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CMSC420: Skip Lists

CMSC420: Skip Lists. Kinga Dobolyi Based off notes by Dave Mount. What is the purpose of this class?. Data storage Speed Trees Binary search tree Could degenerate into a linked list AVL tree Better performance, difficult to implement Splay tree Good overall performance -- amortized.

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CMSC420: Skip Lists

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  1. CMSC420: Skip Lists Kinga Dobolyi Based off notes by Dave Mount

  2. What is the purpose of this class? • Data storage • Speed • Trees • Binary search tree • Could degenerate into a linked list • AVL tree • Better performance, difficult to implement • Splay tree • Good overall performance -- amortized

  3. Can we do “better” • Linked list • Simple to implement • Poor runtime: O(n) • Better runtime • Guaranteed O(logn) • Difficult to implement • Can we have both for general data? • What are the tradeoffs?

  4. Skip Lists • Generalization of a linked list • O(logn) runtime of balanced trees • What is the trick? • Randomized data structure • Efficient in the expected case • Unlike BST, the expected case has nothing to do with the order of the keys being inserted • Probability that it performs badly is very small

  5. General idea • How can we make linked lists better? • Why are lists crummy? What takes forever? • What if we could skip over many elements at a time?

  6. Implementation • Start with a sorted linked list • Add another layer linking every other element • Repeat for that layer, etc • Think of as a hierarchy of sorted linked lists

  7. Skip list runtime • How high does this stack go? • Level 0: n • Level 1: n/2 • Level 2: n/4 • Height of stack will be logn • So search through the skiplist will be O(logn) • But can we maintain this guarantee of efficiency?

  8. Perfect vs Random Skip List • What we just described is a perfect skip list • What happens when you add or delete elements? • To maintain perfect balance • Random skip list • Need to decide when to promote a node to some i level • Use randomization, with a chance of 50%, to decide when to promote a node

  9. Random Skip list • If, on average, we promote half of the nodes at every level • Our skiplist will maintain this O(logn) property in the expected case • It is also likely that, overall, nodes will have a uniform distribution and not bunch up at the ends

  10. Insertion

  11. Exercise • Argue that the expected runtime of search in a skiplist is in O(logn) • What about insertion and deletion?

  12. Easy to implement • For details, see Dave Mount’s notes (linked from website)

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