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CSC 172 DATA STRUCTURES

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SKIP LISTS Read Weiss 10.4.2

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SKIP LISTS Dictionary Data Structure Efficient

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

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

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

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

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

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EXAMPLE

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

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

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

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

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Lookup on a skip list

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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)

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

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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)

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

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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))

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BALLANCED SKIP LISTS Ideal as long as structure is maintained

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

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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.

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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

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INSERTION ON SKIP LISTS EXAMPLE

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

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1 Searching Searching in a sorted linked list takes linear time in the worst and average case. Searching in a sorted array takes logarithmic time in the.

1 Searching Searching in a sorted linked list takes linear time in the worst and average case. Searching in a sorted array takes logarithmic time in the.

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