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
We strongly recommend that you click here and practice it, before moving on to the solution.
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.
PHP
<?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. ?> |
Output:
Subarray between [3, 5] has minimum average
Time Complexity: O(n)
Auxiliary Space: O(1)
Please refer complete article on Find the subarray with least average for more details!
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