# Maximize element at index K in an array with at most sum M and difference between adjacent elements at most 1

Given a positive integer N, the task is to construct an array of length N and find the maximize the value at index K such that the sum of all the array elements is at most M and the absolute difference between any two consecutive array elements is at most 1.

Examples:

Input: N = 3, M = 7, K = 1
Output: 3
Explanation:
According to the given constraints, the array with values {2, 3, 2}maximizes the value at index 1. Therefore, the required output is 3.

Input: N = 3, M = 8, K = 1
Output: 3

Approach: The idea is to achieve the maximum value at index K and to decrease the sum of other elements to meet the criteria of the sum of the array to be at most M. Follow the steps below:

• Let the value at index K be X. So the element at K – 1 is X – 1, at K – 2 is X – 2 and so on.
• At index 0 the value is X – K. Similarly, at index K + 1 the value is X – 1 and so on upto X – (N – K – 1) at index N – 1.
• So to achieve the maximum value at index K, the array structure would be X – K, X – (K – 1), …., X – 2, X – 1, X, X – 1, X – 2, ….., X – (N – K – 1).
• So after arranging the equation, it becomes K * X – (1 + 2 + …. + K) + X + (N – K – 1) * X – (1 + 2 + …. + (N – K – 1)) ? M.

Follow the steps to solve the above equation:

• Calculate (1 + 2 + …. + K) using K * (K + 1) / 2 and (1 + 2 + ….. + (N – K – 1)) using (N – K – 1) * (N – K) / 2 and store in S1 and S2 respectively.
• This reduces the equation to X * (K + 1 + N – K – 1) ? M + S1 + S2.
• Now, the maximum value of X can be obtained by calculating (M + S1 + S2) / N.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach`   `#include ` `using` `namespace` `std;`   `// Function to calculate the maximum` `// possible value at index K` `void` `maxValueAtIndexK(``int` `N, ``int` `K, ``int` `M)` `{` `    ``// Stores the sum of elements` `    ``// in the left and right of index K` `    ``int` `S1 = 0, S2 = 0;`   `    ``S1 = K * (K + 1) / 2;` `    ``S2 = (N - K - 1) * (N - K) / 2;`   `    ``// Stores the maximum` `    ``// possible value at index K` `    ``int` `X = (M + S1 + S2) / N;`   `    ``// Print the answer` `    ``cout << X;` `}`   `// Driver Code` `int` `main()` `{` `    ``// Given N, K & M` `    ``int` `N = 3, K = 1, M = 7;` `    ``maxValueAtIndexK(N, K, M);`   `    ``return` `0;` `}`

## Java

 `// Java program for the above approach` `import` `java.io.*;`   `class` `GFG{`   `// Function to calculate the maximum` `// possible value at index K` `static` `void` `maxValueAtIndexK(``int` `N, ``int` `K,` `                             ``int` `M)` `{` `    `  `    ``// Stores the sum of elements` `    ``// in the left and right of index K` `    ``int` `S1 = ``0``, S2 = ``0``;`   `    ``S1 = K * (K + ``1``) / ``2``;` `    ``S2 = (N - K - ``1``) * (N - K) / ``2``;`   `    ``// Stores the maximum` `    ``// possible value at index K` `    ``int` `X = (M + S1 + S2) / N;`   `    ``// Print the answer` `    ``System.out.println(X);` `}`   `// Driver Code` `public` `static` `void` `main(String[] args)` `{` `    `  `    ``// Given N, K & M` `    ``int` `N = ``3``, K = ``1``, M = ``7``;` `    `  `    ``maxValueAtIndexK(N, K, M);` `}` `}`   `// This code is contributed by Dharanendra L V`

## Python3

 `# Python program for the above approach`   `# Function to calculate the maximum` `# possible value at index K` `def` `maxValueAtIndexK(N, K, M):`   `    ``# Stores the sum of elements` `    ``# in the left and right of index K` `    ``S1 ``=` `0``; S2 ``=` `0``;` `    ``S1 ``=` `K ``*` `(K ``+` `1``) ``/``/` `2``;` `    ``S2 ``=` `(N ``-` `K ``-` `1``) ``*` `(N ``-` `K) ``/``/` `2``;`   `    ``# Stores the maximum` `    ``# possible value at index K` `    ``X ``=` `(M ``+` `S1 ``+` `S2) ``/``/` `N;`   `    ``# Print the answer` `    ``print``(X);`   `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:`   `    ``# Given N, K & M` `    ``N ``=` `3``; K ``=` `1``; M ``=` `7``;` `    ``maxValueAtIndexK(N, K, M);`   `# This code is contributed by 29AjayKumar`

## C#

 `// C# program for the above approach` `using` `System;`   `class` `GFG{`   `// Function to calculate the maximum` `// possible value at index K` `static` `void` `maxValueAtIndexK(``int` `N, ``int` `K,` `                             ``int` `M)` `{` `    `  `    ``// Stores the sum of elements` `    ``// in the left and right of index K` `    ``int` `S1 = 0, S2 = 0;`   `    ``S1 = K * (K + 1) / 2;` `    ``S2 = (N - K - 1) * (N - K) / 2;`   `    ``// Stores the maximum` `    ``// possible value at index K` `    ``int` `X = (M + S1 + S2) / N;`   `    ``// Print the answer` `    ``Console.WriteLine(X);` `}`   `// Driver Code` `static` `public` `void` `Main()` `{` `    `  `    ``// Given N, K & M` `    ``int` `N = 3, K = 1, M = 7;` `    `  `    ``maxValueAtIndexK(N, K, M);` `}` `}`   `// This code is contributed by Dharanendra L V`

## Javascript

 ``

Output:

`3`

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

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