1. 15-211 Fundamental Structures of Computer Science
Binomial Heaps
March 02, 2006
Ananda Guna

2. In this Lecture Binomial Trees Binomial Heaps Implementation
Definition
properties
Binomial Heaps
efficient merging
Implementation
Operations
About Midterm

3. Binary Heaps Binary heap is a data structure that allows
insert in O(log n)
deleteMin in O(log n)
findMin in O(1)
How about merging two heaps
complexity is O(n)
So we discuss a data structure that allows merge in O(log n)

4. Applications of Heaps Binary Heaps Binomial Heaps
efficient findMin, deleteMin
many applications
Binomial Heaps
Efficient merge of two heaps
Merging two heap based data structures
Binomial Heap is build using a structure called Binomial Trees

5. Binomial Trees A Binomial Tree Bk of order k is defined as follows
B0 is a tree with one node
Bk is a pair of Bk-1 trees, where root of one Bk-1 becomes the left most child of the other (for all k ≥ 1) B0 B3 B1 B2

6. Merging two binomial trees
Merging two equal binomial trees of order j = +
New tree has order j + 1

7. Properties of Binomial trees
The following properties hold for a binomial tree of order k
Bk has 2k nodes
The height of Bk is k
Bk has kCi nodes at level i for i = 0,1,…k
The root of Bk has k-children B0, B1, …Bk-1 (in that order) where the ith child is a binomial tree of order i.
If binomial tree of order k has n nodes, then k ≤ log n

8. Proofs Lemma 1: BK has 2k nodes
Proof: (by induction). True for k=0, assume true for k=r. Consider Br+1
Br+1 has 2r + 2r = 2r+1 nodes
Lemma 2: Bk has height k
Proof: homework
Lemma 3: Bk has kCi nodes at level i for i = 0,1,…k
Proof: Let T(k,i) be the number of nodes at depth i. Then T(k,i) = T(k-1,i) + T(k-1,i-1)
= k-1Ci + k-1Ci-1 = kCi

9. Binomial Heap
Binomial Heap is a collection of binomial trees that satisfies the following properties
No two binomial trees in the collection have the same size
Each node in the collection has a key
Each binomial tree in the collection satisfies the heap order property
Roots of the binomial trees are connected and are in increasing order

10. Example
A binomial heap of n=15 nodes containing B0, B1, B2 and B3 binomial trees
What is the connection between n and number of binary trees in the heap?

11. Lemma
Given any integer n, there exists a binomial heap that contain n nodes
Proof:

12. implementation

13. Implementation – Binary Tree Node
Fields in a binomial tree node
Key
number of children (or degree)
Left most child
Right most sibling
A pointer P to parent

14. Implementation – Binary Heap
head
root1
root2
root3
root4

15. Operations

16. Operations on Binomial Heaps
Merge is the key operation on binomial heaps
merge()
insert()
findMin()
find the min of all children O(log n)
deleteRoot()
deleteNode()
decreaseKey()

17. Merging two binomial heaps
Suppose H1 and H2 are two binomial heaps
Merge H1 and H2 into a new heap H
Algorithm:
Let A and B be pointers to H1 and H2
for all orders i
If there is one order i tree, merge it to H
If there are two order i trees, merge them into a new tree of order i+1 and store them in a temp tree T
If there are three order i trees in H1,H2 and T, merge two of them, store as T and add the remainder to H

18. Example

19. Binary Heap Operations
Insert
make a new heap H0 with the new node
Merge(H0, H)
FindMin
min is one of the children connected to the root
cost is O(log n)

20. Binary Heap Operations
DeleteRoot()
Find the tree with the given root
Split the heap into two heaps H1 and H2

21. Binary Heap Operations
DeleteRoot() ctd..
Rearrange binomial trees in heap H2
Merge the two heaps v v

22. Example

23. DeleteNode()
To delete a node, decrease its key to -∞, percolate up to root, then delete the root
DecreaseKey(): Decrease the key and percolate up until heap order property is satisfied

24. Summary Two heaps Next Week Binary Heaps Binomial Heaps
deleteMin()
Binomial Heaps
mergeHeaps()
Next Week
Midterm on Tuesday
Strings and Tries on thursday