Binomial heap
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In computer science, a binomial heap is a heap similar to a binary heap but also supporting the operation of merging two heaps quickly. This is achieved by using a special tree structure. 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.
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[edit] Binomial tree
A binomial heap is implemented as a collection of binomial trees (compare with a binary heap, which has a shape of a single binary tree). A binomial tree is defined recursively:
- A binomial tree of order 0 is a single node
- A binomial tree of order k has a root node whose children are roots of binomial trees of orders k-1, k-2, ..., 2, 1, 0 (in this order).
A binomial tree of order k has 2k nodes, height k.
Because of its unique structure, a binomial tree of order k can be constructed from two trees of order k-1 trivially by attaching one of them as the leftmost child of the other one. This feature is central to the merge operation of a binomial heap, which is its major advantage over other conventional heaps.
[edit] Structure of a binomial heap
A binomial heap is implemented as a set of binomial trees that satisfy the binomial heap properties:
- Each binomial tree in a heap obeys the minimum-heap property: the key of a node is greater than or equal to the key of its parent.
- There can only be either one or zero binomial trees for each order, including zero order.
The first property ensures that the root of each binomial tree contains the smallest key in the tree, which applies to the entire heap.
The second property implies that a binomial heap with n elements consists of at most log n + 1 binomial trees. In fact, the number and orders of these trees are uniquely determined by the number of elements n: each binomial tree corresponds to digit one in the binary representation of number n. For example number 13 is 1101 in binary, 23 + 22 + 20, and thus a binomial heap with 13 elements will consist of three binomial trees of orders 3, 2, and 0 (see figure below).
Example of a binomial heap containing 13 elements with distinct keys.
The heap consists of three binomial trees with orders 0, 2, and 3.
[edit] 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.
[edit] Merge
As mentioned above, the simplest and most important operation is the merging of two binomial trees of the same order within two binomial heaps. Due to the structure of binomial trees, they can be merged trivially. As their root node is the smallest element within the tree, by comparing the two keys, the smaller of them is the minimum key, and becomes the new root node. Then the other tree become a subtree of the combined tree. This operation is basic to the complete merging of two binomial heaps.
function mergeTree(p, q)
if p.root <= q.root
return p.addSubTree(q)
else
return q.addSubTree(p)
The operation of merging two heaps is perhaps the most interesting and can be used as a subroutine in most other operations. The lists of roots of both heaps are traversed simultaneously, similarly as in the merge algorithm.
If only one of the heaps contains a tree of order j, this tree is moved to the merged heap. If both heaps contain a tree of order j, the two trees are merged to one tree of order j+1 so that the minimum-heap property is satisfied. Note that it may later be necessary to merge this tree with some other tree of order j+1 present in one of the heaps. In the course of the algorithm, we need to examine at most three trees of any order (two from the two heaps we merge and one composed of two smaller trees).
Each tree has order at most log n and therefore the running time is O(log n).
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)
else
heap.addTree(tree)
heap.next() p.next() q.next()
[edit] 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 in O(log n) time.
[edit] Find minimum
To find the minimum element of the heap, find the minimum among the roots of the binomial trees. This can again be done easily in O(log n) time, as there are just O(log n) trees and hence 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 O(1). The pointer must be updated when performing any operation other than Find minimum. This can be done in O(log n) without raising the running time of any operation.
[edit] Delete minimum
To delete the minimum element from the heap, first find this element, remove it from its binomial tree, and obtain a list of its subtrees. Then transform this list of subtrees into a separate binomial heap by reordering them from largest to smallest order. Then merge this heap with the original heap.
function deleteMin(heap)
min = heap.trees().first()
for each current in heap.trees()
if current.root < min then min = current
for each tree in min.subTrees()
tmp.addTree(tree)
heap.removeTree(min)
merge(heap, tmp)
[edit] 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 log n, so this takes O(log n) time.
[edit] Delete
To delete an element from the heap, decrease its key to minus infinity (that is, some value lower than any element in the heap) and then delete the minimum in the heap.
[edit] Performance
All of the following operations work in O(log n) time on a binomial heap with n elements:
- Insert a new element to the heap
- Find the element with minimum key
- Delete the element with minimum key from the heap
- Decrease key of a given element
- Delete given element from the heap
- Merge two given heaps to one heap
Finding the element with minimum key can also be done in O(1) by using an additional pointer to the minimum.
[edit] See also
[edit] References
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 19: Binomial Heaps, pp.455–475.
- Vuillemin, J. (1978). A data structure for manipulating priority queues. Communications of the ACM 21, 309-314.
