Binomial heap

In computer science, a binomial heap is a data structure that acts as a priority queue but also allows pairs of heaps to be merged.
It is important as an implementation of the mergeable heap abstract data type (also called meldable heap), which is a priority queue supporting merge operation. It is implemented as a heap similar to a binary heap but using a special tree structure that is different from the complete binary trees used by binary heaps. Binomial heaps were invented in 1978 by Jean Vuillemin.
Binomial heap.
A binomial heap is implemented as a set of binomial trees (compare with a binary heap, which has a shape of a single binary tree), which are defined recursively as follows:
A binomial tree of order formula_1 has formula_5 nodes, and height formula_1. 
The name comes from the shape: a binomial tree of order formula_1 has formula_8 nodes at depth formula_9, a binomial coefficient.
Because of its structure, a binomial tree of order formula_1 can be constructed from two trees of order formula_2 by attaching one of them as the leftmost child of the root of the other tree. This feature is central to the "merge" operation of a binomial heap, which is its major advantage over other conventional heaps.
Structure of a binomial heap.
A binomial heap is implemented as a set of binomial trees that satisfy the "binomial heap properties":
The first property ensures that the root of each binomial tree contains the smallest key in the tree. It follows that the smallest key in the entire heap is one of the roots.
The second property implies that a binomial heap with formula_12 nodes consists of at most formula_13 binomial trees, where formula_14 is the binary logarithm. The number and orders of these trees are uniquely determined by the number of nodes formula_12: there is one binomial tree for each nonzero bit in the binary representation of the number formula_12. For example, the decimal number 13 is 1101 in binary, formula_17, and thus a binomial heap with 13 nodes will consist of three binomial trees of orders 3, 2, and 0 (see figure below).
<br>"Example of a binomial heap containing 13 nodes with distinct keys.The heap consists of three binomial trees with orders 0, 2, and 3."
The number of different ways that formula_12 items with distinct keys can be arranged into a binomial heap equals the largest odd divisor of formula_19. For formula_20 these numbers are
If the formula_12 items are inserted into a binomial heap in a uniformly random order, each of these arrangements is equally likely.
Implementation.
Because no operation requires random access to the root nodes of the binomial trees, the roots of the binomial trees can be stored in a linked list, ordered by increasing order of the tree. Because the number of children for each node is variable, it does not work well for each node to have separate links to each of its children, as would be common in a binary tree; instead, it is possible to implement this tree using links from each node to its highest-order child in the tree, and to its sibling of the next smaller order than it. These sibling pointers can be interpreted as the next pointers in a linked list of the children of each node, but with the opposite order from the linked list of roots: from largest to smallest order, rather than vice versa. This representation allows two trees of the same order to be linked together, making a tree of the next larger order, in constant time.
Merge.
The operation of merging two heaps is used as a subroutine in most other operations.
A basic subroutine within this procedure merges pairs of binomial trees of the same order. This may be done by comparing the keys at the roots of the two trees (the smallest keys in both trees). The root node with the larger key is made into a child of the root node with the smaller key, increasing its order by one:
 function mergeTree(p, q)
 if p.root.key <= q.root.key
 return p.addSubTree(q)
 else
 return q.addSubTree(p)
To merge two heaps more generally, the lists of roots of both heaps are traversed simultaneously in a manner similar to that of the merge algorithm,
in a sequence from smaller orders of trees to larger orders. When only one of the two heaps being merged contains a tree of order formula_22, this tree is moved to the output heap. When both of the two heaps contain a tree of order formula_22, the two trees are merged to one tree of order formula_24 so that the minimum-heap property is satisfied. It may later become necessary to merge this tree with some other tree of order formula_24 in one of the two input heaps. In the course of the algorithm, it will examine at most three trees of any order, two from the two heaps we merge and one composed of two smaller trees.
 function merge(p, q)
 while not (p.end() and q.end())
 tree = mergeTree(p.currentTree(), q.currentTree())
 if not heap.currentTree().empty()
 tree = mergeTree(tree, heap.currentTree())
 heap.addTree(tree)
 heap.next(); p.next(); q.next()
Because each binomial tree in a binomial heap corresponds to a bit in the binary representation of its size, there is an analogy between the merging of two heaps and the binary addition of the "sizes" of the two heaps, from right-to-left. Whenever a carry occurs during addition, this corresponds to a merging of two binomial trees during the merge.
Each tree has order at most formula_26 and therefore the running time is formula_27.
Insert.
Inserting a new element to a heap can be done by simply creating a new heap containing only this element and then merging it with the original heap. Because of the merge, a single insertion takes time formula_27. However, this can be sped up using a merge procedure that shortcuts the merge after it reaches a point where only one of the merged heaps has trees of larger order. With this speedup, across a series of formula_1 consecutive insertions, the total time for the insertions is formula_30. Another way of stating this is that (after logarithmic overhead for the first insertion in a sequence) each successive insert has an "amortized" time of formula_31 (i.e. constant) per insertion.
A variant of the binomial heap, the skew binomial heap, achieves constant worst case insertion time by using forests whose tree sizes are based on the skew binary number system rather than on the binary number system.
Find minimum.
To find the minimum element of the heap, find the minimum among the roots of the binomial trees. This can be done in formula_27 time, as there are just formula_27 tree roots to examine.
By using a pointer to the binomial tree that contains the minimum element, the time for this operation can be reduced to formula_31. The pointer must be updated when performing any operation other than finding the minimum. This can be done in formula_27 time per update, without raising the overall asymptotic running time of any operation.
Delete minimum.
To delete the minimum element from the heap, first find this element, remove it from the root of its binomial tree, and obtain a list of its child subtrees (which are each themselves binomial trees, of distinct orders). Transform this list of subtrees into a separate binomial heap by reordering them from smallest to largest order. Then merge this heap with the original heap. Since each root has at most formula_26 children, creating this new heap takes time formula_27. Merging heaps takes time formula_27, so the entire delete minimum operation takes time formula_27.
 function deleteMin(heap)
 min = heap.trees().first()
 for each current in heap.trees()
 if current.root < min.root then min = current
 for each tree in min.subTrees()
 tmp.addTree(tree)
 heap.removeTree(min)
 merge(heap, tmp)
Decrease key.
After decreasing the key of an element, it may become smaller than the key of its parent, violating the minimum-heap property. If this is the case, exchange the element with its parent, and possibly also with its grandparent, and so on, until the minimum-heap property is no longer violated. Each binomial tree has height at most formula_26, so this takes formula_27 time. However, this operation requires that the representation of the tree include pointers from each node to its parent in the tree, somewhat complicating the implementation of other operations.
Delete.
To delete an element from the heap, decrease its key to negative infinity (or equivalently, to some value lower than any element in the heap) and then delete the minimum in the heap.