1. CS 261 - Fall 2009 Skip Lists

2. Skip Lists Advantages Ordinary linked lists and arrays have fast (O(1)) addition, but slow search Sorted Arrays have fast search (O(log n)) but slow insertion Skip Lists have it all - fast (O(log n)) addition, search and removal times! Cost - Slightly more complicated

3. Start with an ordered list Lets begin with a simple ordered list, sentinel on the left, single links.

4. Start of the struct definition struct skiplist { struct skiplink * sentinel; int size; };

5. Operations Add (Object newValue) - find correct location, add the value Contains (Object testValue) - find correct location, see if its what we want Remove (Object testValue) - find correct location, possibly remove it See anything in common?

6. Slide Right struct skiplink * slideRight (struct skiplink * current, EleType testValue) { while ((current->next != 0) && LT(current->next->value, testValue)) current = current->next; return current; } Finds the link RIGHT BEFORE the place the value we want should be, if it is there

7. Add (EleType newValue) void add (struct skiplist* lst, EleType newValue) { struct skiplink * current = slideRight(lst->sentinel, newValue); struct link * newLink = (struct link *) malloc(sizeof(struct link)); assert (newLink != 0); newLink->value = newValue; newLink->next = current->next; current->next = newLink; lst->size++; }

8. Contains (EleType testValue) int contains (struct skip *lst, EleType testValue) { struct skiplink * current = slideRight (lst->leftSentinel, testValue); if ((current->next != 0) && (EQ(testValue, current->next->value))) return true; return false; } /* reminder, this is one level list, not real skiplist */

9. Remove (Object testValue) void remove (struct skiplist *lst, EleType testValue) { struct skiplink * current = slideRight (lst->leftSentinel, testValue); if ((current->next != 0) && (EQ(testValue, current->next->value))) { current->next = current->next->next; free(current->next); lst->size--; }

10. Problem with Sorted List What’s the use? Add is still O(n), so is contains, so is remove. No better than an ordinary list. Major problem with list - sequential access

11. Why no binary search? What if we kept a link to middle of list?

12. But need to keep going

13. And going and going

14. In theory it would work….. In theory this would work Would give us nice O(log n) search But would be really hard to maintain as values were inserted and removed What about if we relax the rules, and don’t try to be so precise…..

15. The power of probability Instead of being precise, keep a hierarchy of ordered lists with downward pointers Each list has approximately half the elements of the one below So, if the bottom list has n elements, how many levels are there??

16. Picture of Skip List

17. Slightly bigger list

18. Contains (Object testValue) Starting at topmost sentinel, slide right If next value is it, return true If next value is not what we are looking for, move down and slide right If we get to bottom without finding it, return false

19. Notes about contains See how it makes a zig-zag from top to bottom On average, slide only moves one or two positions So basically proportional to height of structure Which is????? O( ?? )

20. Remove Algorithm Start at topmost sentinal Loop as follows –Slide right. If next element is the one we want, remove it –If no down element, reduce size –Move down

21. Notice about Remove Only want to decrement size counter if at bottom level Again, makes zig-zag motion to bottom, is proportional to height So it is again O(log n)

22. Add (Object newElement) Now we come to addition - the most complex operation. Intuitive idea - add to bottom row (remember to increment count) Move upwards, flipping a coin As long as heads, add to next higher row At top, again flip, if heads make new row

23. Add Example Insert the following: 9 14 7 3 20 Coin toss: T H T H H T H T (insert if heads)

24. The Power of Chance At each step we flip a coin So the exact structure is not predictable, and will be different even if you add the same elements in same order over again But the gross structure is predictable, because if you flip a coin N times as N gets large you expect N/2 to be heads.

25. Notes on Add Again proportional to height of structure, not number of nodes in list So also O(log n) So all three operations are O(log n) ! By far the fastest overall Bag structure we have seen! (Downside - does use a lot of memory as values are repeated. But who cares about memory? )

26. int skipListContains(struct skipList* slst, EleType d) { struct skipLink *tmp = slst->topSentinel; while (tmp != 0) { tmp = slideRight(tmp, d); if ((tmp->next != 0) && EQ(d, tmp->next- >value)) return 1; tmp = tmp->down; } return 0; }

27. void skipListAdd(struct skipList * slst, EleType d) { struct skipLink *downLink, *newLink; downLink = skipLinkAdd(slideRight(slst- >topSentinel,d),d); if (downLink != 0 && skipFlip()) { newLink = newSkipLink(d, 0, downLink); slst->topSentinel = newSkipLink(0, newLink, slst->topSentinel); } slst->size++; }

28. struct skipLink* skipLinkAdd(struct skipLink * current, EleType d) { struct skipLink *newLink, *downLink; newLink = 0; if (current->down == 0) { newLink = newSkipLink(d, current->next, 0); current->next = newLink; } else { downLink = skipLinkAdd(slideRight(current->down, d), d); if (downLink != 0 && skipFlip()) { newLink = newSkipLink(d, current->next, downLink); current->next = newLink; } return newLink; }

29. struct skipLink* newSkipLink(EleType d, struct skipLink * nlnk, struct skipLink* dlnk) { struct skipLink * tmp = (struct skipLink *) malloc(sizeof(struct skipLink)); assert(tmp != 0); tmp->value = d; tmp->next = nlnk; tmp->down = dlnk; return tmp; }

30. int skipFlip() { return rand() % 2; }

31. void skipListRemove(struct skipList* slst, EleType d) { struct skipLink *current, *tmp; current = slst->topSentinel; while (current != 0) { current = slideRight(current, d); if (current->next != 0 && EQ(d, current->next->value)) { tmp = current->next; current->next = current->next->next; free(tmp); if (current->down != 0) slst->size--; } current = current->down; }