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#

   // 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); 
 
  

C#

   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)

Special two digit numbers in a Binary Search Tree

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

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