Min-max heaps were introduced in [ASSS86] as an efficient way to support heap operations for both minimum and maximum values. Structurally, the min-max heap levels alternate between min-heap condition and max-heap, and hence evaluates grandchildren/grandparents during insertion or search. Min-max heaps can also be generalized to find the k-th smallest element in O(1) time.
An interesting application is finding the running median of a stream of numbers. [ASSS86] describe a simple extension called min-max-median heap that can find the running median in log-linear time complexity (indexed skip lists can too in amortized time complexity). The following code implements a method for finding the running median on each insertion. The trick is to maintain a min-max and a max-min heap such that either heap has at most one more element than the other.
public double add(int a) { if (minHeap.size() == 0) { minHeap.add(a); return a; } // add new element to appropriate heap if (a < minHeap.findMax()) { minHeap.add(a); } else { maxHeap.add(a); } int minSize = minHeap.size(); int maxSize = maxHeap.size(); // resize heaps to enforce size constraint if (maxSize == minSize - 2) { maxHeap.add(minHeap.removeMax()); } else if (minSize == maxSize - 2) { minHeap.add(maxHeap.removeMin()); } minSize = minHeap.size(); maxSize = maxHeap.size(); // calculate median if (minSize > maxSize) { return minHeap.findMax(); } else if (maxSize > minSize) { return maxHeap.findMin(); } return (minHeap.findMax() + maxHeap.findMin()) / 2.0; }
[ASSS86] M. D. Atkinson, J.-R. Sack, N. Santoro, and T. Strothotte. Min-max Heaps and Generalized Priority Queues. Communications of the ACM, Vol. 29 No. 10, 1986.