Binomial heap (a concept of Data Structure)

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

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).

Binary heap supports various operations
quickly:
extract-min, insert, decrease-key
If we already have two min-heaps, A and B,
there is no efficient way to combine them
into a single min-heap
•Introduce Binomial Heap
It can support efficient union operation

A data structure that supports the following 5
operations:
•Make-Heap( ) : return an empty heap
•Insert(H , x , k) : insert an item x with
key 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

Examples:
Binomial Heap
Fibonacci Heap
Decrease-Key(H,x,k) : assign item x
with a smaller key k
Delete(H,x) : remove item x

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 a
single tree, a binomial heap consists of a
small set of component trees
 no need to rebuild everything when
union is perform
Each component tree is in a special
format, called a binomial tree

Binomial tree
A binomial tree of order k, denoted
by
Bk, is defined recursively as follows:
B0 is a tree with a single node
For k ≥1, Bk is formed by joining
two
Bk-1, such that the root of one tree
becomes the leftmost child of the
root of the other

Properties of Binomial Tree
For a binomial tree Bk
There are 2k nodes
height = k
deg(root) = k ; deg(other node) ˂k
Children 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 trees
Each binomial tree satisfies min-heap
ordering: for each node x,
key(x) ≥ key(parent(x))
For each k, at most one binomial tree
whose root has degree k
(i.e., for each k, at most one Bk)

Binomial heap with 13 elements
12 8
35
25 13
19
33
32
41
52
15
16
21

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 to
the 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 binomial
trees in the two heaps with same order
together, starting with smaller order
first
•Let k be the order of the set of binomial
trees we currently process
There are three cases:
1. If there is only one Bk →done
2. If there are two Bk
→ Merge together, forming Bk+1
3. If there are three Bk
→ Leave one, merge remaining to Bk+1
After that, process next k

Union of two BH with 5 and 13 nodes
12
8
35
25 13 19
33
32
41 52
15
16
21
4
9
14 11
31
H1
H2

12
8
35
25 13 19
33
32
41 52
15
16
21
4
9
14 11
31
After processing
K=0

12 8
35
25 13
19
33
32
41
52
15
16
21
4
9
14 11
31
After processing
K=1,2

12
8
35
25 13
19
33
32 41
52
15
16
21
4
9
14 11
31
After processing
K=3

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 x
with 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);

Extract- min (H)
Step1: select the tree Bj containing the min value
12 8
35
25 13
19
33
32
41
52
15
16
21
Bj with Min

Extract- min (H)
Step2: forming two heaps & extracting “8” from H2
12
8
35
25 13
19
33
32
41
52
15
16
21
H2
H1

Extract- min (H)
Step3: union of H1 and H2
12
35
25 13
19
33
32
41
52
15
16
21

Performance:
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.

Applications:
Discrete event simulation
Priority queues

Binomial heap (a concept of Data Structure)

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