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Minimum Sum of a pair at least K distance apart from an Array
• Last Updated : 11 May, 2021

Given an array of integers A[] of size N, the task is to find the minimum sum that can be obtained by any pair of array elements that are at least K indices apart from each other.

Examples:

Input: A[] = {1, 2, 3, 4, 5, 6}, K = 2
Output:
Explanation:
The minimum sum that can be obtained is by adding 1 and 3 that are at a distance of 2.
Input: A[] = {4, 2, 5, 4, 3, 2, 5}, K = 3
Output:
Explanation:
The minimum sum that can be obtained is by adding 2 and 2 that are at a distance of 4.

Naive Approach:
The simplest approach is to solve the problem is to iterate over the indices [i + K, N – 1] for every ith index and find the minimum element, say min. Check if min + A[i] is less than the minimum sum obtained so far and update minimum_sum accordingly. Finally, print the minimum_sum.

Below is the implementation of the above approach:

## C++

 `// C++ program to implement``// the above approach``#include``using` `namespace` `std;` `// Function to find the minimum``// sum of two elements that``// are atleast K distance apart``void` `findMinSum(``int` `A[], ``int` `K, ``int` `n)``{``    ``int` `minimum_sum = INT_MAX;` `    ``// Iterate over the array``    ``for``(``int` `i = 0; i < n; i++)``    ``{``        ` `        ``// Initialize the min value``        ``int` `mini = INT_MAX;` `        ``// Iterate from i + k to N``        ``for``(``int` `j = i + K; j < n; j++)` `            ``// Find the minimum``            ``mini = min(mini, A[j]);` `        ``if` `(mini == INT_MAX)``            ``continue``;` `        ``// Update the minimum sum``        ``minimum_sum = min(minimum_sum,``                          ``A[i] + mini);``    ``}` `    ``// Print the answer``    ``cout << (minimum_sum);``}` `// Driver Code``int` `main()``{``    ``int` `A[] = { 4, 2, 5, 4, 3, 2, 5 };``    ``int` `K = 3;``    ``int` `n = ``sizeof``(A) / ``sizeof``(A);` `    ``findMinSum(A, K, n);``    ``return` `0;``}` `// This code is contributed by chitranayal`

## Java

 `// Java Program to implement``// the above approach` `import` `java.util.*;``class` `GFG {` `    ``// Function to find the minimum``    ``// sum of two elements that``    ``// are atleast K distance apart``    ``public` `static` `void``    ``findMinSum(``int` `A[], ``int` `K)``    ``{``        ``// Length of the array``        ``int` `n = A.length;` `        ``int` `minimum_sum``            ``= Integer.MAX_VALUE;` `        ``// Iterate over the array``        ``for` `(``int` `i = ``0``; i < n; i++) {` `            ``// Initialize the min value``            ``int` `min = Integer.MAX_VALUE;` `            ``// Iterate from i + k to N``            ``for` `(``int` `j = i + K; j < n; j++)` `                ``// Find the minimum``                ``min = Math.min(min, A[j]);` `            ``if` `(min == Integer.MAX_VALUE)``                ``continue``;` `            ``// Update the minimum sum``            ``minimum_sum = Math.min(minimum_sum,``                                   ``A[i] + min);``        ``}` `        ``// Print the answer``        ``System.out.println(minimum_sum);``    ``}` `    ``// Driver Code``    ``public` `static` `void``        ``main(String[] args)``    ``{` `        ``int` `A[] = { ``4``, ``2``, ``5``, ``4``, ``3``, ``2``, ``5` `};``        ``int` `K = ``3``;` `        ``findMinSum(A, K);``    ``}``}`

## Python3

 `# Python3 Program to implement``# the above approach``import` `sys` `# Function to find the minimum``# sum of two elements that``# are atleast K distance apart``def` `findMinSum(A, K):``  ` `    ``# Length of the array``    ``n ``=` `len``(A);` `    ``minimum_sum ``=` `sys.maxsize;` `    ``# Iterate over the array``    ``for` `i ``in` `range``(n):` `        ``# Initialize the min value``        ``minmum ``=` `sys.maxsize;` `        ``# Iterate from i + k to N``        ``for` `j ``in` `range``(i ``+` `K, n, ``1``):` `            ``# Find the minimum``            ``minmum ``=` `min``(minmum, A[j]);` `        ``if` `(minmum ``=``=` `sys.maxsize):``            ``continue``;` `        ``# Update the minimum sum``        ``minimum_sum ``=` `min``(minimum_sum, A[i] ``+` `minmum);` `    ``# Prthe answer``    ``print``(minimum_sum);` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``A ``=` `[``4``, ``2``, ``5``, ``4``, ``3``, ``2``, ``5``];``    ``K ``=` `3``;` `    ``findMinSum(A, K);` `# This code is contributed by sapnasingh4991`

## C#

 `// C# Program to implement``// the above approach``using` `System;``class` `GFG{` `  ``// Function to find the minimum``  ``// sum of two elements that``  ``// are atleast K distance apart``  ``public` `static` `void` `findMinSum(``int` `[]A,``                                ``int` `K)``  ``{``    ``// Length of the array``    ``int` `n = A.Length;` `    ``int` `minimum_sum = ``int``.MaxValue;` `    ``// Iterate over the array``    ``for` `(``int` `i = 0; i < n; i++)``    ``{` `      ``// Initialize the min value``      ``int` `min = ``int``.MaxValue;` `      ``// Iterate from i + k to N``      ``for` `(``int` `j = i + K; j < n; j++)` `        ``// Find the minimum``        ``min = Math.Min(min, A[j]);` `      ``if` `(min == ``int``.MaxValue)``        ``continue``;` `      ``// Update the minimum sum``      ``minimum_sum = Math.Min(minimum_sum,``                             ``A[i] + min);``    ``}` `    ``// Print the answer``    ``Console.WriteLine(minimum_sum);``  ``}` `  ``// Driver Code``  ``public` `static` `void` `Main(String[] args)``  ``{``    ``int` `[]A = { 4, 2, 5, 4, 3, 2, 5 };``    ``int` `K = 3;` `    ``findMinSum(A, K);``  ``}``}` `// This code is contributed by Rohit_ranjan`

## Javascript

 ``
Output:
`4`

Time Complexity: O(N2)
Auxiliary Space: O(1)

Efficient Approach:
The above approach can be optimized using a Suffix Array. Follow the steps below:

• Initialize a suffix array(say suffix[]), where suffix[i] stores the minimum of all the elements from index N-1 to i.
• For any ith index, the minimum element which is K distance apart is stored at index i + K in the suffix array.
• For i ranging from 0 to N – 1, check if A[i] + suffix[i + k] < minimum_sum or not and update minimum_sum accordingly.
• Finally, print minimum_sum as the required answer.

Below is the implementation of the above approach:

## C++

 `// C++ Program to implement``//the above approach``#include ``using` `namespace` `std;` `// Function to find the minimum``// sum of two elements that``// are atleast K distance apart``void` `findMinSum(``int` `A[], ``int` `K, ``int` `len)``{` `  ``// Length of the array``  ``int` `n = len;``  ``int` `suffix_min[n] = {0};` `  ``suffix_min[n - 1] = A[n - 1];` `  ``// Find the suffix array``  ``for` `(``int` `i = n - 2; i >= 0; i--)``    ``suffix_min[i] = min(suffix_min[i + 1], A[i]);` `  ``int` `min_sum = INT_MAX;` `  ``// Iterate in the array``  ``for` `(``int` `i = 0; i < n; i++)``  ``{``    ``if` `(i + K < n)` `      ``// Update minimum sum``      ``min_sum = min(min_sum, A[i] +``                    ``suffix_min[i + K]);``  ``}` `  ``// Print the answer``  ``cout << min_sum;``}`  `// Driver Code``int` `main()``{``    ``int` `A[] = { 1, 2, 3, 4, 5, 6 };``    ``int` `K = 2;``    ``int` `n = ``sizeof``(A) / ``sizeof``(A);``    ``findMinSum(A, K, n);``    ``return` `0;``}` `// This code is contributed by Rohit_ranjan`

## Java

 `// Java Program to implement``// the above approach` `import` `java.util.*;``class` `GFG {` `    ``// Function to find the minimum``    ``// sum of two elements that``    ``// are atleast K distance apart``    ``public` `static` `void``    ``findMinSum(``int` `A[], ``int` `K)``    ``{` `        ``// Length of the array``        ``int` `n = A.length;``        ``int` `suffix_min[] = ``new` `int``[n];` `        ``suffix_min[n - ``1``] = A[n - ``1``];` `        ``// Find the suffix array``        ``for` `(``int` `i = n - ``2``; i >= ``0``; i--)``            ``suffix_min[i]``                ``= Math.min(suffix_min[i + ``1``],``                           ``A[i]);` `        ``int` `min_sum = Integer.MAX_VALUE;` `        ``// Iterate in the array``        ``for` `(``int` `i = ``0``; i < n; i++) {` `            ``if` `(i + K < n)` `                ``// Update minimum sum``                ``min_sum = Math.min(``                    ``min_sum, A[i]``                                 ``+ suffix_min[i + K]);``        ``}` `        ``// Print the answer``        ``System.out.println(min_sum);``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `A[] = { ``1``, ``2``, ``3``, ``4``, ``5``, ``6` `};``        ``int` `K = ``2``;` `        ``findMinSum(A, K);``    ``}``}`

## Python3

 `# Python3 program to implement``# the above approach``import` `sys` `# Function to find the minimum``# sum of two elements that``# are atleast K distance apart``def` `findMinSum(A, K):``    ` `    ``# Length of the array``    ``n ``=` `len``(A);``    ` `    ``suffix_min ``=` `[``0``] ``*` `n;``    ``suffix_min[n ``-` `1``] ``=` `A[n ``-` `1``];` `    ``# Find the suffix array``    ``for` `i ``in` `range``(n ``-` `2``, ``-``1``, ``-``1``):``        ``suffix_min[i] ``=` `min``(suffix_min[i ``+` `1``], A[i]);` `    ``min_sum ``=` `sys.maxsize;` `    ``# Iterate in the array``    ``for` `i ``in` `range``(n):``        ``if` `(i ``+` `K < n):` `            ``# Update minimum sum``            ``min_sum ``=` `min``(min_sum, A[i] ``+``                          ``suffix_min[i ``+` `K]);` `    ``# Print the answer``    ``print``(min_sum);` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ` `    ``A ``=` `[ ``1``, ``2``, ``3``, ``4``, ``5``, ``6` `];``    ``K ``=` `2``;` `    ``findMinSum(A, K);` `# This code is contributed by Amit Katiyar`

## C#

 `// C# program to implement``// the above approach``using` `System;` `class` `GFG{` `// Function to find the minimum``// sum of two elements that``// are atleast K distance apart``public` `static` `void` `findMinSum(``int` `[]A, ``int` `K)``{``    ` `    ``// Length of the array``    ``int` `n = A.Length;``    ``int` `[]suffix_min = ``new` `int``[n];` `    ``suffix_min[n - 1] = A[n - 1];` `    ``// Find the suffix array``    ``for``(``int` `i = n - 2; i >= 0; i--)``        ``suffix_min[i] = Math.Min(suffix_min[i + 1],``                                          ``A[i]);` `    ``int` `min_sum = ``int``.MaxValue;` `    ``// Iterate in the array``    ``for``(``int` `i = 0; i < n; i++)``    ``{``        ``if` `(i + K < n)` `            ``// Update minimum sum``            ``min_sum = Math.Min(min_sum, A[i] +``                               ``suffix_min[i + K]);``    ``}` `    ``// Print the answer``    ``Console.WriteLine(min_sum);``}` `// Driver Code``public` `static` `void` `Main(String[] args)``{``    ``int` `[]A = { 1, 2, 3, 4, 5, 6 };``    ``int` `K = 2;` `    ``findMinSum(A, K);``}``}` `// This code is contributed by 29AjayKumar`

## Javascript

 ``
Output:
`4`

Time Complexity: O(N)
Auxiliary Space: O(N)

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