C Binary Heap

A binary heap is a heap a 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 an almost 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 to the entire data structure.

“Greater than” means according to whatever comparison function is chosen to sort the heap, not necessarily “greater than” in the mathematical sense (since the quantities are not always numerical). Heaps where the comparison function is mathematical “greater than” are called max-heaps; those where the comparison function is mathematical “less than” 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. To do this the rows alternate between min heap and max heap. The algorithms are roughly the same, but in each step must consider the alternating rows with alternating comparisons. The performance is roughly the same as a normal single direction heap. This idea can be generalized to a min-max-median heap.

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