Find maximum average subarray of k length

Given an array with positive and negative numbers, find the maximum average subarray of given length.

Example:

Input:  arr[] = {1, 12, -5, -6, 50, 3}, k = 4
Output: Maximum average subarray of length 4 begins
        at index 1.
Maximum average is (12 - 5 - 6 + 50)/4 = 51/4

A Simple Solution is to run two loops. The outer loop picks starting point, the inner loop goes till length ‘k’ from the starting point and computes average of elements. Time complexity of this solution is O(n*k).



A Better Solution is to create an auxiliary array of size n. Store cumulative sum of elements in this array. Let the array be csum[]. csum[i] stores sum of elements from arr[0] to arr[i]. Once we have csum[] array with us, we can compute sum between two indexes in O(1) time.
Below is the implementation of this idea. One observation is, a subarray of given length has maximum average if it has maximum sum. So we can avoid floating point arithmetic by just comparing sum.

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ program to find maximum average subarray
// of given length.
#include<bits/stdc++.h>
using namespace std;
  
// Returns beginning index of maximum average
// subarray of length 'k'
int findMaxAverage(int arr[], int n, int k)
{
    // Check if 'k' is valid
    if (k > n)
        return -1;
  
    // Create and fill array to store cumulative
    // sum. csum[i] stores sum of arr[0] to arr[i]
    int *csum = new int[n];
    csum[0] = arr[0];
    for (int i=1; i<n; i++)
       csum[i] = csum[i-1] + arr[i];
  
    // Initialize max_sm as sum of first subarray
    int max_sum = csum[k-1], max_end = k-1;
  
    // Find sum of other subarrays and update
    // max_sum if required.
    for (int i=k; i<n; i++)
    {
        int curr_sum = csum[i] - csum[i-k];
        if (curr_sum > max_sum)
        {
            max_sum = curr_sum;
            max_end = i;
        }
    }
  
    delete [] csum; // To avoid memory leak
  
    // Return starting index
    return max_end - k + 1;
}
  
// Driver program
int main()
{
    int arr[] = {1, 12, -5, -6, 50, 3};
    int k = 4;
    int n = sizeof(arr)/sizeof(arr[0]);
    cout << "The maximum average subarray of "
         "length "<< k << " begins at index "
         << findMaxAverage(arr, n, k);
    return 0;
}
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// Java program to find maximum average
// subarray of given length.
import java .io.*;
  
class GFG {
  
    // Returns beginning index 
    // of maximum average
    // subarray of length 'k'
    static int findMaxAverage(int []arr, 
                           int n, int k)
    {
          
        // Check if 'k' is valid
        if (k > n)
            return -1;
      
        // Create and fill array 
        // to store cumulative
        // sum. csum[i] stores 
        // sum of arr[0] to arr[i]
        int []csum = new int[n];
          
        csum[0] = arr[0];
        for (int i = 1; i < n; i++)
        csum[i] = csum[i - 1] + arr[i];
      
        // Initialize max_sm as 
        // sum of first subarray
        int max_sum = csum[k - 1], 
                    max_end = k - 1;
      
        // Find sum of other 
        // subarrays and update
        // max_sum if required.
        for (int i = k; i < n; i++)
        {
            int curr_sum = csum[i] - 
                    csum[i - k];
            if (curr_sum > max_sum)
            {
                max_sum = curr_sum;
                max_end = i;
            }
        }
      
        // To avoid memory leak
        //delete [] csum; 
          
        // Return starting index
        return max_end - k + 1;
    }
  
    // Driver Code
    static public void main (String[] args)
    {
        int []arr = {1, 12, -5, -6, 50, 3};
        int k = 4;
        int n = arr.length;
          
        System.out.println("The maximum "
          + "average subarray of length "
                + k + " begins at index "
            + findMaxAverage(arr, n, k));
    }
}
  
// This code is contributed by anuj_67.
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// C# program to find maximum average
// subarray of given length.
using System;
class GFG{
  
// Returns beginning index 
// of maximum average
// subarray of length 'k'
static int findMaxAverage(int []arr, 
                       int n, int k)
{
      
    // Check if 'k' is valid
    if (k > n)
        return -1;
  
    // Create and fill array 
    // to store cumulative
    // sum. csum[i] stores 
    // sum of arr[0] to arr[i]
    int []csum = new int[n];
      
    csum[0] = arr[0];
    for (int i = 1; i < n; i++)
    csum[i] = csum[i - 1] + arr[i];
  
    // Initialize max_sm as 
    // sum of first subarray
    int max_sum = csum[k - 1], 
              max_end = k - 1;
  
    // Find sum of other 
    // subarrays and update
    // max_sum if required.
    for (int i = k; i < n; i++)
    {
        int curr_sum = csum[i] - 
                   csum[i - k];
        if (curr_sum > max_sum)
        {
            max_sum = curr_sum;
            max_end = i;
        }
    }
  
    // To avoid memory leak
    //delete [] csum; 
      
    // Return starting index
    return max_end - k + 1;
}
  
    // Driver Code
    static public void Main ()
    {
        int []arr = {1, 12, -5, -6, 50, 3};
        int k = 4;
        int n = arr.Length;
        Console.WriteLine("The maximum average subarray of "+
                            "length "+ k + " begins at index "
                                    + findMaxAverage(arr, n, k));
    }
}
  
// This code is contributed by anuj_67.
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

# Python program to find maximum average subarray
# of given length.
  
# Returns beginning index of maximum average
# subarray of length 'k'
def findMaxAverage(arr, n, k):
    # Check if 'k' is valid
    if k > n:
        return -1
  
    # Create and fill array to store cumulative
    # sum. csum[i] stores sum of arr[0] to arr[i]
    csum = [0]*n
    csum[0] = arr[0]
    for i in range(1, n):
        csum[i] = csum[i-1] + arr[i];
  
    # Initialize max_sm as sum of first subarray
    max_sum = csum[k-1]
    max_end = k-1
  
    # Find sum of other subarrays and update
    # max_sum if required.
    for i in range(k, n):
      
        curr_sum = csum[i] - csum[i-k]
        if curr_sum > max_sum:
          
            max_sum = curr_sum
            max_end = i
          
    # Return starting index
    return max_end - k + 1
  
# Driver program
arr = [1, 12, -5, -6, 50, 3]
k = 4
n = len(arr)
print("The maximum average subarray of length",k,
"begins at index",findMaxAverage(arr, n, k))
  
#This code is contributed by
#Smitha Dinesh Semwal
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

<?php
// PHP program to find maximum
// average subarray of given length.
  
// Returns beginning index of
// maximum average subarray of 
// length 'k'
function findMaxAverage($arr, $n, $k)
{
      
    // Check if 'k' is valid
    if ($k > $n)
        return -1;
  
    // Create and fill array to
    // store cumulative sum. 
    // csum[i] stores sum of 
    // arr[0] to arr[i]
    $csum = array();
    $csum[0] = $arr[0];
    for($i = 1; $i < $n; $i++)
    $csum[$i] = $csum[$i - 1] + 
                $arr[$i];
  
    // Initialize max_sm as sum
    // of first subarray
    $max_sum = $csum[$k - 1]; 
    $max_end = $k - 1;
  
    // Find sum of other subarrays 
    // and update max_sum if required.
    for($i = $k; $i < $n; $i++)
    {
        $curr_sum = $csum[$i] - 
                    $csum[$i - $k];
        if ($curr_sum > $max_sum)
        {
            $max_sum = $curr_sum;
            $max_end = $i;
        }
    }
  
    // Return starting index
    return $max_end - $k + 1;
}
  
    // Driver Code
    $arr = array(1, 12, -5, -6, 50, 3);
    $k = 4;
    $n = count($arr);
    echo "The maximum average subarray of "
        ,"length ", $k , " begins at index "
        , findMaxAverage($arr, $n, $k);
          
// This code is contributed by anuj_67.
?>
chevron_right


Output:
The maximum average subarray of length 4 begins at index 1

Time Complexity of above solution is O(n), but it requires O(n) auxiliary space.

We can avoid need of extra space by using below Efficient Method.
1) Compute sum of first ‘k’ elements, i.e., elements arr[0..k-1]. Let this sum be ‘sum’. Initialize ‘max_sum’ as ‘sum’
2) Do following for every element arr[i] where i varies from ‘k’ to ‘n-1’
…….a) Remove arr[i-k] from sum and add arr[i], i.e., do sum += arr[i] – arr[i-k]
…….b) If new sum becomes more than max_sum so far, update max_sum.
3) Return ‘max_sum’

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ program to find maximum average subarray
// of given length.
#include<bits/stdc++.h>
using namespace std;
  
// Returns beginning index of maximum average
// subarray of length 'k'
int findMaxAverage(int arr[], int n, int k)
{
    // Check if 'k' is valid
    if (k > n)
        return -1;
  
    // Compute sum of first 'k' elements
    int sum = arr[0];
    for (int i=1; i<k; i++)
        sum += arr[i];
  
    int max_sum = sum, max_end = k-1;
  
    // Compute sum of remaining subarrays
    for (int i=k; i<n; i++)
    {
        int sum = sum + arr[i] - arr[i-k];
        if (sum > max_sum)
        {
            max_sum = sum;
            max_end = i;
        }
    }
  
    // Return starting index
    return max_end - k + 1;
}
  
// Driver program
int main()
{
    int arr[] = {1, 12, -5, -6, 50, 3};
    int k = 4;
    int n = sizeof(arr)/sizeof(arr[0]);
    cout << "The maximum average subarray of "
         "length "<< k << " begins at index "
         << findMaxAverage(arr, n, k);
    return 0;
}
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// Java program to find maximum average subarray
// of given length.
  
import java.io.*;
  
class GFG {
  
    // Returns beginning index of maximum average
    // subarray of length 'k'
    static int findMaxAverage(int arr[], int n, int k)
    {
          
        // Check if 'k' is valid
        if (k > n)
            return -1;
      
        // Compute sum of first 'k' elements
        int sum = arr[0];
        for (int i = 1; i < k; i++)
            sum += arr[i];
      
        int max_sum = sum, max_end = k-1;
      
        // Compute sum of remaining subarrays
        for (int i = k; i < n; i++)
        {
            sum = sum + arr[i] - arr[i-k];
            if (sum > max_sum)
            {
                max_sum = sum;
                max_end = i;
            }
        }
      
        // Return starting index
        return max_end - k + 1;
    }
  
    // Driver program
    public static void main (String[] args)
    {
        int arr[] = {1, 12, -5, -6, 50, 3};
        int k = 4;
        int n = arr.length;
        System.out.println( "The maximum average"
                     + " subarray of length " + k 
                     + " begins at index "
                    + findMaxAverage(arr, n, k));
    }
}
  
// This code is contributed by anuj_67.
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

# Python 3 program to find maximum
# average subarray of given length.
  
# Returns beginning index of maximum
# average subarray of length 'k'
def findMaxAverage(arr, n, k):
  
    # Check if 'k' is valid
    if (k > n):
        return -1
  
    # Compute sum of first 'k' elements
    sum = arr[0
      
    for i in range(1, k):
        sum += arr[i] 
  
    max_sum = sum
    max_end = k - 1
  
    # Compute sum of remaining subarrays
    for i in range(k, n):
      
        sum = sum + arr[i] - arr[i - k] 
          
        if (sum > max_sum):
          
            max_sum = sum
            max_end =
          
    # Return starting index
    return max_end - k + 1
  
# Driver program
arr = [1, 12, -5, -6, 50, 3
k = 4
n = len(arr) 
  
print("The maximum average subarray of length", k,
                                "begins at index"
                        findMaxAverage(arr, n, k))
  
# This code is contributed by
# Smitha Dinesh Semwal
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// C# program to find maximum average 
// subarray of given length.
using System;
  
class GFG {
  
    // Returns beginning index of 
    // maximum average subarray of
    // length 'k'
    static int findMaxAverage(int []arr,
                           int n, int k)
    {
          
        // Check if 'k' is valid
        if (k > n)
            return -1;
      
        // Compute sum of first 'k' 
        // elements
        int sum = arr[0];
        for (int i = 1; i < k; i++)
            sum += arr[i];
      
        int max_sum = sum;
        int max_end = k-1;
      
        // Compute sum of remaining 
        // subarrays
        for (int i = k; i < n; i++)
        {
            sum = sum + arr[i] - arr[i-k];
            if (sum > max_sum)
            {
                max_sum = sum;
                max_end = i;
            }
        }
      
        // Return starting index
        return max_end - k + 1;
    }
  
    // Driver program
    public static void Main ()
    {
        int []arr = {1, 12, -5, -6, 50, 3};
        int k = 4;
        int n = arr.Length;
        Console.WriteLine( "The maximum "
          + "average subarray of length "
                + k + " begins at index "
            + findMaxAverage(arr, n, k));
    }
}
  
// This code is contributed by anuj_67.
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

<?php
// PHP program to find maximum
// average subarray of given length.
  
// Returns beginning index
// of maximum average
// subarray of length 'k'
function findMaxAverage($arr, $n, $k)
{
      
    // Check if 'k' is valid
    if ($k > $n)
        return -1;
  
    // Compute sum of first
    // 'k' elements
    $sum = $arr[0];
    for($i = 1; $i < $k; $i++)
        $sum += $arr[$i];
  
    $max_sum = $sum;
    $max_end = $k-1;
  
    // Compute sum of
    // remaining subarrays
    for($i = $k; $i < $n; $i++)
    {
        $sum = $sum + $arr[$i] - 
                 $arr[$i - $k];
        if ($sum > $max_sum)
        {
            $max_sum = $sum;
            $max_end = $i;
        }
    }
  
    // Return starting index
    return $max_end - $k + 1;
}
  
    // Driver Code
    $arr = array(1, 12, -5, -6, 50, 3);
    $k = 4;
    $n = count($arr);
    echo "The maximum average subarray of ",
         "length ", $k , " begins at index "
        , findMaxAverage($arr, $n, $k);
          
// This code is contributed by anuj_67.
?>
chevron_right

Output:

The maximum average subarray of length 4 begins at index 1

Time complexity of this method is also O(n), but it requires constant extra space.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above




Improved By : vt_m



Article Tags :
Practice Tags :