How do I find the median of numbers in linear time using heaps?

Question

Wikipedia says:

Selection algorithms: Finding the min, max, both the min and max, median, or even the k-th largest element can be done i

Answer 1

You would use a min-max-median heap to find the min, max and median in constant time (and take linear time to build the heap). You can use order-statistics trees to find the kth smallest/largest value. Both of these data structures are described in this paper on min-max heaps [pdf link]. Min-max heaps are binary heaps that alternate between min-heaps and max-heaps.

From the paper: A min-max-median heap is a binary heap with the following properties:

1) The median of all elements is located at the root

2) The left subtree of the root is a min-max heap Hl of size ceiling[((n-1)/2)] containing elements less than or equal to the median. The right subtree is a max-min heap Hr of size floor[((n-1)/2)] containing only elements greater than or equal to the median.

The paper goes on to explain how to build such a heap.

Edit: Upon reading the paper more thoroughly it appears as though building the min-max-median heaps requires that you first find the median (FTA: "Find the median of all n elements using any one of the known linear-time algorithms"). That said, once you have built the heap you can maintain the median simply by maintaining the balance between the min-max heap on the left and the max-min heap on the right. DeleteMedian replaces the root with either the min of the max-min heap or the max of the min-max heap (whichever maintains the balance).

So if you plan on using a min-max-median heap to find the median of a fixed data set you're SOL but if you are using it on a changing data set it is possible.