David Stotts Computer Science Department - PowerPoint Presentation

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David StottsComputer Science DepartmentUNC Chapel Hill

Data Structures and Analysis(COMP 410)

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Skip Lists( some slides adapted from UMd CMSC 420, Bill Pugh et al. )

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Generalization of sorted linked listInvented by Bill Pugh (UMd) in 1990Original CACM paper: Pugh, W. (1990). "Skip lists: A probabilistic alternative to balanced trees" (PDF). Communications of the ACM. 33 (6): 668. An alternative to balanced BSTProbabilistic Data Structure

we roll dice, flip coins to build the structureSkip List

but first …

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“Perfect” skip list

first… no dice123

Multiple lists, multi-level list,

O(log N) levels

Level

i+1

has half as many items as level

i

12

3

4

13

14

17

7

7

17

14

13

12

4

3

x

x

x

x

x

Level

i+1

divides level

i

in half

sentinels

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“Perfect” skip list

first… no dice123

List nodes are variable size… each with 1 to O(log N) pointers

Sentinels at each end of “empty list”

12

3

4

13

14

17

7

7

17

14

13

12

4

3

x

x

x

x

x

Each level lets you “skip over” many items below in one hop

sentinels

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“Perfect” skip list

4712

x

x

x

x

x

1

2

3

Search path…

find(7)

Start at top level, move right on a level, down as compare

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Now add probabilityImperfect list, randomized order of cell sizesCell size is chosen on add by roll-of-dice

xxx

x

x

Level

i

not guaranteed to have twice level i+1, but random numbers will make it near twice

3

2

1

Level

i

not guaranteed to split level i-1 into equal parts

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Example insert

9

X

X

3

29

74

42

12

X

X

X

insert(9)

Roll dice: get a

1

new node should be a one-level cell

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Example insert

9

X

X

3

29

74

42

12

X

X

X

insert(35)

Roll dice: get a

3

new node should be a three-level cell

35

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Example insert

9

X

X

3

29

74

42

12

X

X

X

insert(35)

Roll dice: get a

3

new node should be a three-level cell

35

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Another example

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Another example

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Another example

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Another example

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function genRandomLevel ( ) { // generate an integer 0 or larger // following a distribution where 0 is 0.5 likely // 1 is .025 likely, 2 is .0125 likely, 3 is .00625, // etc.

var ranLev=0; while (

Math.random

() > 0.5 )

ranLev

++;

return

ranLev

;

}

Rolling the Dice

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var nLevels = 16;var nTrials = 1000; var SL = makeSkipList(nLevels);var levHits

= [ ];for (var i=0; i<nLevels; i++) { levHits

[

i

]=0; }

for (

var

i

=0;

i

<

nTrials

;

i

++) {

levHits

[

SL.dice

(

nLevels) ]++; }

alert(levHits);

Gives these node level counts:

9965, 5037, 2519, 1263, 619, 289, 149, 84, 30, 24, 14, 6, 0, 1, 0, 0

Checking the distribution

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OO Signaturenew:  SKLISTinsert: Elt  remove: Elt 

find: Elt  Boolean (searching)size:  Int

+

(non-negative integers)

empty:  Boolean

ADT: SKLIST of

Elt

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Time complexity of operationsinsert worst: O(n), avg: O(log n)remove worst: O(n), avg: O(log n) find worst: O(n), avg: O(log n)

empty O(1)size O(1)iterator O(n) (traversal)

SKLIST Implementation

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Beyond this is just templatesEND

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