Explanation of the Median of Medians algorithm

Question

The Median of medians approach is very popular in quicksort type partitioning algorithms to yield a fairly good pivot, such that it partitions the

Answer 1

Think of the following set of numbers:

5 2 6 3 1

The median of these numbers is 3. Now if you have a number n, if n > 3, then it is bigger than at least half of the numbers above. If n < 3, then it is smaller than at least half of the numbers above.

So that is the idea. That is, for each set of 5 numbers, you get their median. Now you have n / 5 numbers. This is obvious.

Now if you get the median of those numbers (call it m), it is bigger than half of them and smaller than the other half (by definition of median!). In other words, m is bigger than n / 10 numbers (which themselves were medians of small 5 element groups) and bigger than another n / 10 numbers (which again were medians of small 5 element groups).

In the example above, we saw that if the median is k and you have m > k, then m is also bigger than 2 other numbers (that were themselves smaller than k). This means that for each of those smaller 5 element groups where m was bigger than its median, m is also bigger than two other numbers. This makes it at least 3 numbers (2 numbers + the median itself) in each of those n / 10 small 5 element groups, that are smaller than m. Hence, m is at least bigger than 3n/10 numbers.

Similar logic for the number of elements m is bigger than.