Binary heap

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A binary heap is a heap data structure created using a binary tree. It can be seen as a binary tree with two additional constraints:

  • The shape property: the tree is a complete binary tree; that is, all levels of the tree, except possibly the last one (deepest) are fully filled, and, if the last level of the tree is not complete, the nodes of that level are filled from left to right.
  • The heap property: each node is greater than or equal to each of its children according to some comparison predicate which is fixed for the entire data structure.

"Greater than or equal to" means according to whatever comparison function is chosen to sort the heap, not necessarily "greater than or equal to" in the mathematical sense (since the quantities are not always numerical). Heaps where the comparison function is mathematical "greater than or equal to" are called max-heaps; those where the comparison function is mathematical "less than or equal to" are called "min-heaps". Conventionally, min-heaps are used, since they are readily applicable for use in priority queues.

Note that the ordering of siblings in a heap is not specified by the heap property, so the two children of a parent can be freely interchanged, as long as this does not violate the shape and heap properties (compare with treap).

The binary heap is a special case of the d-ary heap in which d = 2.

It is possible to modify the heap structure to allow extraction of both the smallest and largest element in O(logn) time.[1] To do this, the rows alternate between min heap and max heap. The algorithms are roughly the same, but, in each step, one must consider the alternating rows with alternating comparisons. The performance is roughly the same as a normal single direction heap. This idea can be generalised to a min-max-median heap.

Contents

Heap operations

Adding to the heap

If we have a heap, and we add an element, we can perform an operation known as up-heap, bubble-up, percolate-up, sift-up, or heapify-up in order to restore the heap property. We can do this in O(log n) time, using a binary heap, by following this algorithm:

We do this at maximum for each level in the tree—the height of the tree, which is O(log n). However, since approximately 50% of the elements are leaves and 75% are in the bottom two levels, it is likely that the new element to be inserted will only move a few levels upwards to maintain the heap. Thus, binary heaps support insertion in average constant time, O(1).

Say we have a max-heap

and we want to add the number 15 to the heap. We first place the 15 in the position marked by the X. However, the heap property is violated since 15 is greater than 8, so we need to swap the 15 and the 8. So, we have the heap looking as follows after the first swap:

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