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: 4
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: 4
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++ program to implement // the above approach #include<bits/stdc++.h> 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[0]);
findMinSum(A, K, n);
return 0;
} // This code is contributed by chitranayal |
// 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 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
minimum = sys.maxsize;
# Iterate from i + k to N
for j in range (i + K, n, 1 ):
# Find the minimum
minimum = min (minimum, A[j]);
if (minimum = = sys.maxsize):
continue ;
# Update the minimum sum
minimum_sum = min (minimum_sum, A[i] + minimum);
# Print 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# 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 |
<script> // Javascript program to implement
// the above approach
// Function to find the minimum
// sum of two elements that
// are atleast K distance apart
function findMinSum(A, K, n)
{
let minimum_sum = Number.MAX_VALUE;
// Iterate over the array
for (let i = 0; i < n; i++)
{
// Initialize the min value
let mini = Number.MAX_VALUE;
// Iterate from i + k to N
for (let j = i + K; j < n; j++)
// Find the minimum
mini = Math.min(mini, A[j]);
if (mini == Number.MAX_VALUE)
continue ;
// Update the minimum sum
minimum_sum = Math.min(minimum_sum,
A[i] + mini);
}
// Print the answer
document.write(minimum_sum);
}
let A = [ 4, 2, 5, 4, 3, 2, 5 ];
let K = 3;
let n = A.length;
findMinSum(A, K, n);
// This code is contributed by divyeshrabadiya07.
</script> |
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++ Program to implement //the above approach #include <bits/stdc++.h> 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[0]);
findMinSum(A, K, n);
return 0;
} // This code is contributed by Rohit_ranjan |
// 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 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# 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 |
<script> // JavaScript program for the above approach // Function to find the minimum
// sum of two elements that
// are atleast K distance apart
function
findMinSum(A, K)
{
// Length of the array
let n = A.length;
let suffix_min = Array.from({length: n}, (_, i) => 0);
suffix_min[n - 1] = A[n - 1];
// Find the suffix array
for (let i = n - 2; i >= 0; i--)
suffix_min[i]
= Math.min(suffix_min[i + 1],
A[i]);
let min_sum = Number.MAX_VALUE;
// Iterate in the array
for (let 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
document.write(min_sum);
}
// Driver Code let A = [ 1, 2, 3, 4, 5, 6 ];
let K = 2;
findMinSum(A, K);
</script> |
4
Time Complexity: O(N)
Auxiliary Space: O(N)