CSC 172 DATA STRUCTURES

SKIP LISTSRead Weiss 10.4.2

SKIP LISTS Dictionary Data Structure Efficient

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

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

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

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

LISTS

How much time does it take to search a sorted linked list?

How can this be improved?

EXAMPLE

EXAMPLE What is this sequence?

14,23,28,34,42,50,59,66,72,79,86,96,103,110

EXAMPLE What is this sequence?

14,23,34,42,50,59,66,72,79,86,96,103,110

EXAMPLE What is this sequence?

14,23,34,42,50,59,66,72,79,86,96,103,110,116,125

SKIP LISTS Use two lists L

2 stores all element

L1 stores some elements

Links between shared elements

Lookup on a skip list

Lookup on a skip list

1)Take L1 until you go too far2)Back up one3)Transfer to L24)Take L2 until you find element

(or go too far – not found – or insert)

Lookup on a skip list

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

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)

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

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))

BALLANCED SKIP LISTS Ideal as long as structure is maintained

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

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.

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 level¼ of the elements go up 2 levels1/8 of the elements go up 3 levels

INSERTION ON SKIP LISTS EXAMPLE

ANALYSIS Intuitively:

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