15-211Fundamental Structures of Computer Science

March 02, 2006Ananda Guna

Binomial Heaps

In this Lecture

Binomial Trees Definition properties

Binomial Heaps efficient merging

Implementation Operations About Midterm

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)

Applications of Heaps

Binary 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

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 B

1B

2B

3

Merging two binomial trees

Merging two equal binomial trees of order j

+=

New tree has order j + 1

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

Proofs

Lemma 1: BK has 2k nodesProof: (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 kProof: homework

Lemma 3: Bk has kCi nodes at level i for i = 0,1,…kProof: 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

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

Example

A binomial heap of n=15 nodes containing B0, B1, B2 and B3 binomial trees

What is the connection between n and the binomial trees in the heap?

Lemma

Given any integer n, there exists a binomial heap that contain n nodes

Proof:

implementation

Implementation – Binomial 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

Implementation – Binomial Heap

head

root1 root2 root3 root4

Operations

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()

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

Example

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)

Binary Heap Operations

DeleteRoot() Find the tree with the given root Split the heap into two heaps H1 and H2

Binary Heap Operations

DeleteRoot() ctd.. Rearrange binomial trees in heap H2

Merge the two heaps

vv

Example

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

Summary

Two heaps Binary Heaps

deleteMin()

Binomial Heaps mergeHeaps()

Next Week Midterm on Tuesday Strings and Tries on thursday