Skip ListsEduardo Nakamura

nakamura@tamu.edu

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Outline and Reading

Skip list (§9.4)

Search and update operations in a skip list (§9.4.1)

Probabilistic analysis of skip lists (§9.4.2)

Skip List

1. A list of h+1 lists Sh ⊆ Sh–1 ⊆ … ⊆ S1 ⊆ S0

2. h is the height of the skip list

3. Si+1 has roughly half of the elements from Si, randomly chosen

S0 has all entries (key;value) in non-decreasing order

Sh has no entry (key;value)

4. Search “mimics” a binary search

Sketching the idea

12–∞ +∞17 20 25 31 38 39 44 50 55S0

Building a Skip List

lower sentinel upper sentinel

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

S0

S1

Building a Skip List

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

S0

S1

S2

Building a Skip List

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

S0

S1

S2

S3

Building a Skip List

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

S0

S1

S2

S3

S4

Building a Skip List

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Building a Skip List

h = 5 (height of the skip list)

“Quadruply linked list” (quad-nodes)

key and value Pointer to up, down, left, and right nodes

Traverse horizontally and vertically using

above() and below()

before() and after()

Bounded by sentinels (–∞ and +∞)

Implementing a skip list

above()

below()

before() after()0 B

righ

t

left

up

down

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

50

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Searching a Skip List

skipSearch(50)

Pseudo-code for skipSearch()

Algorithm skipSearch(k)Input: A search key kOutput: Position p in the list S0 such that the entry at p has key ≤ k

p ← Sh.begin();

while below(p) ≠ null do

p ← below(p); //drop down

while k ≥ key(after(p)) do

p ← after(p); //scan forward

end while

end while

return p;

Pseudo-code for find()

Algorithm find(k)Input: A search key kOutput: Position p in the list S0 such that the entry at p has key = k, null otherwise

p ← skipSearch();

if key(p) ≠ k thenreturn null;

elsereturn p;

end if

Exercise

1. Search for key 55 (show the visited nodes and links)

2. Search for key 20 (show the visited nodes and links)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Exercise

1. Search for key 55 (show the visited nodes and links)

2. Search for key 20 (show the visited nodes and links)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Exercise

1. Search for key 55 (show the visited nodes and links)

2. Search for key 20 (show the visited nodes and links)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Removing from a Skip List

erase(31)

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Removing from a Skip List

erase(31)

12–∞ +∞17 20 25 31 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Removing from a Skip List

erase(31)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 31 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Removing from a Skip List

erase(31)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 31 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Removing from a Skip List

erase(31)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Removing from a Skip List

erase(31)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Removing from a Skip List

erase(31)

Pseudo-code for erase()

Algorithm erase(k)Input: Key kOutput: None

current ← skipSearch(k);

if key(current) ≠ k then

return;

end if

current ← above(current);

doerase(current.below()); //erase lower levelcurrent ← above(current); //move to upper level

until current = null

Exercise

1. Remove for key 55 (show the visited nodes and links)

2. Remove for key 17 (show the visited nodes and links)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Inserting in a Skip List

insert(41)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Inserting in a Skip List

insert(41)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Inserting in a Skip List

insert(41)

41

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Inserting in a Skip List

insert(41)

41

41

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Inserting in a Skip List

insert(41)

41

41

41

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Inserting in a Skip List

insert(41)

41

41

41

41

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Inserting in a Skip List

insert(41)

41

41

41

41

41

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Inserting in a Skip List

insert(41)

41

41

41

41

41

In this case (tails), the

algorithm would stop…

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Inserting in a Skip List

insert(41)

41

41

41

41

41

41

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Inserting in a Skip List

41

41

41

41

41

–∞ +∞S5

41

Pseudo-code for insert(k,v)Algorithm insert(k,v):

Input: Key k and value vOutput: Topmost position of the entry inserted in the listp ← skipSearch(k);q ← null;e ← (k,v);i ← −1;repeat

i ← i + 1;if i ≥ h then

h ← h + 1; // add a new level to the skip list t ← after(s); // s is Sh.begin() s ← insertAfterAbove(null,s,(−∞,null));insertAfterAbove(s,t,(+∞,null));

end ifwhile above(p) = null do

p ← before(p); //scan backwardp ← above(p); //jump up to higher level q ← insertAfterAbove(p,q,e); //add to the new tower

end while until coinFlip() = tails;n ← n + 1;return q;

Exercise

1. Insert key 22 (show the visited nodes and links)

2. Insert for key 57 (show the visited nodes and links)

12–∞ +∞17 20 25 38 39 44 50 55

12–∞ +∞17 25 38 44 55

–∞ +∞17 25 55

–∞ +∞17 55

–∞ +∞17

–∞ +∞

S0

S1

S2

S3

S4

S5

Space analysis

Fact 1:The probability of i consecutive heads by flipping a coin is 1/2i

Fact 2: If each of n items is present in a set with probability P, the expected

size of the set is nP

By Fact 1: we insert an item in list Si with probability 1/2i

# of nodes =

By Fact 2: the expected size of list Si is n/2i , where n is the # of items

∑h

i = 0

n2i

= ∑h

i = 0

n 2–i < 2n

< 1

= 2n 1 –1

2h+1

Space analysis

Space is O(n)

# of nodes = ∑h

i = 0

n2i

= ∑h

i = 0

n 2–i < 2n

< 1

= 2n 1 –1

2h+1

Time analysis (find)

Fact 3: The expected number of coin tosses required to get tails is 2

Two things are accounted for time analysis: 1. the number of drop-down steps, plus 2. the number of scan-forward steps

Drop-down: O(h) = O(log n)

Scan-forward: O(log n) = O(1) (fact 3) for each O(log n) levels

Expected time is O(log n)

A skip list is a data structure for dictionaries

Randomized insertion algorithm

In a skip list with n items

The expected space used is O(n)

The expected search, insertion and deletion time is O(log n)

Skip lists are fast and simple to implement in practice

Summary