Maximum sum of pairs with specific difference

Given an array of integers and a number k. We can pair two numbers of the array if the difference between them is strictly less than k. The task is to find the maximum possible sum of disjoint pairs. Sum of P pairs is the sum of all 2P numbers of pairs.

Examples:

Input  : arr[] = {3, 5, 10, 15, 17, 12, 9}, K = 4
Output : 62
Explanation:
Then disjoint pairs with difference less than K are, (3, 5), (10, 12), (15, 17)  
So maximum sum which we can get is 3 + 5 + 12 + 10 + 15 + 17 = 62
Note that an alternate way to form disjoint pairs is, (3, 5), (9, 12), (15, 17), but this pairing produces lesser sum.

Input  : arr[] = {5, 15, 10, 300}, k = 12
Output : 25

Approach: First, we sort the given array in increasing order. Once array is sorted, we traverse the array. For every element, we try to pair it with its previous element first. Why do we prefer previous element? Let arr[i] can be paired with arr[i-1] and arr[i-2] (i.e. arr[i] – arr[i-1] < K and arr[i]-arr[i-2] < K). Since the array is sorted, value of arr[i-1] would be more than arr[i-2]. Also, we need to pair with difference less than k, it means if arr[i-2] can be paired, then arr[i-1] can also be paired in a sorted array. 

Now observing the above facts, we can formulate our dynamic programming solution as below, 

Let dp[i] denotes the maximum disjoint pair sum we can achieve using first i elements of the array. Assume currently, we are at i’th position, then there are two possibilities for us. 

  Pair up i with (i-1)th element, i.e. 
      dp[i] = dp[i-2] + arr[i] + arr[i-1]
  Don't pair up, i.e. 
      dp[i] = dp[i-1] 

Above iteration takes O(N) time and sorting of array will take O(N log N) time so total time complexity of the solution will be O(N log N) 

Implementation:

C++

// C++ program to find maximum pair sum whose
// difference is less than K
#include <bits/stdc++.h>
using namespace std;

// method to return maximum sum we can get by
// finding less than K difference pair
int maxSumPairWithDifferenceLessThanK(int arr[], int N, int K)
{
    // Sort input array in ascending order.
    sort(arr, arr+N);

    // dp[i] denotes the maximum disjoint pair sum
    // we can achieve using first i elements
    int dp[N];

    //  if no element then dp value will be 0
    dp[0] = 0;

    for (int i = 1; i < N; i++)
    {
        // first give previous value to dp[i] i.e.
        // no pairing with (i-1)th element
        dp[i] = dp[i-1];

        // if current and previous element can form a pair
        if (arr[i] - arr[i-1] < K)
        {
            // update dp[i] by choosing maximum between
            // pairing and not pairing
            if (i >= 2)
                dp[i] = max(dp[i], dp[i-2] + arr[i] + arr[i-1]);
            else
                dp[i] = max(dp[i], arr[i] + arr[i-1]);
        }
    }

    //  last index will have the result
    return dp[N - 1];
}

//  Driver code to test above methods
int main()
{
    int arr[] = {3, 5, 10, 15, 17, 12, 9};
    int N = sizeof(arr)/sizeof(int);

    int K = 4;
    cout << maxSumPairWithDifferenceLessThanK(arr, N, K);
    return 0;
}

Java

// Java program to find maximum pair sum whose
// difference is less than K

import java.io.*;
import java.util.*;

class GFG {

    // method to return maximum sum we can get by
    // finding less than K difference pair
    static int maxSumPairWithDifferenceLessThanK(int arr[],
                                               int N, int K)
    {

        // Sort input array in ascending order.
        Arrays.sort(arr);

        // dp[i] denotes the maximum disjoint pair sum
        // we can achieve using first i elements
        int dp[] = new int[N];

        // if no element then dp value will be 0
        dp[0] = 0;

        for (int i = 1; i < N; i++)
        {
            // first give previous value to dp[i] i.e.
            // no pairing with (i-1)th element
            dp[i] = dp[i-1];

            // if current and previous element can form a pair
            if (arr[i] - arr[i-1] < K)
            {

                // update dp[i] by choosing maximum between
                // pairing and not pairing
                if (i >= 2)
                    dp[i] = Math.max(dp[i], dp[i-2] + arr[i] +
                                                    arr[i-1]);
                else
                    dp[i] = Math.max(dp[i], arr[i] + arr[i-1]);
            }
        }

        // last index will have the result
        return dp[N - 1];
    }

    // Driver code to test above methods
    public static void main (String[] args) {

        int arr[] = {3, 5, 10, 15, 17, 12, 9};
        int N = arr.length;
        int K = 4;

        System.out.println ( maxSumPairWithDifferenceLessThanK(
                                                    arr, N, K));

    }
}

//This code is contributed by vt_m.

Python3

# Python3 program to find maximum pair
# sum whose difference is less than K

# method to return maximum sum we can
# get by get by finding less than K
# difference pair
def maxSumPairWithDifferenceLessThanK(arr, N, K):

    # Sort input array in ascending order.
    arr.sort()

    # dp[i] denotes the maximum disjoint
    # pair sum we can achieve using first
    # i elements
    dp = [0] * N

    # if no element then dp value will be 0
    dp[0] = 0

    for i in range(1, N):

        # first give previous value to
        # dp[i] i.e. no pairing with
        # (i-1)th element
        dp[i] = dp[i-1]

        # if current and previous element
        # can form a pair
        if (arr[i] - arr[i-1] < K):

            # update dp[i] by choosing
            # maximum between pairing
            # and not pairing
            if (i >= 2):
                dp[i] = max(dp[i], dp[i-2] + arr[i] + arr[i-1]);
            else:
                dp[i] = max(dp[i], arr[i] + arr[i-1]);

    # last index will have the result
    return dp[N - 1]

# Driver code to test above methods
arr = [3, 5, 10, 15, 17, 12, 9]
N = len(arr)
K = 4
print(maxSumPairWithDifferenceLessThanK(arr, N, K))

# This code is contributed by Smitha Dinesh Semwal

C#

// C# program to find maximum pair sum whose
// difference is less than K
using System;

class GFG {

    // method to return maximum sum we can get by
    // finding less than K difference pair
    static int maxSumPairWithDifferenceLessThanK(int []arr,
                                              int N, int K)
    {

        // Sort input array in ascending order.
        Array.Sort(arr);

        // dp[i] denotes the maximum disjoint pair sum
        // we can achieve using first i elements
        int []dp = new int[N];

        // if no element then dp value will be 0
        dp[0] = 0;

        for (int i = 1; i < N; i++)
        {
            // first give previous value to dp[i] i.e.
            // no pairing with (i-1)th element
            dp[i] = dp[i-1];

            // if current and previous element can form
            // a pair
            if (arr[i] - arr[i-1] < K)
            {

                // update dp[i] by choosing maximum
                // between pairing and not pairing
                if (i >= 2)
                    dp[i] = Math.Max(dp[i], dp[i-2]
                                + arr[i] + arr[i-1]);
                else
                    dp[i] = Math.Max(dp[i], arr[i]
                                        + arr[i-1]);
            }
        }

        // last index will have the result
        return dp[N - 1];
    }

    // Driver code to test above methods
    public static void Main () {

        int []arr = {3, 5, 10, 15, 17, 12, 9};
        int N = arr.Length;
        int K = 4;

        Console.WriteLine(
          maxSumPairWithDifferenceLessThanK(arr, N, K));

    }
}

// This code is contributed by anuj_67.

PHP

<?php
// Php program to find maximum pair sum whose
// difference is less than K

// method to return maximum sum we can get by
// finding less than K difference pair
function maxSumPairWithDifferenceLessThanK($arr, $N, $K)
{
    // Sort input array in ascending order.
    sort($arr) ;

    // dp[i] denotes the maximum disjoint pair sum
    // we can achieve using first i elements
    $dp = array() ;

    // if no element then dp value will be 0
    $dp[0] = 0;

    for ($i = 1; $i < $N; $i++)
    {
        // first give previous value to dp[i] i.e.
        // no pairing with (i-1)th element
        $dp[$i] = $dp[$i-1];

        // if current and previous element can form a pair
        if ($arr[$i] - $arr[$i-1] < $K)
        {
            // update dp[i] by choosing maximum between
            // pairing and not pairing
            if ($i >= 2)
                $dp[$i] = max($dp[$i], $dp[$i-2] + $arr[$i] + $arr[$i-1]);
            else
                $dp[$i] = max($dp[$i], $arr[$i] + $arr[$i-1]);
        }
    }

    // last index will have the result
    return $dp[$N - 1];
}

    // Driver code

    $arr = array(3, 5, 10, 15, 17, 12, 9);
    $N = sizeof($arr) ;

    $K = 4;
    echo maxSumPairWithDifferenceLessThanK($arr, $N, $K);


    // This code is contributed by Ryuga
?>

Javascript

<script>

// Javascript program to find maximum pair sum whose
// difference is less than K

    // method to return maximum sum we can get by
    // finding less than K difference pair
    function maxSumPairWithDifferenceLessThanK(arr,
                                               N, K)
    {

        // Sort input array in ascending order.
        arr.sort();

        // dp[i] denotes the maximum disjoint pair sum
        // we can achieve using first i elements
        let dp = [];

        // if no element then dp value will be 0
        dp[0] = 0;

        for (let i = 1; i < N; i++)
        {
            // first give previous value to dp[i] i.e.
            // no pairing with (i-1)th element
            dp[i] = dp[i-1];

            // if current and previous element can form a pair
            if (arr[i] - arr[i-1] < K)
            {

                // update dp[i] by choosing maximum between
                // pairing and not pairing
                if (i >= 2)
                    dp[i] = Math.max(dp[i], dp[i-2] + arr[i] +
                                                    arr[i-1]);
                else
                    dp[i] = Math.max(dp[i], arr[i] + arr[i-1]);
            }
        }

        // last index will have the result
        return dp[N - 1];
    }

// Driver code to test above methods

        let arr = [3, 5, 10, 15, 17, 12, 9];
        let N = arr.length;
        let K = 4;

        document.write( maxSumPairWithDifferenceLessThanK(
                                                    arr, N, K));
// This code is contributed by avijitmondal1998.
</script>

Output

62

Time complexity: O(N Log N) 
Auxiliary Space: O(N)

An optimised solution contributed by Amit Sane is given below, 

Implementation:

C++

// C++ program to find maximum pair sum whose
// difference is less than K
#include <bits/stdc++.h>
using namespace std;

// Method to return maximum sum we can get by
// finding less than K difference pairs
int maxSumPair(int arr[], int N, int k)
{
    int maxSum = 0;

    // Sort elements to ensure every i and i-1 is closest
    // possible pair
    sort(arr, arr + N);

    // To get maximum possible sum,
    // iterate from largest to
    // smallest, giving larger
    // numbers priority over smaller
    // numbers.
    for (int i = N - 1; i > 0; --i)
    {
        // Case I: Diff of arr[i] and arr[i-1]
        //  is less than K,add to maxSum
        // Case II: Diff between arr[i] and arr[i-1] is not
        // less than K, move to next i since with
        // sorting we know, arr[i]-arr[i-1] <
        // rr[i]-arr[i-2] and so on.
        if (arr[i] - arr[i - 1] < k)
        {
            // Assuming only positive numbers.
            maxSum += arr[i];
            maxSum += arr[i - 1];

            // When a match is found skip this pair
            --i;
        }
    }

    return maxSum;
}

// Driver code
int main()
{
    int arr[] = { 3, 5, 10, 15, 17, 12, 9 };
    int N = sizeof(arr) / sizeof(int);

    int K = 4;
    cout << maxSumPair(arr, N, K);
    return 0;
}

Java

// Java program to find maximum pair sum whose
// difference is less than K

import java.io.*;
import java.util.*;

class GFG {

    // Method to return maximum sum we can get by
    // finding less than K difference pairs
    static int maxSumPairWithDifferenceLessThanK(int arr[],
                                                 int N,
                                                 int k)
    {
        int maxSum = 0;

        // Sort elements to ensure every i and i-1 is
        // closest possible pair
        Arrays.sort(arr);

        // To get maximum possible sum,
        // iterate from largest
        // to smallest, giving larger
        // numbers priority over
        // smaller numbers.
        for (int i = N - 1; i > 0; --i)
        {
            // Case I: Diff of arr[i] and arr[i-1] is less
            // than K, add to maxSum
            // Case II: Diff between arr[i] and arr[i-1] is
            // not less than K, move to next i
            // since with sorting we know, arr[i]-arr[i-1] <
            // arr[i]-arr[i-2] and so on.
            if (arr[i] - arr[i - 1] < k)
            {
                // Assuming only positive numbers.
                maxSum += arr[i];
                maxSum += arr[i - 1];

                // When a match is found skip this pair
                --i;
            }
        }

        return maxSum;
    }

    // Driver code
    public static void main(String[] args)
    {

        int arr[] = { 3, 5, 10, 15, 17, 12, 9 };
        int N = arr.length;
        int K = 4;

        System.out.println(
            maxSumPairWithDifferenceLessThanK(arr, N, K));
    }
}

// This code is contributed by vt_m.

Python3

# Python3 program to find maximum pair sum
# whose difference is less than K

# Method to return maximum sum we can
# get by finding less than K difference
# pairs


def maxSumPairWithDifferenceLessThanK(arr, N, k):

    maxSum = 0

    # Sort elements to ensure every i and
    # i-1 is closest possible pair
    arr.sort()

    # To get maximum possible sum, iterate
    # from largest to smallest, giving larger
    # numbers priority over smaller numbers.
    i = N - 1
    while (i > 0):

        # Case I: Diff of arr[i] and arr[i-1]
        #     is less than K, add to maxSum
        # Case II: Diff between arr[i] and
        #     arr[i-1] is not less than K,
        #     move to next i since with sorting
        #     we know, arr[i]-arr[i-1] < arr[i]-arr[i-2]
        #     and so on.
        if (arr[i] - arr[i - 1] < k):

            # Assuming only positive numbers.
            maxSum += arr[i]
            maxSum += arr[i - 1]

            # When a match is found skip this pair
            i -= 1
        i -= 1

    return maxSum


# Driver Code
arr = [3, 5, 10, 15, 17, 12, 9]
N = len(arr)

K = 4
print(maxSumPairWithDifferenceLessThanK(arr, N, K))

# This code is contributed by mits

C#

// C# program to find maximum pair sum whose
// difference is less than K
using System;
class GFG {

    // Method to return maximum sum we can get by
    // finding less than K difference pairs
    static int maxSumPairWithDifferenceLessThanK(int []arr,
                                               int N, int k)
    {
        int maxSum = 0;

        // Sort elements to ensure
        // every i and i-1 is closest
        // possible pair
        Array.Sort(arr);

        // To get maximum possible sum,
        // iterate from largest
        // to smallest, giving larger
        // numbers priority over
        // smaller numbers.
        for (int i = N-1; i > 0; --i)
        {

            /* Case I: Diff of arr[i] and
                       arr[i-1] is less than K,
                       add to maxSum
               Case II: Diff between arr[i] and
                        arr[i-1] is not less
                        than K, move to next i
                        since with sorting we
                        know, arr[i]-arr[i-1] <
                        arr[i]-arr[i-2] and
                        so on.*/
            if (arr[i] - arr[i-1] < k)
            {

                // Assuming only positive numbers.
                maxSum += arr[i];
                maxSum += arr[i - 1];

                // When a match is found
                // skip this pair
                --i;
            }
        }

        return maxSum;
    }

    // Driver Code
    public static void Main ()
    {
        int []arr = {3, 5, 10, 15, 17, 12, 9};
        int N = arr.Length;
        int K = 4;

        Console.Write( maxSumPairWithDifferenceLessThanK(arr,
                                                         N, K));
    }
}

// This code is contributed by nitin mittal.

PHP

<?php
// PHP program to find maximum pair sum
// whose difference is less than K

// Method to return maximum sum we can
// get by finding less than K difference
// pairs
function maxSumPairWithDifferenceLessThanK($arr, $N, $k)
{
    $maxSum = 0;

    // Sort elements to ensure every i and
    // i-1 is closest possible pair
    sort($arr);

    // To get maximum possible sum, iterate
    // from largest to smallest, giving larger
    // numbers priority over smaller numbers.
    for ($i = $N - 1; $i > 0; --$i)
    {
        // Case I: Diff of arr[i] and arr[i-1]
        //           is less than K, add to maxSum
        // Case II: Diff between arr[i] and
        //            arr[i-1] is not less than K,
        //          move to next i since with sorting
        //          we know, arr[i]-arr[i-1] < arr[i]-arr[i-2]
        //            and so on.
        if ($arr[$i] - $arr[$i - 1] < $k)
        {
            // Assuming only positive numbers.
            $maxSum += $arr[$i];
            $maxSum += $arr[$i - 1];

            // When a match is found skip this pair
            --$i;
        }
    }

    return $maxSum;
}

// Driver Code
$arr = array(3, 5, 10, 15, 17, 12, 9);
$N = sizeof($arr);

$K = 4;
echo maxSumPairWithDifferenceLessThanK($arr, $N, $K);

// This code is contributed
// by Sach_Code
?>

Javascript

<script>

// Javascript program to find
// maximum pair sum whose
// difference is less than K
// Method to return maximum sum we can get by
// finding less than K difference pairs
function maxSumPairWithDifferenceLessThanK(arr, N, k)
{
    var maxSum = 0;

    // Sort elements to ensure every i and i-1 is
    // closest possible pair
    arr.sort((a,b)=>a-b);

    // To get maximum possible sum,
    // iterate from largest
    // to smallest, giving larger
    // numbers priority over
    // smaller numbers.
    for (i = N - 1; i > 0; --i)
    {
        // Case I: Diff of arr[i] and arr[i-1] is less
        // than K, add to maxSum
        // Case II: Diff between arr[i] and arr[i-1] is
        // not less than K, move to next i
        // since with sorting we know, arr[i]-arr[i-1] <
        // arr[i]-arr[i-2] and so on.
        if (arr[i] - arr[i - 1] < k)
        {
            // Assuming only positive numbers.
            maxSum += arr[i];
            maxSum += arr[i - 1];

            // When a match is found skip this pair
            --i;
        }
    }

    return maxSum;
}

// Driver code
var arr = [ 3, 5, 10, 15, 17, 12, 9 ];
var N = arr.length;
var K = 4;

document.write(maxSumPairWithDifferenceLessThanK(arr, N, K));

// This code is contributed by 29AjayKumar

</script>

Output

62

Time complexity: O(N Log N) 
Auxiliary Space: O(1)