Find the subarray with least average

Given an array arr[] of size n and integer k such that k <= n.

Examples :

Input:  arr[] = {3, 7, 90, 20, 10, 50, 40}, k = 3
Output: Subarray between indexes 3 and 5
The subarray {20, 10, 50} has the least average 
among all subarrays of size 3.

Input:  arr[] = {3, 7, 5, 20, -10, 0, 12}, k = 2
Output: Subarray between [4, 5] has minimum average



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A Simple Solution is to consider every element as beginning of subarray of size k and compute sum of subarray starting with this element. Time complexity of this solution is O(nk).

An Efficient Solution is to solve the above problem in O(n) time and O(1) extra space. The idea is to use sliding window of size k. Keep track of sum of current k elements. To compute sum of current window, remove first element of previous window and add current element (last element of current window).

1) Initialize res_index = 0 // Beginning of result index
2) Find sum of first k elements. Let this sum be 'curr_sum'
3) Initialize min_sum = sum
4) Iterate from (k+1)'th to n'th element, do following
   for every element arr[i]
      a) curr_sum = curr_sum + arr[i] - arr[i-k]
      b) If curr_sum < min_sum
           res_index = (i-k+1)
5) Print res_index and res_index+k-1 as beginning and ending
   indexes of resultant subarray.

Below is the implementation of above algorithm.

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// A Simple C++ program to find minimum average subarray
#include <bits/stdc++.h>
using namespace std;
  
// Prints beginning and ending indexes of subarray
// of size k with minimum average
void findMinAvgSubarray(int arr[], int n, int k)
{
    // k must be smaller than or equal to n
    if (n < k)
        return;
  
    // Initialize  beginning index of result
    int res_index = 0;
  
    // Compute sum of first subarray of size k
    int curr_sum = 0;
    for (int i = 0; i < k; i++)
        curr_sum += arr[i];
  
    // Initialize minimum sum as current sum
    int min_sum = curr_sum;
  
    // Traverse from (k+1)'th element to n'th element
    for (int i = k; i < n; i++) {
        // Add current item and remove first item of
        // previous subarray
        curr_sum += arr[i] - arr[i - k];
  
        // Update result if needed
        if (curr_sum < min_sum) {
            min_sum = curr_sum;
            res_index = (i - k + 1);
        }
    }
  
    cout << "Subarray between [" << res_index << ", "
         << res_index + k - 1 << "] has minimum average";
}
  
// Driver program
int main()
{
    int arr[] = { 3, 7, 90, 20, 10, 50, 40 };
    int k = 3; // Subarray size
    int n = sizeof arr / sizeof arr[0];
    findMinAvgSubarray(arr, n, k);
    return 0;
}
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// A Simple Java program to find 
// minimum average subarray
  
class Test {
      
    static int arr[] = new int[] { 3, 7, 90, 20, 10, 50, 40 };
  
    // Prints beginning and ending indexes of subarray
    // of size k with minimum average
    static void findMinAvgSubarray(int n, int k)
    {
        // k must be smaller than or equal to n
        if (n < k)
            return;
  
        // Initialize beginning index of result
        int res_index = 0;
  
        // Compute sum of first subarray of size k
        int curr_sum = 0;
        for (int i = 0; i < k; i++)
            curr_sum += arr[i];
  
        // Initialize minimum sum as current sum
        int min_sum = curr_sum;
  
        // Traverse from (k+1)'th element to n'th element
        for (int i = k; i < n; i++) 
        {
            // Add current item and remove first
            // item of previous subarray
            curr_sum += arr[i] - arr[i - k];
  
            // Update result if needed
            if (curr_sum < min_sum) {
                min_sum = curr_sum;
                res_index = (i - k + 1);
            }
        }
  
        System.out.println("Subarray between [" +
                            res_index + ", " + (res_index + k - 1) +
                            "] has minimum average");
    }
  
    // Driver method to test the above function
    public static void main(String[] args)
    {
        int k = 3; // Subarray size
        findMinAvgSubarray(arr.length, k);
    }
}
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# Python3 program to find
# minimum average subarray
  
# Prints beginning and ending 
# indexes of subarray of size k
# with minimum average
def findMinAvgSubarray(arr, n, k):
  
    # k must be smaller than or equal to n
    if (n < k): return 0
  
    # Initialize beginning index of result
    res_index = 0
  
    # Compute sum of first subarray of size k
    curr_sum = 0
    for i in range(k):
        curr_sum += arr[i]
  
    # Initialize minimum sum as current sum
    min_sum = curr_sum
  
    # Traverse from (k + 1)'th
    # element to n'th element
    for i in range(k, n):
      
        # Add current item and remove first 
        # item of previous subarray
        curr_sum += arr[i] - arr[i-k]
  
        # Update result if needed
        if (curr_sum < min_sum):
          
            min_sum = curr_sum
            res_index = (i - k + 1)
          
    print("Subarray between [", res_index,
          ", ", (res_index + k - 1),
          "] has minimum average")
  
# Driver Code
arr = [3, 7, 90, 20, 10, 50, 40]
k = 3 # Subarray size
n = len(arr)
findMinAvgSubarray(arr, n, k)
  
# This code is contributed by Anant Agarwal.
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// A Simple C# program to find 
// minimum average subarray
using System;
  
class Test {
      
    static int[] arr = new int[] { 3, 7, 90, 20, 10, 50, 40 };
  
    // Prints beginning and ending indexes of subarray
    // of size k with minimum average
    static void findMinAvgSubarray(int n, int k)
    {
        // k must be smaller than or equal to n
        if (n < k)
            return;
  
        // Initialize beginning index of result
        int res_index = 0;
  
        // Compute sum of first subarray of size k
        int curr_sum = 0;
        for (int i = 0; i < k; i++)
            curr_sum += arr[i];
  
        // Initialize minimum sum as current sum
        int min_sum = curr_sum;
  
        // Traverse from (k+1)'th element to n'th element
        for (int i = k; i < n; i++) 
        {
            // Add current item and remove first item of
            // previous subarray
            curr_sum += arr[i] - arr[i - k];
  
            // Update result if needed
            if (curr_sum < min_sum) {
                min_sum = curr_sum;
                res_index = (i - k + 1);
            }
        }
  
        Console.Write("Subarray between [" + res_index + ", " +
                     (res_index + k - 1) + 
                     "] has minimum average");
    }
  
    // Driver method to test the above function
    public static void Main()
    {
        int k = 3; // Subarray size
        findMinAvgSubarray(arr.Length, k);
    }
}
  
// This code is contributed by nitin mittal.
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<?php
// A Simple PHP program to find 
// minimum average subarray
  
// Prints beginning and ending
// indexes of subarray of size 
// k with minimum average
function findMinAvgSubarray($arr, $n, $k)
{
      
    // k must be smaller 
    // than or equal to n
    if ($n < $k)
        return;
  
    // Initialize beginning 
    // index of result
    $res_index = 0;
  
    // Compute sum of first
    // subarray of size k
    $curr_sum = 0;
    for ($i = 0; $i < $k; $i++)
        $curr_sum += $arr[$i];
  
    // Initialize minimum sum 
    // as current sum
    $min_sum = $curr_sum;
  
    // Traverse from (k+1)'th element
    // to n'th element
    for ( $i = $k; $i < $n; $i++) 
    {
          
        // Add current item and 
        // remove first item of
        // previous subarray
        $curr_sum += $arr[$i] - $arr[$i - $k];
  
        // Update result if needed
        if ($curr_sum < $min_sum) {
            $min_sum = $curr_sum;
            $res_index = ($i - $k + 1);
        }
    }
  
    echo "Subarray between [" ,$res_index 
          , ", " ,$res_index + $k - 1, "] has minimum average";
}
  
    // Driver Code
    $arr = array(3, 7, 90, 20, 10, 50, 40);
      
    // Subarray size
    $k = 3; 
    $n = sizeof ($arr) / sizeof ($arr[0]);
    findMinAvgSubarray($arr, $n, $k);
    return 0;
  
// This code is contributed by nitin mittal.
?>
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Output:
Subarray between [3, 5] has minimum average

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

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Improved By : nitin mittal



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