Find maximum average subarray of k length

Given an array with positive and negative numbers, find the maximum average subarray of the 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, and the inner loop goes to length ‘k’ from the starting point and computes the average of elements. 

Time Complexity: O(n*k), as we are using nested loops to traverse n*k times.
Auxiliary Space: O(1), as we are not using any extra space.

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 the csum[] array with us, we can compute the sum between two indexes in O(1) time. 
Below is the implementation of this idea. One observation is, that a subarray of a given length has a maximum average if it has a maximum sum. So we can avoid floating-point arithmetic by just comparing sums.

C++

// 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;
}

Java

// 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.

Python3

# 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

C#

// 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.

PHP

<?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.
?>

Javascript

<script>

// Javascript 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]
    let csum = new Array(n);

    csum[0] = arr[0];
    for(let i = 1; i < n; i++)
        csum[i] = csum[i - 1] + arr[i];

    // Initialize max_sm as
    // sum of first subarray
    let max_sum = csum[k - 1],
        max_end = k - 1;

    // Find sum of other
    // subarrays and update
    // max_sum if required.
    for(let i = k; i < n; i++)
    {
        let 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
let arr = [ 1, 12, -5, -6, 50, 3 ];
let k = 4;
let n = arr.length;
document.write("The maximum average subarray of "+
               "length "+ k + " begins at index " +
               findMaxAverage(arr, n, k));

// This code is contributed by divyeshrabadiya07

</script>

Output

The maximum average subarray of length 4 begins at index 1

Time Complexity: O(n), as we are using a loop to traverse n times. Where n is the number of elements in the array.
Auxiliary Space: O(n), as we are using extra space for the array csum.

We can avoid the need for extra space by using the 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’ 
   • Remove arr[i-k] from sum and add arr[i], i.e., do sum += arr[i] – arr[i-k] 
   • If new sum becomes more than max_sum so far, update max_sum. 
3. Return ‘max_sum’

C++

// 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++)
    {
        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;
}

Java

// 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.

Python3

# 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 = 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

C#

// 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.

PHP

<?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.
?>

Javascript

<script>
    // Javascript 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
        let sum = arr[0];
        for (let i = 1; i < k; i++)
            sum += arr[i];

        let max_sum = sum;
        let max_end = k-1;

        // Compute sum of remaining
        // subarrays
        for (let 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;
    }

    let arr = [1, 12, -5, -6, 50, 3];
    let k = 4;
    let n = arr.length;
    document.write( "The maximum "
                      + "average subarray of length "
                      + k + " begins at index "
                      + findMaxAverage(arr, n, k));

    // This code is contributed by suresh07.
</script>

Output

The maximum average subarray of length 4 begins at index 1

Time Complexity: O(n), as we are using a loop to traverse n times. Where n is the number of elements in the array.
Auxiliary Space: O(1), as we are not using any extra space.

Approach#3: Using sliding window

We can use a sliding window approach to solve this problem, which reduces the time complexity to O(n). We can start by calculating the sum of the first k elements and then move the window one element at a time, subtracting the element that is no longer in the window and adding the new element that is now in the window

Algorithm

1. Initialize a variable max_sum to the sum of the first k elements and max_index to 0.
2. Initialize a variable window_sum to max_sum.
3. Loop through the array from index k to n-1:
a. Subtract the element that is no longer in the window from window_sum.
b. Add the new element that is now in the window to window_sum.
c. If window_sum is greater than max_sum, update max_sum to window_sum and max_index to the starting index of the current window.
4. Return max_index.

Python3

def max_avg_subarray(arr, k):
    n = len(arr)
    window_sum = sum(arr[:k])
    max_sum = window_sum
    max_index = 0
    for i in range(k, n):
        window_sum += arr[i] - arr[i-k]
        if window_sum > max_sum:
            max_sum = window_sum
            max_index = i - k + 1
    return max_index
arr = [1, 12, -5, -6, 50, 3]
k = 4
print(max_avg_subarray(arr, k))

Javascript

function max_avg_subarray(arr, k) {
    // Get the length of the array
    let n = arr.length;

    // Calculate the sum of the first window of size k
    let window_sum = arr.slice(0, k).reduce((a, b) => a + b, 0);
    let max_sum = window_sum;
    let max_index = 0;

    // Slide the window and update the maximum sum
    for (let i = k; i < n; i++) {
        window_sum += arr[i] - arr[i - k];
        if (window_sum > max_sum) {
            max_sum = window_sum;
            max_index = i - k + 1;
        }
    }
    return max_index;
}

let arr = [1, 12, -5, -6, 50, 3];
let k = 4;
console.log(max_avg_subarray(arr, k));

Output

1

Time complexity: O(n) , where n is length of array

Auxiliary Space is O(1).