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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
ProofsLemma 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