1. CSC 172 DATA STRUCTURES

2. SKIP LISTS Read Weiss 10.4.2

3. SKIP LISTS  Dictionary Data Structure  Efficient

4. SKIP LISTS  Dictionary Data Structure (insert delete lookup)  Efficient O(lg n)

5. SKIP LISTS  Dictionary Data Structure (insert delete lookup)  Efficient O(lg n)

6. SKIP LISTS  Dictionary Data Structure  Efficient (with high probability)  Randomized

7. SKIP LISTS  Dictionary Data Structure  Efficient (with high probability)  Randomized  Easy to implement

8. LISTS  How much time does it take to search a sorted linked list?  How can this be improved?

9. EXAMPLE

10.  What is this sequence? 14,23,28,34,42,50,59,66,72, 79,86,96,103,110

11. EXAMPLE  What is this sequence? 14,23,34,42,50,59,66,72, 79,86,96,103,110

12. EXAMPLE  What is this sequence? 14,23,34,42,50,59,66,72, 79,86,96,103,110,116,125

13. SKIP LISTS  Use two lists  L 2 stores all element  L 1 stores some elements  Links between shared elements

14. Lookup on a skip list

15. 1)Take L1 until you go too far 2)Back up one 3)Transfer to L2 4)Take L2 until you find element (or go too far – not found – or insert)

16. Lookup on a skip list How should we distribute the L1 list? What is the time cost of a search?

17. Lookup on a skip list How should we distribute the L1 list? What is the time cost of a search? Minimize : L1.length + (L2.length/L1.length)

18. NEXT STEP  2 linked lists 2(n^(1/2))  Can we improve this further?

19. NEXT STEP  2 linked lists 2(n^(1/2))  Can we improve this further?  3 linked lists 3(n^(1/3))  k linked lists k(n^(1/k))  N linked lists ????  lg n linked lists lg n (n^(1/lg n))

20. BALLANCED SKIP LISTS  Ideal as long as structure is maintained

21. BALLANCED SKIP LISTS  Ideal as long as structure is maintained  Insertions and deletions mess up structure

22. INSERTION ON SKIP LISTS  Search to find location  Must insert on bottom list  Which other lists?  FLIP A COIN  If heads add to level above and flip again.  If tails done.

23. INSERTION ON SKIP LISTS  FLIP A COIN  If heads add to level above and flip again.  If tails done. ½ of the elements go up one level ¼ of the elements go up 2 levels 1/8 of the elements go up 3 levels

24. INSERTION ON SKIP LISTS  EXAMPLE

25. ANALYSIS  Intuitively:  Height of the structure is O(lg n)  How many coin flips do we need to get lg n heads?