Binomial Heap

Data Structures &

Algorithms

By Saliha Bilal

Contents:Binomial Heap

Binomial tree

Implementations

Heap making

Insertion

Find minimum

Extract minimumUnion of heaps

Delete key

Decrease key

Performance

Aplications

A binomial heap is a heap similar to a binary  heap but  also  supports  quick  merging  of  two heaps.

A  binomial  heap  is  implemented  as  a collection of binomial trees

Binomial Heap

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 nodeA 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).

Binomial Tree

Binomial Trees of order 0 to 3

Binary heap supports various operationsquickly: extract-min, insert, decrease-key

If we already have two min-heaps, A and B,there is no efficient way to combine theminto a single min-heap•Introduce Binomial Heap It can support efficient union operation

Binomial Heap

Mergeable Heaps

A data structure that supports the following 5 operations:•Make-Heap( ) : return an empty heap•Insert(H , x , k) : insert an item x withkey k into a heap H•Find-Min(H) : return item with min key•Extract-Min(H) : return and remove•Union(H1, H2) : merge heaps H1 and H2

Mergeable Heaps

Examples:Binomial HeapFibonacci HeapBoth heaps also supportDecrease-Key(H,x,k) : assign item x with a smaller key kDelete(H,x) : remove item x

Binomial heap

Operation ↓

Binomial heap

Make heap O(1)

Find minimum

O(log n)

Extract minimum

O(log n)

Insert  O(log n)

Delete  O(log n)

Decrease key

O(log n)

Union  O(log n)

Binomial Heap

Unlike binary heap which consists of asingle tree, a binomial heap consists of asmall set of component trees no need to rebuild everything whenunion is perform

Each component tree is in a specialformat, called a binomial tree

Binomial treeA binomial tree of order k, denoted byBk, is defined recursively as follows:

B0 is a tree with a single node

For k ≥1, Bk is formed by joining twoBk-1, such that the root of one treebecomes the leftmost child of theroot of the other

Binomial treeB0

B1 B2

B3

B4

Properties of Binomial Tree

For a binomial tree Bk

There are 2k nodesheight = kdeg(root) = k ; deg(other node) ˂kChildren of root, from left to right,are Bk-1, Bk-2, …, B1, B0

Binomial Heap

Binomial  heap  of  n  elements  consists  of  a  specific set of binomial treesEach binomial tree satisfies min-heapordering: for each node x,key(x) ≥ key(parent(x))For each k, at most one binomial treewhose root has degree k(i.e., for each k, at most one Bk)

Binomial heap with 13 elements

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Binomial Heap Operations

With the binomial heap,•Make-Heap( ): O(1) time

•Find-Min( ): O(log n) time

•Decrease-Key( ): O(log n) time[ Decrease-Key assumes we have the pointer tothe item x in which its key is changed ]

•Remaining operations : Based on Union( )

Union Operation

•Let H1 and H2 be two binomial heaps•To Union them, we process all binomialtrees in the two heaps with same ordertogether, starting with smaller orderfirst•Let k be the order of the set of binomialtrees we currently processThere are three cases:1. If there is only one Bk →done2. If there are two Bk

 → Merge together, forming Bk+13. If there are three Bk

 → Leave one, merge remaining to Bk+1After that, process next k

Union of two BH with 5 and 13 nodes 

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Union of two BH with 5 and 13 nodes 

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After processingK=0

Union of two BH with 5 and 13 nodes 

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After processingK=1,2

Union of two BH with 5 and 13 nodes 

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After processingK=3

Binomial Heap Operations

So, Union( ) takes O(log n) time

For remaining operations,Insert( ), Extract-Min( ), Delete( )how can they be done with Union?

Insert(H, x, k):→Create new heap H’, storing the item xwith key k; then, Union(H, H’)

Extract-Min(H):→Find the tree Bj containing the min;Detach Bj from H → forming a heap H1 ;Remove root of Bj → forming a heap H2 ;Finally, Union(H, H’)

Delete(H, x):→Decrease-Key(H,x,-∞); Extract-Min(H);

Binomial Heap Operations

Extract- min (H)Step1: select the tree Bj containing the min value

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Bj with Min

Extract- min (H)Step2: forming two heaps & extracting “8” from H2

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H2

H1

Extract- min (H)Step3: union of H1 and H2

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Performance:Insert a new element to the heapFind the element with minimum keyDelete the element with minimum key from the heapDecrease key of a given elementDelete given element from the heapMerge two given heaps to one heapFinding the element with minimum key can also be done in O(1) by using an additional pointer to the minimum.

Applications:

Discrete event simulation

Priority queues

THANK

YOU!