Mergeable heap
# Mergeable heap

In computer science, a mergeable heap (also called a meldable heap) is an abstract data type, which is a heap supporting a merge operation.

## Definition

A mergeable heap supports the usual heap operations:

* Make-Heap, create an empty heap.
* Insert(H,x), insert an element x into the heap H.
* Min(H), return the minimum element, or Nil if no such element exists.
* Extract-Min(H), extract and return the minimum element, or Nil if no such element exists.

And one more that distinguishes it:

* Merge(H1,H2), combine the elements of H1 and H2 into a single heap.

## Trivial implementation

It is straightforward to implement a mergeable heap given a simple heap:

Merge(H1,H2):

* x ← Extract-Min(H2)
* while x ≠ Nil
   * Insert(H1, x)
   * x ← Extract-Min(H2)

This can however be wasteful as each Extract-Min(H) and Insert(H,x) typically have to maintain the heap property.

## More efficient implementations

Examples of mergeable heap data structures include:

* Binomial heap
* Fibonacci heap
* Leftist tree
* Pairing heap
* Skew heap

A more complete list with performance comparisons can be found at Heap (data structure) § Comparison of theoretic bounds for variants.

In most mergeable heap structures, merging is the fundamental operation on which others are based. Insertion is implemented by merging a new single-element heap with the existing heap. Deletion is implemented by merging the children of the deleted node.

