1. Fundamental Structures of Computer Science II
Priority Queues - 2
CS 202 – Fundamental Structures of Computer Science II
Bilkent University
Computer Engineering Department
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

2. Fundamental Structures of Computer Science II
Binomial Queues
Binamial Queue
A collection of heap-ordered trees (a forest).
A tree in the forest is a binomial tree.
There is at most one binomial tree of every height in the forest.
A binomial tree of height 0 has one node.
A binomial tree of height k has 2k nodes.
A binomial tree of height k, Bk, is formed by attaching a binomial tree of height k-1, Bk-1’ to another binomial tree of height k-1, Bk-1 .
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

3. Fundamental Structures of Computer Science II
Binomial Trees B0 B1 B2 B3
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

4. Fundamental Structures of Computer Science IIB4
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

5. Fundamental Structures of Computer Science II
A binomial tree Bk consists of root with children B0, B1, …, Bk-1
Binomial trees of height k have exactly 2k nodes.
The number of nodes at depth d is binomial coefficient
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

6. Fundamental Structures of Computer Science II
k = 4
d=0
d=1
d=2
d=3
d=4 B4
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

7. Heaps and binomial trees
Assume all binary trees in a forest are in heap order.
Assume we have at most one binomial tree of any height.
Then we can represent a heap of any size uniquely by a binomial tree forest.
Example: heap of size 13 could be represented by forest: B3’ B2, B0,
We call this heap ordered binomial forest a binomial queue.
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

8. Example H1 A binomial queue H1 of size 6 is shown above16 12 18 21 24 65 H1
A binomial queue H1 of size 6 is shown above
Can be represented as: 110
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

9. Binomial Queue Operations
FindMin
Merge
Insert
DeleteMin
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

10. Fundamental Structures of Computer Science II
FindMin
The minimum element in the binomial queue can be found by scanning the roots of all trees in the forest.
There are at most logN different trees
The cost of findMin is therefore O(logN) in the worst case.
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

11. Fundamental Structures of Computer Science II
Merge
Merging two binomial queues is conceptually simple.
Merge H1 and H2
Just add them in binary.
Assume H1 has 6 nodes. H2 has 7 nodes.
H3 = node (H1) + nodes(H3) = 13.
H1: 0110
H2: 0111
H3: 1101 (this is the resulting bin. heap).
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

12. Fundamental Structures of Computer Science II
Merge - example 1 2 H1 16 12 18 21 24 65 13 14 23 H2 26 51 24 65
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

13. Fundamental Structures of Computer Science II
Merge - example - steps B0 H1
Merge 1 node tree with empty tree.
The result is the one node tree itself. B0 H3 13 13 H2
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

14. Fundamental Structures of Computer Science II
Merge - example - steps B2 B1 H1 16 14 18 26 16 18 14 H2 26 H2
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

15. Fundamental Structures of Computer Science IIH1 12 21 24 12 65 23 21 24 23 H2 51 24 65 51 24 65 65 14 B3 H3 26 16 H3 B2 18
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

16. Fundamental Structures of Computer Science II
Final Binomial Queue H3 13 14 12 23 B0 26 16 21 24 51 24 18 65 65 B2 B3
H3 = 1101
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

17. Fundamental Structures of Computer Science II
Insert
Insertion is a special case of merge
Insert an item to a binomial heap H1
Make a new heap, H2, of one node (item to be inserted)
Merge H1 and H2.
Example
Insert items 1, 2, 3, 4, 5, 6, 7, in the given order into an empty binomial heap.
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

18. Fundamental Structures of Computer Science II1 1 3 1 2 2
After 1 is
inserted
After 2 is
inserted
After 3 is
inserted 1 5 1 2 3 2 3 4 4
After 4 is inserted
After 5 is inserted
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

19. Fundamental Structures of Computer Science II5 1 6 2 3 4
After 6 is inserted 5 1 7 6 2 3 4
After 7 is inserted
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

20. Fundamental Structures of Computer Science II
DeleteMin
Sketch of the Algorithm
Assume we want to delete minimum item from binomial queue H.
Find the binomial tree in H that has the minimum root. Let say this is Bk in H.
Take tree Bk from H. Let the remaining trees in H make a new binomial queue H1
Remove the root from Bk (root is the minimum that you should return as a result of algorithm).
The children of root of Bk make a new heap H2 that can consists of tree B0 through Bk-1
Merge H1 and H2.
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

21. Fundamental Structures of Computer Science II
DeletMin - Example 13 14 12 B0 23 26 16 21 24 51 24 18 65 B2 B3 65 H
Delete the minimum item from the binomial queue above.
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

22. Fundamental Structures of Computer Science II
DeletMin - Example 13 14 12 B0 23 26 16 21 24 51 24 18 65 B2 B3 65 H
The root that has the minimum item is 12 and belongs to tree B3
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

23. Fundamental Structures of Computer Science II13 14 12 B0 23 26 16 21 24 51 24 18 65 B2 H B3 65 13 14 21 23 24 B0 B0 26 16 51 24 65 B1 18 65 B2 B2
MERGE H1 H2
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

24. Fundamental Structures of Computer Science II13 14 21 24 23 B0 B0 26 16 51 24 65 B1 18 65 B2 B2 H1 H2
101
111
1100
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

25. Fundamental Structures of Computer Science II13 14 21 24 23 B0 B0 26 16 51 24 65 B1 18 65 B2 B2 H1 H2 13 13 21 21 24 65 B2
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University

26. Fundamental Structures of Computer Science II14 13 23 26 16 51 24 21 24 18 65 65 13 23 51 24 21 24 14 B2 65 65 26 16 B3
Remaining Bino. Heap
After DeleteMin 18
CS 202, Spring 2003
Fundamental Structures of Computer Science II
Bilkent University