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# [PPT]15-211 Fundamental Structures of Computer Science · Web view15-211 Fundamental Structures of Computer Science Binomial Heaps March 02, 2006 Ananda Guna In this Lecture Binomial

May 18, 2018

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• 15-211Fundamental Structures of Computer Science

March 02, 2006

Ananda 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

B1

B2

B3

• 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 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

• 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

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

v

v

• 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

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