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 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 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 sum of elements less than 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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