# Maximum subarray size, such that all subarrays of that size have sum less than k

Given an array of n positive integers and a positive integer k, the task is to find the maximum subarray size such that all subarrays of that size have sum of elements less than or equals to k.

Examples :

```Input :  arr[] = {1, 2, 3, 4} and k = 8.
Output : 2
Sum of subarrays of size 1: 1, 2, 3, 4.
Sum of subarrays of size 2: 3, 5, 7.
Sum of subarrays of size 3: 6, 9.
Sum of subarrays of size 4: 10.
So, maximum subarray size such that all subarrays
of that size have the sum of elements less than 8 is 2.

Input :  arr[] = {1, 2, 10, 4} and k = 8.
Output : -1
There is an array element with value greater than k,
so subarray sum cannot be less than k.

Input :  arr[] = {1, 2, 10, 4} and K = 14
Output : 2
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

First of all, required subarray size must lie between 1 to n.

Now, since all the array element are positive integers, we can say that the prefix sum of any subarray shall be strictly increasing.
Thus, we can say that

```if arr[i] + arr[i + 1] + ..... + arr[j - 1] + arr[j] <= K
then arr[i] + arr[i + 1] + ..... + arr[j - 1] <= K, as
arr[j] is a positive integer. ```

So, we perform Binary Search over the range 1 to n and find the highest subarray size such that all the subarrays of that size have the sum of elements less than or equals to k.

Below is implementation of this approach.

## C/C++

 `// C++ program to find maximum  ` `// subarray size, such that all  ` `// subarrays of that size have  ` `// sum less than K. ` `#include ` `using` `namespace` `std; ` ` `  `// Search for the maximum length of  ` `// required subarray. ` `int` `bsearch``(``int` `prefixsum[], ``int` `n,  ` `                             ``int` `k) ` `{ ` `    ``// Initialize result ` `    ``int` `ans = -1;  ` ` `  `    ``// Do Binary Search for largest  ` `    ``// subarray size  ` `    ``int` `left = 1, right = n; ` `    ``while` `(left <= right) ` `    ``{ ` `        ``int` `mid = (left + right) / 2; ` ` `  `        ``// Check for all subarrays after mid ` `        ``int` `i; ` `        ``for` `(i = mid; i <= n; i++) ` `        ``{ ` `            ``// Checking if all the subarrays ` `            ``//  of a size less than k. ` `            ``if` `(prefixsum[i] - prefixsum[i - mid] > k) ` `                ``break``; ` `        ``} ` ` `  `        ``// All subarrays of size mid have  ` `        ``// sum less than or equal to k ` `        ``if` `(i == n + 1) ` `        ``{ ` `            ``left = mid + 1; ` `            ``ans = mid; ` `        ``} ` ` `  `        ``// We found a subrray of size mid  ` `        ``// with sum greater than k ` `        ``else` `            ``right = mid - 1; ` `    ``} ` `    ``return` `ans; ` `} ` ` `  `// Return the maximum subarray size, ` `// such that all subarray of that size ` `// have sum less than K. ` `int` `maxSize(``int` `arr[], ``int` `n, ``int` `k) ` `{ ` `    ``// Initialize prefix sum array as 0. ` `    ``int` `prefixsum[n + 1]; ` `    ``memset``(prefixsum, 0, ``sizeof``(prefixsum)); ` ` `  `    ``// Finding prefix sum of the array. ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``prefixsum[i + 1] = prefixsum[i] +  ` `                           ``arr[i]; ` ` `  `    ``return` `bsearch``(prefixsum, n, k); ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `arr[] = {1, 2, 10, 4}; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` `    ``int` `k = 14; ` `    ``cout << maxSize(arr, n, k) << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java program to find maximum  ` `// subarray size, such that all  ` `// subarrays of that size have ` `// sum less than K. ` `import` `java.util.Arrays; ` ` `  `class` `GFG  ` `{ ` `     `  `    ``// Search for the maximum length  ` `    ``// of required subarray. ` `    ``static` `int` `bsearch(``int` `prefixsum[],  ` `                       ``int` `n, ``int` `k) ` `    ``{ ` `        ``// Initialize result ` `        ``int` `ans = -``1``;  ` ` `  `        ``// Do Binary Search for largest  ` `        ``// subarray size ` `        ``int` `left = ``1``, right = n; ` `         `  `        ``while` `(left <= right)  ` `        ``{ ` `            ``int` `mid = (left + right) / ``2``; ` ` `  `            ``// Check for all subarrays after mid ` `            ``int` `i; ` `            ``for` `(i = mid; i <= n; i++)  ` `            ``{ ` `                 `  `                ``// Checking if all the subarrays  ` `                ``// of a size is less than k. ` `                ``if` `(prefixsum[i] - prefixsum[i - mid] > k) ` `                    ``break``; ` `            ``} ` ` `  `            ``// All subarrays of size mid have  ` `            ``// sum less than or equal to k ` `            ``if` `(i == n + ``1``) ` `            ``{ ` `                ``left = mid + ``1``; ` `                ``ans = mid; ` `            ``} ` ` `  `            ``// We found a subrray of size mid  ` `            ``// with sum greater than k ` `            ``else` `                ``right = mid - ``1``; ` `        ``} ` ` `  `        ``return` `ans; ` `    ``} ` ` `  `    ``// Return the maximum subarray size, such  ` `    ``// that all subarray of that size have  ` `    ``// sum less than K. ` `    ``static` `int` `maxSize(``int` `arr[], ``int` `n, ``int` `k) ` `    ``{ ` `         `  `        ``// Initialize prefix sum array as 0. ` `        ``int` `prefixsum[] = ``new` `int``[n + ``1``]; ` `        ``Arrays.fill(prefixsum, ``0``); ` ` `  `        ``// Finding prefix sum of the array. ` `        ``for` `(``int` `i = ``0``; i < n; i++) ` `            ``prefixsum[i + ``1``] = prefixsum[i] + arr[i]; ` ` `  `        ``return` `bsearch(prefixsum, n, k); ` `    ``} ` `     `  `    ``// Driver code ` `    ``public` `static` `void` `main(String arg[]) ` `    ``{ ` `        ``int` `arr[] = { ``1``, ``2``, ``10``, ``4` `}; ` `        ``int` `n = arr.length; ` `        ``int` `k = ``14``; ` `        ``System.out.println(maxSize(arr, n, k)); ` `    ``} ` `} ` ` `  `// This code is contributed by Anant Agarwal. `

## Python3

 `# Python program to find maximum  ` `# subarray size, such that all  ` `# subarrays of that size have ` `# sum less than K. ` ` `  `# Search for the maximum length of  ` `# required subarray. ` `def` `bsearch(prefixsum, n, k): ` ` `  `    ``# Initialize result ` `    ``# Do Binary Search for largest ` `    ``# subarray size ` `    ``ans, left, right ``=` `-``1``, ``1``, n ` ` `  `    ``while` `(left <``=` `right): ` ` `  `        ``# Check for all subarrays after mid ` `        ``mid ``=` `(left ``+` `right)``/``/``2` ` `  `        ``for` `i ``in` `range``(mid, n ``+` `1``): ` ` `  `            ``# Checking if all the subarray of  ` `            ``# a size is less than k. ` `            ``if` `(prefixsum[i] ``-` `prefixsum[i ``-` `mid] > k): ` `                ``i ``=` `i ``-` `1` `                ``break` `        ``i ``=` `i ``+` `1` `        ``if` `(i ``=``=` `n ``+` `1``): ` `            ``left ``=` `mid ``+` `1` `            ``ans ``=` `mid ` `        ``# We found a subrray of size mid with sum ` `        ``# greater than k ` `        ``else``: ` `            ``right ``=` `mid ``-` `1` ` `  `    ``return` `ans; ` ` `  `# Return the maximum subarray size, such  ` `# that all subarray of that size have  ` `# sum less than K. ` `def` `maxSize(arr, n, k): ` `    ``prefixsum ``=` `[``0` `for` `x ``in` `range``(n ``+` `1``)] ` `     `  `    ``# Finding prefix sum of the array. ` `    ``for` `i ``in` `range``(n): ` `        ``prefixsum[i ``+` `1``] ``=` `prefixsum[i] ``+` `arr[i] ` ` `  `    ``return` `bsearch(prefixsum, n, k); ` ` `  `# Driver Code ` `arr ``=` `[ ``1``, ``2``, ``10``, ``4` `] ` `n ``=` `len``(arr) ` `k ``=` `14` `print` `(maxSize(arr, n, k)) ` ` `  `# This code is contributed by Afzal `

## C#

 `// C# program to find maximum  ` `// subarray size, such that all  ` `// subarrays of that size have ` `// sum less than K. ` `using` `System; ` ` `  `class` `GFG { ` `     `  `    ``// Search for the maximum length  ` `    ``// of required subarray. ` `    ``static` `int` `bsearch(``int` `[]prefixsum,  ` `                          ``int` `n, ``int` `k) ` `    ``{ ` `         `  `        ``// Initialize result ` `        ``int` `ans = -1;  ` ` `  `        ``// Do Binary Search for  ` `        ``// largest subarray size ` `        ``int` `left = 1, right = n; ` `         `  `        ``while` `(left <= right)  ` `        ``{ ` `            ``int` `mid = (left + right) / 2; ` ` `  `            ``// Check for all subarrays  ` `            ``// after mid ` `            ``int` `i; ` `            ``for` `(i = mid; i <= n; i++)  ` `            ``{ ` `                 `  `                ``// Checking if all the  ` `                ``// subarrays of a size is ` `                ``// less than k. ` `                ``if` `(prefixsum[i] -  ` `                     ``prefixsum[i - mid] > k) ` `                    ``break``; ` `            ``} ` ` `  `            ``// All subarrays of size mid have  ` `            ``// sum less than or equal to k ` `            ``if` `(i == n + 1) ` `            ``{ ` `                ``left = mid + 1; ` `                ``ans = mid; ` `            ``} ` ` `  `            ``// We found a subrray of size mid  ` `            ``// with sum greater than k ` `            ``else` `                ``right = mid - 1; ` `        ``} ` ` `  `        ``return` `ans; ` `    ``} ` ` `  `    ``// Return the maximum subarray size, such  ` `    ``// that all subarray of that size have  ` `    ``// sum less than K. ` `    ``static` `int` `maxSize(``int` `[]arr, ``int` `n, ``int` `k) ` `    ``{ ` `         `  `        ``// Initialize prefix sum array as 0. ` `        ``int` `[]prefixsum = ``new` `int``[n + 1]; ` `        ``for``(``int` `i=0;i

## PHP

 ` ``\$k``) ` `                ``break``; ` `        ``} ` ` `  `        ``// All subarrays of size mid have  ` `        ``// sum less than or equal to k ` `        ``if` `(``\$i` `== ``\$n` `+ 1) ` `        ``{ ` `            ``\$left` `= ``\$mid` `+ 1; ` `            ``\$ans` `= ``\$mid``; ` `        ``} ` ` `  `        ``// We found a subrray of size mid  ` `        ``// with sum greater than k ` `        ``else` `            ``\$right` `= ``\$mid` `- 1; ` `    ``} ` `    ``return` `\$ans``; ` `} ` ` `  `// Return the maximum subarray size, ` `// such that all subarray of that size ` `// have sum less than K. ` `function` `maxSize(&``\$arr``, ``\$n``, ``\$k``) ` `{ ` `    ``// Initialize prefix sum array as 0. ` `    ``\$prefixsum` `= ``array_fill``(0, ``\$n` `+ 1, NULL); ` ` `  `    ``// Finding prefix sum of the array. ` `    ``for` `(``\$i` `= 0; ``\$i` `< ``\$n``; ``\$i``++) ` `        ``\$prefixsum``[``\$i` `+ 1] = ``\$prefixsum``[``\$i``] +  ` `                             ``\$arr``[``\$i``]; ` ` `  `    ``return` `bsearch(``\$prefixsum``, ``\$n``, ``\$k``); ` `} ` ` `  `// Driver code ` `\$arr` `= ``array``(1, 2, 10, 4); ` `\$n` `= sizeof(``\$arr``); ` `\$k` `= 14; ` `echo` `maxSize(``\$arr``, ``\$n``, ``\$k``) . ``"\n"``; ` ` `  `// This code is contributed  ` `// by ChitraNayal ` `?> `

Output :

```2
```

Time Complexity : O(n log n).

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