Maximum subsequence sum such that all elements are K distance apart
Last Updated :
12 Jul, 2025
Given an array arr[] of N integers and another integer K. The task is to find the maximum sum of a subsequence such that the difference of the indices of all consecutive elements in the subsequence in the original array is exactly K. For example, if arr[i] is the first element of the subsequence then the next element has to be arr[i + k] then arr[i + 2k] and so on.
Examples:
Input: arr[] = {2, -3, -1, -1, 2}, K = 2
Output: 3
Input: arr[] = {2, 3, -1, -1, 2}, K = 3
Output: 5
Approach: There are K sequences possible where the elements are K distance apart and these can be of the forms:
0, 0 + k, 0 + 2k, ..., 0 + n*k
1, 1 + k, 1 + 2k, ..., 1 + n*k
2, 2 + k, 2 + 2k, ..., 2 + n*k
k-1, k-1 + k, k-1 + 2k, ..., k-1 + n*k
Now, any subarray of the sequences is a subsequence of the original array where elements are K distance apart from each other.
So, the task now reduces to find the maximum subarray sum of these sequences which can be found by Kadane's algorithm.
Below is the implementation of the above approach:
C++
// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
// Function to return the maximum subarray sum
// for the array {a[i], a[i + k], a[i + 2k], ...}
int maxSubArraySum(int a[], int n, int k, int i)
{
int max_so_far = INT_MIN, max_ending_here = 0;
while (i < n) {
max_ending_here = max_ending_here + a[i];
if (max_so_far < max_ending_here)
max_so_far = max_ending_here;
if (max_ending_here < 0)
max_ending_here = 0;
i += k;
}
return max_so_far;
}
// Function to return the sum of
// the maximum required subsequence
int find(int arr[], int n, int k)
{
// To store the result
int maxSum = 0;
// Run a loop from 0 to k
for (int i = 0; i <= min(n, k); i++) {
int sum = 0;
// Find the maximum subarray sum for the
// array {a[i], a[i + k], a[i + 2k], ...}
maxSum = max(maxSum,
maxSubArraySum(arr, n, k, i));
}
// Return the maximum value
return maxSum;
}
// Driver code
int main()
{
int arr[] = { 2, -3, -1, -1, 2 };
int n = sizeof(arr) / sizeof(arr[0]);
int k = 2;
cout << find(arr, n, k);
return 0;
}
// Java implementation of the approach
import java.util.*;
class GFG
{
// Function to return the maximum subarray sum
// for the array {a[i], a[i + k], a[i + 2k], ...}
static int maxSubArraySum(int a[], int n,
int k, int i)
{
int max_so_far = Integer.MIN_VALUE,
max_ending_here = 0;
while (i < n)
{
max_ending_here = max_ending_here + a[i];
if (max_so_far < max_ending_here)
max_so_far = max_ending_here;
if (max_ending_here < 0)
max_ending_here = 0;
i += k;
}
return max_so_far;
}
// Function to return the sum of
// the maximum required subsequence
static int find(int arr[], int n, int k)
{
// To store the result
int maxSum = 0;
// Run a loop from 0 to k
for (int i = 0; i <= Math.min(n, k); i++)
{
int sum = 0;
// Find the maximum subarray sum for the
// array {a[i], a[i + k], a[i + 2k], ...}
maxSum = Math.max(maxSum,
maxSubArraySum(arr, n, k, i));
}
// Return the maximum value
return maxSum;
}
// Driver code
public static void main(String[] args)
{
int arr[] = {2, -3, -1, -1, 2};
int n = arr.length;
int k = 2;
System.out.println(find(arr, n, k));
}
}
// This code is contributed by Rajput-Ji
# Python3 implementation of the approach
# Function to return the maximum subarray sum
# for the array {a[i], a[i + k], a[i + 2k], ...}
import sys
def maxSubArraySum(a, n, k, i):
max_so_far = -sys.maxsize;
max_ending_here = 0;
while (i < n):
max_ending_here = max_ending_here + a[i];
if (max_so_far < max_ending_here):
max_so_far = max_ending_here;
if (max_ending_here < 0):
max_ending_here = 0;
i += k;
return max_so_far;
# Function to return the sum of
# the maximum required subsequence
def find(arr, n, k):
# To store the result
maxSum = 0;
# Run a loop from 0 to k
for i in range(0, min(n, k) + 1):
sum = 0;
# Find the maximum subarray sum for the
# array {a[i], a[i + k], a[i + 2k], ...}
maxSum = max(maxSum,
maxSubArraySum(arr, n, k, i));
# Return the maximum value
return maxSum;
# Driver code
if __name__ == '__main__':
arr = [ 2, -3, -1, -1, 2 ];
n = len(arr);
k = 2;
print(find(arr, n, k));
# This code is contributed by Princi Singh
// C# implementation of the approach
using System;
class GFG
{
// Function to return the maximum subarray sum
// for the array {a[i], a[i + k], a[i + 2k], ...}
static int maxSubArraySum(int []a, int n,
int k, int i)
{
int max_so_far = int.MinValue,
max_ending_here = 0;
while (i < n)
{
max_ending_here = max_ending_here + a[i];
if (max_so_far < max_ending_here)
max_so_far = max_ending_here;
if (max_ending_here < 0)
max_ending_here = 0;
i += k;
}
return max_so_far;
}
// Function to return the sum of
// the maximum required subsequence
static int find(int []arr, int n, int k)
{
// To store the result
int maxSum = 0;
// Run a loop from 0 to k
for (int i = 0; i <= Math.Min(n, k); i++)
{
// Find the maximum subarray sum for the
// array {a[i], a[i + k], a[i + 2k], ...}
maxSum = Math.Max(maxSum,
maxSubArraySum(arr, n, k, i));
}
// Return the maximum value
return maxSum;
}
// Driver code
public static void Main()
{
int []arr = {2, -3, -1, -1, 2};
int n = arr.Length;
int k = 2;
Console.WriteLine(find(arr, n, k));
}
}
// This code is contributed by AnkitRai01
<script>
// Javascript implementation of the approach
// Function to return the maximum subarray sum
// for the array {a[i], a[i + k], a[i + 2k], ...}
function maxSubArraySum(a, n, k, i)
{
let max_so_far = Number.MIN_VALUE,
max_ending_here = 0;
while (i < n) {
max_ending_here = max_ending_here + a[i];
if (max_so_far < max_ending_here)
max_so_far = max_ending_here;
if (max_ending_here < 0)
max_ending_here = 0;
i += k;
}
return max_so_far;
}
// Function to return the sum of
// the maximum required subsequence
function find(arr, n, k)
{
// To store the result
let maxSum = 0;
// Run a loop from 0 to k
for (let i = 0; i <= Math.min(n, k); i++) {
let sum = 0;
// Find the maximum subarray sum for the
// array {a[i], a[i + k], a[i + 2k], ...}
maxSum = Math.max(maxSum,
maxSubArraySum(arr, n, k, i));
}
// Return the maximum value
return maxSum;
}
// Driver code
let arr = [ 2, -3, -1, -1, 2 ];
let n = arr.length;
let k = 2;
document.write(find(arr, n, k));
</script>
Time Complexity: O(min(n,k)*n)
Auxiliary Space: O(1)
Related Topic: Subarrays, Subsequences, and Subsets in Array
Approach(Using dynamic programming):
- Initialize an array dp of size N, where dp[i] represents the maximum sum of a subsequence ending at the ith index.
- Initialize the dp array with the values of the original array.
- Iterate over the array from the Kth index to the end and for each index i, calculate dp[i] as the maximum value of dp[j] + arr[i], where j is the index that is K positions behind i.
- Return the maximum value in the dp array.
Here is the implementation of the above approach:
C++
#include <iostream>
#include <vector>
#include <algorithm>
int maxSubsequenceSum(std::vector<int>& arr, int K) {
int N = arr.size();
std::vector<int> dp = arr;
for (int i = K; i < N; ++i) {
for (int j = i - K; j >= i - K; j -= K) { // Corrected loop condition here
if (j >= 0) {
dp[i] = std::max(dp[i], dp[j] + arr[i]);
}
}
}
return *std::max_element(dp.begin(), dp.end());
}
int main() {
std::vector<int> arr = {2, -3, -1, -1, 2};
int K = 2;
std::cout << maxSubsequenceSum(arr, K) << std::endl; // Output: 3
return 0;
}
import java.util.Arrays;
public class Main {
// Function to find the maximum subsequence sum with a
// constraint
public static int maxSubsequenceSum(int[] arr, int K)
{
int N = arr.length;
int[] dp = Arrays.copyOf(
arr,
N); // Initialize a dynamic programming array
// Iterate through the elements
for (int i = K; i < N; ++i) {
// Find the maximum sum considering the
// constraint
for (int j = i - K; j >= i - K && j >= 0;
j -= K) {
dp[i] = Math.max(dp[i], dp[j] + arr[i]);
}
}
// Find and return the maximum value in the dynamic
// programming array
int max = Arrays.stream(dp).max().getAsInt();
return max;
}
public static void main(String[] args)
{
int[] arr = { 2, -3, -1, -1, 2 };
int K = 2;
System.out.println(
maxSubsequenceSum(arr, K)); // Output: 3
}
}
def maxSubsequenceSum(arr, K):
N = len(arr)
dp = arr.copy()
for i in range(K, N):
for j in range(i-K, i-2*K, -K):
if j >= 0:
dp[i] = max(dp[i], dp[j]+arr[i])
return max(dp)
# Example usage
arr = [2, -3, -1, -1, 2]
K = 2
print(maxSubsequenceSum(arr, K)) # Output: 3
using System;
using System.Collections.Generic;
using System.Linq;
class Program {
// Function to find the maximum subsequence sum with a
// limit K
static int MaxSubsequenceSum(List<int> arr, int K)
{
int N = arr.Count;
List<int> dp = arr.ToList(); // Create a copy of the
// input list
for (int i = K; i < N; ++i) {
for (int j = i - K; j <= i - 1;
j++) // Loop corrected here
{
if (j >= 0) {
dp[i] = Math.Max(dp[i], dp[j] + arr[i]);
}
}
}
return dp.Max();
}
static void Main(string[] args)
{
List<int> arr = new List<int>{ 2, -3, -1, -1, 2 };
int K = 2;
Console.WriteLine(
MaxSubsequenceSum(arr, K)); // Output: 3
}
}
function maxSubsequenceSum(arr, K) {
const N = arr.length;
const dp = [...arr]; // Create a copy of the input array
for (let i = K; i < N; ++i) {
for (let j = i - K; j <= i - 1; j += K) { // Corrected loop condition here
if (j >= 0) {
dp[i] = Math.max(dp[i], dp[j] + arr[i]);
}
}
}
return Math.max(...dp);
}
const arr = [2, -3, -1, -1, 2];
const K = 2;
console.log(maxSubsequenceSum(arr, K)); // Output: 3
Time Complexity: O(NK), where N is the length of the input array arr and K is the value passed as an argument to the function. This is because there are two nested loops, and the inner loop runs K times for each element in the array, so the overall time complexity is O(NK).
Auxiliary Space: O(N), since the size of the dp array is the same as the size of the input array arr. Therefore, the space complexity is linear in the size of the input.
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem