Maximum sum subsequence with at-least k distant elements

3.2

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

  1. Sum of elements in subsequence is maximum
  2. Indices of elements of subsequence differ atleast by k
    1. 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:

      1. 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.
      2. 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.

      These two cases can be written as:

      Let MS[i] denotes the maximum sum of subsequence 
      from i to N. 
      
      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
      


      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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