1. Skip Lists – Why? BSTs –Worse case insertion, search O(n) –Best case insertion, search O(log n) –Where your run fits in O(n) – O(log n) depends on the order of the data –Hard to keep balanced 2-3 trees –Guaranteed to be balanced –Complicated to implement

2. Skip List Still no guarantee to be O(log n) for insertion, search, deletion, in every situation. Will give O(log n) performance with high probability (in most runs). Good compromise between performance and difficulty of implementation.

3. What is it? Recall the ordered linked list with one pointer to the next item. (worst case O(n), avg O(n/2) ) Add a pointer to every other node that points to the second next (current+2). (worst case O(n/2) ) 2152129 3749

4. What (cont.) Now add in a third level so that every 4 th item points 4 beyond it  Add a (log n) level so that every (n/2) th item points n/2 away from it. Now search is O(log n) But insertion (and keeping “balanced”) is VERY difficult.

5. Probabilistic Structure Rather than worrying about being perfect, just be good in the long run (especially if it is difficult to be perfect. The number of pointers is the level of the node. Randomly pick a level. –Want an exponential distribution with 50% being level 1, 25% being level 2, …

6. Sidebar – Implementing this Random Function Flipping a coin gives us a 50% chance of getting a head. If we get a tail on the first flip, then flip again. The chance of getting a TH is 25% P(TTH) = 12.5% Use a random number generator that generates uniformly distributed random numbers 0 or 1) –Count the number of times it takes to get a 0 –This will be the level and it will occur with the proper frequency (in the long run, i.e. with big enough data sets.

7. Operations Search – easy Insert –Pick level –Find position Keep track of previous node for each level used in the search. Update next pointers in “previous nodes” from level “picked level” to 0 (as well as the next in the inserted node). i.e., splice the node into the lists that it is part of.