CSC 172 DATA STRUCTURES

1. CSC 172 DATA STRUCTURES

2. SKIP LISTSRead 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. EXAMPLE 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 L2 stores all element L1 stores some elements Links between shared elements

14. Lookup on a skip list

15. Lookup on a skip list Take L1 until you go too far Back up one Transfer to L2 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?