

    // function for finding
    // pairs and min value
    static void pairs(int[] arr, int n, int k)
    {
        // initialize
        // smallest and count
        int smallest = 0;
        int count = 0;

        // iterate over all pairs
        for (int i = 0; i < n; i++)
            for (int j = i + 1; j < n; j++) {
                // is abs value is smaller
                // than smallest update
                // smallest and reset
                // count to 1
                if (Math.Abs(arr[i] + arr[j] - k)
                    < smallest) {
                    smallest
                        = Math.Abs(arr[i] + arr[j] - k);
                    count = 1;
                }

                // if abs value is equal
                // to smallest increment
                // count value
                else if (Math.Abs(arr[i] + arr[j] - k)
                         == smallest)
                    count++;
            }

        // print result
        Console.WriteLine("Minimal Value = " + smallest);
        Console.WriteLine("Total Pairs = " + count);
    }

    // Driver Code
    public static void Main()
    {
        int[] arr = { 3, 5, 7, 5, 1, 9, 9 };
        int k = 12;
        int n = arr.Length;
        pairs(arr, n, k);
    }
}

// This code is contributed
// by anuj_67.

PHP

<?php
// PHP program to find number of
// pairs and minimal possible value

// function for finding pairs
// and min value
function pairs($arr, $n, $k)
{
    
    // initialize smallest and count
    $smallest = PHP_INT_MAX;
    $count = 0;

    // iterate over all pairs
    for ($i = 0; $i < $n; $i++)
        for($j = $i + 1; $j < $n; $j++)
        {
            
            // is abs value is smaller than smallest
            // update smallest and reset count to 1
            if ( abs($arr[$i] + $arr[$j] - $k) < $smallest )
            { 
                $smallest = abs($arr[$i] + $arr[$j] - $k);
                $count = 1;
            }

            // if abs value is equal to smallest
            // increment count value
            else if (abs($arr[$i] + 
                     $arr[$j] - $k) == $smallest)
                $count++;
        }

        // print result
        echo "Minimal Value = " , $smallest , "\n";
        echo "Total Pairs = ", $count , "\n"; 
} 

    // Driver Code
    $arr = array (3, 5, 7, 5, 1, 9, 9);
    $k = 12;
    $n = sizeof($arr);
    pairs($arr, $n, $k);

// This code is contributed by aj_36 
?>

JavaScript

<script>

// Javascript program to find number of pairs  and minimal 
// possible value

// function for finding pairs and min value
function pairs(arr, n, k)
{
    // initialize smallest and count
    var smallest = 1000000000;
    var count=0;

    // iterate over all pairs
    for (var i=0; i<n; i++)
        for(var j=i+1; j<n; j++)
        {
            // is Math.abs value is smaller than smallest
            // update smallest and reset count to 1
            if ( Math.abs(arr[i] + arr[j] - k) < smallest )
            { 
                smallest = Math.abs(arr[i] + arr[j] - k);
                count = 1;
            }

            // if Math.abs value is equal to smallest
            // increment count value
            else if (Math.abs(arr[i] + arr[j] - k) == smallest)
                count++;
        }

        // print result
        document.write( "Minimal Value = " + smallest + "<br>");
        document.write( "Total Pairs = " + count + "<br>");    
} 

// driver program
var arr = [3, 5, 7, 5, 1, 9, 9];
var k = 12;
var n = arr.length;
pairs(arr, n, k);

</script>

Output

Minimal Value = 0
Total Pairs = 4

Time Complexity: O(n²) where n is the number of elements in the array.
Auxiliary Space : O(1)

An efficient solution is to use a self balancing binary search tree (which is implemented in set in C++ and TreeSet in Java). We can find closest element in O(log n) time in map.

C++

// C++ program to find number of pairs
// and minimal possible value
#include <bits/stdc++.h>
using namespace std;

// function for finding pairs and min value
void pairs(int arr[], int n, int k)
{
    // initialize smallest and count
    int smallest = INT_MAX, count = 0;
    set<int> s;

    // iterate over all pairs
    s.insert(arr[0]);
    for (int i = 1; i < n; i++) {
        // Find the closest elements to  k - arr[i]
        int lower
            = *lower_bound(s.begin(), s.end(), k - arr[i]);

        int upper
            = *upper_bound(s.begin(), s.end(), k - arr[i]);

        // Find absolute value of the pairs formed
        // with closest greater and smaller elements.
        int curr_min = min(abs(lower + arr[i] - k),
                           abs(upper + arr[i] - k));

        // is abs value is smaller than smallest
        // update smallest and reset count to 1
        if (curr_min < smallest) {
            smallest = curr_min;
            count = 1;
        }

        // if abs value is equal to smallest
        // increment count value
        else if (curr_min == smallest)
            count++;
        s.insert(arr[i]);

    } // print result

    cout << "Minimal Value = " << smallest << "\n";
    cout << "Total Pairs = " << count << "\n";
}

// driver program
int main()
{
    int arr[] = { 3, 5, 7, 5, 1, 9, 9 };
    int k = 12;
    int n = sizeof(arr) / sizeof(arr[0]);
    pairs(arr, n, k);
    return 0;
}

Python3

# Python program to find number of pairs
# and minimal possible value

from sys import maxsize
from bisect import bisect_left, bisect_right

# function for finding pairs and min value
def pairs(arr, n, k):
    # initialize smallest and count
    smallest = maxsize
    count = 0
    s = set()

    # iterate over all pairs
    s.add(arr[0])
    for i in range(1, n):
        # Find the closest elements to k - arr[i]
        sorted_s = sorted(s)
        index = bisect_left(sorted_s, k - arr[i])
        if index == len(sorted_s):
            lower = sorted_s[index - 1]
        else:
            lower = sorted_s[index]
        index = bisect_right(sorted_s, k - arr[i])
        if index == len(sorted_s):
            upper = sorted_s[index - 1]
        else:
            upper = sorted_s[index]

        # Find absolute value of the pairs formed
        # with closest greater and smaller elements.
        curr_min = min(abs(lower + arr[i] - k), abs(upper + arr[i] - k))

        # is abs value is smaller than smallest
        # update smallest and reset count to 1
        if curr_min < smallest:
            smallest = curr_min
            count = 1
        # if abs value is equal to smallest
        # increment count value
        elif curr_min == smallest:
            count += 1
        s.add(arr[i])

    # print result
    print("Minimal Value = ", smallest)
    print("Total Pairs = ", count)

# driver program
arr = [3, 5, 7, 5, 1, 9, 9]
k = 12
n = len(arr)
pairs(arr, n, k)

# This code is contributed by vikramshirsath177.

Java

import java.util.*;

class Main {
    // function for finding pairs and min value
    static void pairs(final int[] arr,final int n,final int k) {
        // initialize smallest and count
        int smallest = Integer.MAX_VALUE, count = 0;
        Set<Integer> s = new TreeSet<>();

        // iterate over all pairs
        s.add(arr[0]);
        for (int i = 1; i < n; i++) {
            // Find the closest elements to  k - arr[i]
            int lower = Integer.MIN_VALUE;
            int upper = Integer.MAX_VALUE;
            for (Integer x : s) {
                if (x <= (k - arr[i]) && x >= lower) {
                   lower = x;
                 }
                if (x >= (k - arr[i]) && x <= upper) {
                    upper = x;
                 }
             }

            // Find absolute value of the pairs formed
            // with closest greater and smaller elements.
            int curr_min = Math.min(Math.abs(lower + arr[i] - k), Math.abs(upper + arr[i] - k));

            // is abs value is smaller than smallest
            // update smallest and reset count to 1
            if (curr_min < smallest) {
                smallest = curr_min;
                count = 1;
            }

            // if abs value is equal to smallest
            // increment count value
            else if (curr_min == smallest)
                count++;
            s.add(arr[i]);

        } // print result

        System.out.println("Minimal Value = " + smallest);
        System.out.println("Total Pairs = " + count);
    }

    // driver program
    public static void main(String[] args) {
        int[] arr = {3, 5, 7, 5, 1, 9, 9};
        int k = 12;
        int n = arr.length;
        pairs(arr, n, k);
    }
}

JavaScript

function pairs(arr, n, k) {
    // initialize smallest and count
    let smallest = Number.MAX_SAFE_INTEGER;
    let count = 0;
    let s = new Set();
 
    // iterate over all pairs
    s.add(arr[0]);
    for (let i = 1; i < n; i++) {
        // Find the closest elements to  k - arr[i]
        let lower = [...s].find((element) => element >= k - arr[i]);
        let upper = [...s].find((element) => element >= k - arr[i]);
 
        // Find absolute value of the pairs formed
        // with closest greater and smaller elements.
        let curr_min = Math.min(Math.abs(lower + arr[i] - k), Math.abs(upper + arr[i] - k));
 
        // if abs value is smaller than smallest
        // update smallest and reset count to 1
        if (curr_min < smallest) {
            smallest = curr_min;
            count = 1;
        }
        // if abs value is equal to smallest
        // increment count value
        else if (curr_min === smallest)
            count++;
        s.add(arr[i]);
    }
 
    // print result
    console.log(`Minimal Value = ${smallest}`);
    console.log(`Total Pairs = ${count}`);
}
 
// driver program
let arr = [3, 5, 7, 5, 1, 9, 9];
let k = 12;
let n = arr.length;
pairs(arr, n, k);

using System;
using System.Collections.Generic;
using System.Linq;

class Program
{
    // function for finding pairs and min value
    static void pairs(int[] arr, int n, int k)
    {
        // initialize smallest and count
        int smallest = int.MaxValue, count = 0;
        SortedSet<int> s = new SortedSet<int>();

        // iterate over all pairs
        s.Add(arr[0]);
        for (int i = 1; i < n; i++)
        {
            // Find the closest elements to k - arr[i]
            int lower = s.Where(e => e >= k - arr[i]).DefaultIfEmpty(int.MinValue).First();
            int upper = s.Where(e => e > k - arr[i]).DefaultIfEmpty(int.MaxValue).First();

            // Find absolute value of the pairs formed
            // with closest greater and smaller elements.
            int curr_min = Math.Min(Math.Abs(lower + arr[i] - k), Math.Abs(upper + arr[i] - k));

            // if abs value is smaller than smallest
            // update smallest and reset count to 1
            if (curr_min < smallest)
            {
                smallest = curr_min;
                count = 1;
            }
            // if abs value is equal to smallest
            // increment count value
            else if (curr_min == smallest)
            {
                count++;
            }
            s.Add(arr[i]);
        }

        // print result
        Console.WriteLine("Minimal Value = " + smallest);
        Console.WriteLine("Total Pairs = " + count);
    }

    // driver program
    static void Main(string[] args)
    {
        int[] arr = { 3, 5, 7, 5, 1, 9, 9 };
        int k = 12;
        int n = arr.Length;
        pairs(arr, n, k);
    }
}

Output

Minimal Value = 0
Total Pairs = 4

Time Complexity : O(n Log n)
Auxiliary Space: O(n)

Shivam.Pradhan

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