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Binomial Queues

Binomial Queues. Text Read Weiss, § 6.8 Binomial Queue Definition of binomial queue Definition of binary addition Building a Binomial Queue Sequence of inserts What in the world does binary addition have in common with binomial queues?. Motivation.

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Binomial Queues

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  1. Binomial Queues • Text • Read Weiss, §6.8 • Binomial Queue • Definition of binomial queue • Definition of binary addition • Building a Binomial Queue • Sequence of inserts • What in the world does binary addition have in common with binomial queues?

  2. Motivation • A binary heap provides O(log n) inserts and O(log n) deletes but suffers from O(n log n) merges • A binomial queue offers O(log n) inserts and O(log n) deletes and O(log n) merges • Note, however, binomial queue inserts and deletes are more expensive than binary heap inserts and deletes

  3. Definition A Binomial Queue is a collection of heap-ordered trees known as a forest. Each tree is a binomial tree. A recursive definition is: • A binomial tree of height 0 is a one-node tree. • A binomial tree, Bk, of height k is formed by attaching a binomial tree Bk−1 to the root of another binomial tree Bk−1 .

  4. Examples B2 B0 4 B1 4 4 5 8 8 4 12 7 B3 5 8 10 8 12 11 4 6 7 B4 5 8 10 7 8 15 9 10 12 11 10 24 22

  5. Questions • How many nodes does the binomial tree Bkhave? • How many children does the root of Bkhave? • What types of binomial trees are the children of the root of Bk? • Is there a binomial queue with one node? With two nodes? With three nodes? … With n nodes for any positive integer n?

  6. Binary Numbers • Consider binary numbers • Positional notation: • 25 24 23 22 21 20 • 0 1 0 1 1 0 • 0 + 16 + 0 + 4 + 2 + 0 = 22  0101102 = 2210 • What is the decimal value of these binary numbers? • 011 = • 101 = • 10110 = • 1001011 =

  7. Binary Numbers • Consider binary numbers • Positional notation: • 25 24 23 22 21 20 • 0 1 0 1 1 0 • 0 + 16 + 0 + 4 + 2 + 0 = 22  0101102 = 2210 • What is the decimal value of these binary numbers? • 011 = 3 • 101 = 5 • 10110 = 22 • 1001011 = 75

  8. Binary Addition (carry)11 1 0 1 1 0 1 0 + 0 0 1 1 1 0 0 ----------------- 1 1 1 0 1 1 0

  9. Merging Binomial Queues Consider two binomial queues, H1 and H2 3 4 6 H1 15 7 5 5 8 10 27 8 12 11 12 3 1 H2 21 16 6 23

  10. Merging Binomial Queues Merge two B0 trees forming new B1 tree 3 4 H1 15 7 5 5 8 10 27 8 12 11 1 6 12 3 H2 21 16 6 23

  11. Merging Binomial Queues Merge two B1 trees forming new B2 tree 3 4 H1 15 7 5 5 8 10 27 8 12 11 1 12 6 16 3 H2 21 6 23

  12. Merging Binomial Queues Merge two B2 trees forming new B3 tree (but which two B2 trees?) 3 4 H1 15 7 5 5 8 10 27 8 12 11 1 12 6 16 3 H2 21 6 23

  13. Merging Binomial Queues Which two B2 trees? Arbitrary decision: merge two original B2 trees 3 4 H1 15 7 5 5 8 10 27 8 12 11 1 12 6 16 3 H2 21 6 23

  14. Merging Binomial Queues Which two B2 trees? Arbitrary decision: merge two original B2 trees 3 4 H1 15 7 5 5 8 10 27 8 12 11 1 12 6 16 3 H2 21 6 23

  15. Merging Binomial Queues Which root becomes root of merged tree? Arbitrary decision: in case of a tie, make the root of H1 be the root of the merged tree. 3 4 H1 15 7 5 5 8 10 27 8 12 11 1 12 6 16 3 H2 21 6 23

  16. Merging Binomial Queues Merge two B2 trees forming new B3 tree 4 H1 7 5 8 10 8 12 11 1 12 6 3 16 3 15 H2 5 21 6 27 23

  17. Merging Binomial Queues Merge two B3 trees forming a new B4 tree H1 3 3 4 15 5 21 7 6 5 8 10 27 23 8 1 12 11 12 6 16 H2

  18. Merging Binomial Queues Call new binomial queue H3 3 H3 3 4 1 15 5 21 12 7 6 6 5 8 10 27 23 16 8 12 11

  19. Merging Binomial Queues Reconsider the two original binomial queues, H1 and H2 and identify types of trees 3 4 6 H1 15 7 5 5 8 10 27 8 12 11 B0 B3 B2 2 3 1 H2 21 16 6 23 B0 B2 B1

  20. Merging Binomial Queues Represent each binomial queue by a binary number 3 4 6 H1 15 7 5 5 8 10 27 8 B4 B3 B2 B1 B0 0 1 1 0 1 12 11 B0 B3 B2 2 3 1 H2 21 16 6 23 B4 B3 B2 B1 B0 0 0 1 1 1 B0 B2 B1

  21. Merging Binomial Queues Note that the merged binomial queue can be represented by the binary sum 3 4 6 H1 15 7 5 5 B4 B3 B2 B1 B0 0 1 1 0 1 8 10 27 8 12 11 2 3 1 H2 21 16 6 B4 B3 B2 B1 B0 0 0 1 1 1 23 3 H3 3 4 1 15 5 21 2 7 B4 B3 B2 B1 B0 1 0 1 0 0 6 6 5 8 10 27 23 16 8 12 11

  22. This suggests a way to implement binomial queues

  23. Implementing Binomial Queues • Use a k-ary tree to represent each binomial tree • Use a Vector to hold references to the root node of each binomial tree

  24. Implementing Binomial Queues Use a k-ary tree to represent each binomial tree. Use an array to hold references to root nodes of each binomial tree. 3 4 6 H1 15 7 5 5 B4 B3 B2 B1 B0 0 1 1 0 1 8 10 27 8 12 11 H1 B4 B3 B2 B1 B0 ØØ 4 8 5 3 12 7 6 5 8 15 10 27 11

  25. Questions • We now know how to merge two binomial queues. How do you perform an insert? • How do you perform a delete? • What is the order of complexity of a merge? an insert? a delete?

  26. Implementing Binomial Queues Carefully study Java code in Weiss, Figure 6.52 – 6.56

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