We are given a row-wise sorted matrix of size r*c, we need to find the median of the matrix given. It is assumed that r*c is always odd.
Input : 1 3 5 2 6 9 3 6 9 Output : Median is 5 If we put all the values in a sorted array A = 1 2 3 3 5 6 6 9 9) Input: 1 3 4 2 5 6 7 8 9 Output: Median is 5
Simple Method: The simplest method to solve this problem is to store all the elements of the given matrix in an array of size r*c. Then we can either sort the array and find the median element in O(r*clog(r*c)) or we can use the approach discussed here to find the median in O(r*c). Auxiliary space required will be O(r*c) in both cases.
An efficient approach for this problem is to use a binary search algorithm. The idea is that for a number to be median there should be exactly (n/2) numbers which are less than this number. So, we try to find the count of numbers less than all the numbers. Below is the step by step algorithm for this approach:
- First, we find the minimum and maximum elements in the matrix. The minimum element can be easily found by comparing the first element of each row, and similarly, the maximum element can be found by comparing the last element of each row.
- Then we use binary search on our range of numbers from minimum to maximum, we find the mid of the min and max and get a count of numbers less than our mid. And accordingly change the min or max.
- For a number to be median, there should be (r*c)/2 numbers smaller than that number. So for every number, we get the count of numbers less than that by using upper_bound() in each row of the matrix, if it is less than the required count, the median must be greater than the selected number, else the median must be less than or equal to the selected number.
Below is the implementation of the above approach:
Median is 5
Time Complexity: O(32 * r * log(c)). The upper bound function will take log(c) time and is performed for each row. And since the numbers will be max of 32 bit, so binary search of numbers from min to max will be performed in at most 32 ( log2(2^32) = 32 ) operations.
Auxiliary Space : O(1)
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