Given an array and a number k, find a subsequence such that

- Sum of elements in subsequence is maximum
- Indices of elements of subsequence differ atleast by k
- If we select element at index i such that i + k + 1 >= N, then we cannot select any other element as part of the subsequence. Hence we need to decide whether to select this element or one of the elements after it.
- If we select element at index i such that i + k + 1 < N, then the next element we can select is at i + k + 1 index. Thus we need to decide whether to select these two elements, or move on to the next adjacent element.

Examples

Input : arr[] = {4, 5, 8, 7, 5, 4, 3, 4, 6, 5} k = 2 Output: 19 Explanation: The highest value is obtained if you pick indices 1, 4, 7, 10 giving 4 + 7 + 3 + 5 = 19 Input: arr[] = {50, 70, 40, 50, 90, 70, 60, 40, 70, 50} k = 2 Output: 230 Explanation: There are 10 elements and k = 2. If you select 2, 5, and 9 you get a total value of 230, which is the maximum possible.

A

**simple solution**is to consider all subsequences one by one. In every subsequence, check for distance condition and return the maximum sum subsequence.An

**efficient solution**is to use dynamic programming.There are two cases:

These two cases can be written as:

Let MS[i] denotes the maximum sum of subsequence from i = N-2 to 0. Base Case: MS[N-1] = arr[N-1] If i + 1 + k >= N MS[i] = max(arr[i], MS[i+1]), Else MS[i] = max(arr[i] + MS[i+k+1], MS[i+1]) Evidently, the solution to the problem is to find MS[0].

Below is the implementation:

## C++

`// CPP program to find maximum sum subsequence`

`// such that elements are at least k distance`

`// away.`

`#include <bits/stdc++.h>`

`using`

`namespace`

`std;`

`int`

`maxSum(`

`int`

`arr[],`

`int`

`N,`

`int`

`k)`

`{`

`// MS[i] is going to store maximum sum`

`// subsequence in subarray from arr[i]`

`// to arr[n-1]`

`int`

`MS[N];`

`// We fill MS from right to left.`

`MS[N - 1] = arr[N - 1];`

`for`

`(`

`int`

`i = N - 2; i >= 0; i--) {`

`if`

`(i + k + 1 >= N)`

`MS[i] = max(arr[i], MS[i + 1]);`

`else`

`MS[i] = max(arr[i] + MS[i + k + 1], MS[i + 1]);`

`}`

`return`

`MS[0];`

`}`

`// Driver code`

`int`

`main()`

`{`

`int`

`N = 10, k = 2;`

`int`

`arr[] = { 50, 70, 40, 50, 90, 70, 60, 40, 70, 50 };`

`cout << maxSum(arr, N, k);`

`return`

`0;`

`}`

## Java

`// Java program to find maximum sum subsequence`

`// such that elements are at least k distance`

`// away.`

`import`

`java.io.*;`

`class`

`GFG {`

`static`

`int`

`maxSum(`

`int`

`arr[],`

`int`

`N,`

`int`

`k)`

`{`

`// MS[i] is going to store maximum sum`

`// subsequence in subarray from arr[i]`

`// to arr[n-1]`

`int`

`MS[] =`

`new`

`int`

`[N];`

`// We fill MS from right to left.`

`MS[N -`

`1`

`] = arr[N -`

`1`

`];`

`for`

`(`

`int`

`i = N -`

`2`

`; i >=`

`0`

`; i--) {`

`if`

`(i + k +`

`1`

`>= N)`

`MS[i] = Math.max(arr[i], MS[i +`

`1`

`]);`

`else`

`MS[i] = Math.max(arr[i] + MS[i + k +`

`1`

`],`

`MS[i +`

`1`

`]);`

`}`

`return`

`MS[`

`0`

`];`

`}`

`// Driver code`

`public`

`static`

`void`

`main(String[] args)`

`{`

`int`

`N =`

`10`

`, k =`

`2`

`;`

`int`

`arr[] = {`

`50`

`,`

`70`

`,`

`40`

`,`

`50`

`,`

`90`

`,`

`70`

`,`

`60`

`,`

`40`

`,`

`70`

`,`

`50`

`};`

`System.out.println(maxSum(arr, N, k));`

`}`

`}`

`// This code is contributed by Prerna Saini`

## Python3

`# Python3 program to find maximum`

`# sum subsequence such that elements`

`# are at least k distance away.`

`def`

`maxSum(arr, N, k):`

`# MS[i] is going to store maximum sum`

`# subsequence in subarray from arr[i]`

`# to arr[n-1]`

`MS`

`=`

`[`

`0`

`for`

`i`

`in`

`range`

`(N)]`

`# We fill MS from right to left.`

`MS[N`

`-`

`1`

`]`

`=`

`arr[N`

`-`

`1`

`]`

`for`

`i`

`in`

`range`

`(N`

`-`

`2`

`,`

`-`

`1`

`,`

`-`

`1`

`):`

`if`

`(i`

`+`

`k`

`+`

`1`

`>`

`=`

`N):`

`MS[i]`

`=`

`max`

`(arr[i], MS[i`

`+`

`1`

`])`

`else`

`:`

`MS[i]`

`=`

`max`

`(arr[i]`

`+`

`MS[i`

`+`

`k`

`+`

`1`

`],`

`MS[i`

`+`

`1`

`])`

`return`

`MS[`

`0`

`]`

`# Driver code`

`N`

`=`

`10`

`; k`

`=`

`2`

`arr`

`=`

`[`

`50`

`,`

`70`

`,`

`40`

`,`

`50`

`,`

`90`

`,`

`70`

`,`

`60`

`,`

`40`

`,`

`70`

`,`

`50`

`]`

`print`

`(maxSum(arr, N, k))`

`# This code is contributed by Anant Agarwal.`

## C#

`// C# program to find maximum sum`

`// subsequence such that elements`

`// are at least k distance away.`

`using`

`System;`

`class`

`GFG {`

`static`

`int`

`maxSum(`

`int`

`[]arr,`

`int`

`N,`

`int`

`k)`

`{`

`// MS[i] is going to store maximum sum`

`// subsequence in subarray from arr[i]`

`// to arr[n-1]`

`int`

`[]MS =`

`new`

`int`

`[N];`

`// We fill MS from right to left.`

`MS[N - 1] = arr[N - 1];`

`for`

`(`

`int`

`i = N - 2; i >= 0; i--) {`

`if`

`(i + k + 1 >= N)`

`MS[i] = Math.Max(arr[i], MS[i + 1]);`

`else`

`MS[i] = Math.Max(arr[i] + MS[i + k + 1],`

`MS[i + 1]);`

`}`

`return`

`MS[0];`

`}`

`// Driver code`

`public`

`static`

`void`

`Main()`

`{`

`int`

`N = 10, k = 2;`

`int`

`[]arr = { 50, 70, 40, 50, 90, 70, 60,`

`40, 70, 50 };`

`Console.WriteLine(maxSum(arr, N, k));`

`}`

`}`

`// This code is contributed by Anant Agarwal.`

## PHP

`<?php`

`// PHP program to find`

`// maximum sum subsequence`

`// such that elements are`

`// at least k distance`

`// away.`

`function`

`maxSum(`

`$arr`

`,`

`$N`

`,`

`$k`

`)`

`{`

`// MS[i] is going to`

`// store maximum sum`

`// subsequence in`

`// subarray from arr[i]`

`// to arr[n-1]`

`// We fill MS from`

`// right to left.`

`$MS`

`[`

`$N`

`- 1] =`

`$arr`

`[`

`$N`

`- 1];`

`for`

`(`

`$i`

`=`

`$N`

`- 2;`

`$i`

`>= 0;`

`$i`

`--)`

`{`

`if`

`(`

`$i`

`+`

`$k`

`+ 1 >=`

`$N`

`)`

`$MS`

`[`

`$i`

`] = max(`

`$arr`

`[`

`$i`

`],`

`$MS`

`[`

`$i`

`+ 1]);`

`else`

`$MS`

`[`

`$i`

`] = max(`

`$arr`

`[`

`$i`

`] +`

`$MS`

`[`

`$i`

`+`

`$k`

`+ 1],`

`$MS`

`[`

`$i`

`+ 1]);`

`}`

`return`

`$MS`

`[0];`

`}`

`// Driver code`

`$N`

`= 10;`

`$k`

`= 2;`

`$arr`

`=`

`array`

`(50, 70, 40, 50, 90,`

`70, 60, 40, 70, 50);`

`echo`

`(maxSum(`

`$arr`

`,`

`$N`

`,`

`$k`

`));`

`// This code is contributed by Ajit.`

`?>`

Output:230

Time Complexity : O(n)

Auxiliary Space : O(n)This article is contributed by

**Sayan Mahapatra**. 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.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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