CSE 326: Data Structures Binomial Queues

1. CSE 326: Data StructuresBinomial Queues Peter Henry on behalf of James Fogarty Autumn 2007

2. Yet Another Data Structure:Binomial Queues • Structural property • Forest of binomial trees with at mostone tree of any height • Order property • Each binomial tree has the heap-order property What’s a forest? What’s a binomial tree?

3. The Binomial Tree, Bh • Bh has height h and exactly 2h nodes • Bh is formed by making Bh-1 a child of another Bh-1 • Root has exactly h children • Number of nodes at depth d is binomial coeff. • Hence the name; we will not use this last property B0 B1 B2 B3

4. Binomial Queue with n elements Binomial Q with n elements has a unique structural representation in terms of binomial trees! Write n in binary: n = 1101 (base 2) = 13 (base 10) 1 B3 1 B2 No B1 1 B0

5. Properties of Binomial Queue • At most one binomial tree of any height • n nodes  binary representation is of size ?  deepest tree has height ?  number of trees is ? Define: height(forest F) = maxtree T in F { height(T) } Binomial Q with n nodes has height Θ(log n) O(log n)

6. Operations on Binomial Queue • Will again define merge as the base operation • insert, deleteMin, buildBinomialQ will use merge • Can we do increaseKey efficiently?decreaseKey? • What about findMin? Yes! Just likeBinary Heaps log n normally O(1) if you maintain min explicitly over ops

7. Merging Two Binomial Queues Essentially like adding two binary numbers! • Combine the two forests • For k from 0 to maxheight { • m total number of Bk’s in the two BQs • if m=0: continue; • if m=1: continue; • if m=2: combine the two Bk’s to form a Bk+1 • if m=3: retain one Bk and combine the other two to form a Bk+1 } # of 1’s 0+0 = 0 1+0 = 1 1+1 = 0+c 1+1+c = 1+c Claim: When this process ends, the forest has at most one tree of any height

8. Example: Binomial Queue Merge H1: H2: -1 5 1 3 21 3 9 2 7 6 1 11 5 7 8 6

9. Example: Binomial Queue Merge H1: H2: -1 5 1 3 3 9 2 7 6 1 21 11 5 7 8 6

10. Example: Binomial Queue Merge H1: H2: -1 5 1 3 9 2 7 6 1 3 11 5 7 8 21 6

11. Example: Binomial Queue Merge H1: H2: -1 1 3 5 2 7 1 3 11 9 5 6 8 21 6 7

12. Example: Binomial Queue Merge H1: H2: -1 1 3 2 1 5 7 11 3 5 8 9 6 6 21 7

13. Example: Binomial Queue Merge H1: H2: -1 1 3 2 1 5 7 11 3 5 8 9 6 6 21 7

14. Complexity of Merge Constant time for each height Max number of heights is: log n  worst case running time = Θ( ) log n

15. Insert in a Binomial Queue Create a single node BQ and merge into existing BQ (show by picture) Θ(log n) worst case Θ(1) on average: first 0 in binary repr 0/1 with prob 0.5 so expected time: 2 Insert(x): Similar to leftist or skew heap runtime Worst case complexity: same as mergeO( ) Average case complexity: O(1) Why?? Hint: Think of adding 1 to 1101

16. deleteMin in Binomial Queue Similar to leftist and skew heaps…. • find Bi with smallest root • delete root of Bi gives new BinQ BQ’ • Merge BQ’ with (BQ -{Bi})

17. deleteMin: Example BQ 7 3 4 8 5 7 find and deletesmallest root How long does FIND take? O(log N) merge BQ (withoutthe shaded part) and BQ’ 3 8 5 BQ’ Do merge on tabletbefore next slide 7

18. deleteMin: Example Result: 7 4 8 5 7 runtime: Θ(log n)