skip list & skip graph

SKIP LIST & SKIP GRAPH

James AspnesGauri Shah

Many slides have been taken from the original presentation by the authors

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Definition of Skip ListA skip list for a set L of distinct (key, element) items is a series of linked lists L0, L1 , … , Lh such that

Each list Li contains the special keys +∞ and -∞List L0 contains the keys of L in non-decreasing order Each list is a subsequence of the previous one, i.e.,

L0 ⊃ L1 ⊃ … ⊃ Lh

List Lh contains only the two special keys +∞ and -∞

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Skip List

(Idea due to Pugh ’90, CACM paper)Dictionary based on a probabilistic data structure.Allows efficient search, insert, and delete operations.Each element in the dictionary typically stores additionaluseful information beside its search key. Example:

<student id. Transcripts> [for University of Iowa]<date, news> [for Daily Iowan]

Probabilistic alternative to a balanced tree.

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Skip List

A G J M R W

HEAD TAIL

Each node linked at higher level with probability 1/2.

Level 0

A J M

Level 1

J

Level 2

∞∞

5

Another example

56 64 78 ∞31 34 44∞ 12 23 26

∞31∞

64 ∞31 34∞ 23

L0

L1

L2

Each element of Li appears in L.i+1 with probability p.Higher levels denote express lanes.

6

Searching in Skip List

Search for a key x in a skip:

Start at the first position of the top list At the current position P, compare x with y ≥ key after P

x = y -> return element (after (P))x > y -> “scan forward” x < y -> “drop down”

– If we move past the bottom list, then no such key exists

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Example of search for 78

∞∞

L1

L2

L3

∞31∞

64 ∞31 34∞ 23

56 64 78 ∞31 34 44∞ 12 23 26L0

At L1 P is the next element ∞ is bigger than 78, we drop downAt L0, 78 = 78, so the search is over.

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Insertion

• The insert algorithm uses randomization to decide in how many levels the new item <k> should be added to the skip list.

• After inserting the new item at the bottom level flip a coin.

• If it returns tail, insertion is complete. Otherwise, move to next higher level and insert <k> in this level at

the appropriate position, and repeat the coin flip.

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Insertion Example

∞∞ 10 36

∞∞

23

23 ∞∞

L0

L1

L2

∞∞

L0

L1

L2

L3

∞∞ 10 362315

∞∞ 15

∞∞ 2315p0

p1

p2

1) Suppose we want to insert 152) Do a search, and find the spot between 10 and 233) Suppose the coin come up “head” three times

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Deletion

• Search for the given key <k>. If a position with key <k> is not found, then no such key exists.

• Otherwise, if a position with key <k> is found (it will be definitely found on the bottom level), then we remove all occurrences of <k> from every level.

• If the uppermost level is empty, remove it.

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Deletion Example1) Suppose we want to delete 342) Do a search, find the spot between 23 and 453) Remove all occurrences of the key

∞ ∞4512

∞ ∞

23

23∞ ∞

L0

L1

L2

L0

L1

L2

∞ ∞4512 23 34

∞ ∞34

∞ ∞23 34p0

p1

p2

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O(1) pointers per node

Average number of pointers per node = O(1)

Total number of pointers

= 2.n + 2. n/2 + 2. n/4 + 2. n/8 + … = 4.n

So, the average number of pointers per node = 4

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Number of levels

Pr[a given element x is above level c log n] = 1/2c log n = 1/nc

Pr[any element is above level c log n] = n. 1/nc

= 1/nc-1

So the number of levels = O(log n) with high probability

The number of levels = O(log n) w.h.p

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Search time

Consider a skiplist with two levels L0 and L1. To search a key, first search L1 and then search L0.

Cost (i.e. search time) = length (L1) + n / length (L1) Minimum when length (L1) = n / length (L1).

Thus length(L1) = (n) 1/2, and cost = 2. (n) 1/2

(Three lists) minimum cost = 3. (n)1/3

(Log n lists) minimum cost = log n. (n) 1/log n = 2.log n

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Skip lists for P2P?

• Heavily loaded top-level nodes.• Easily susceptible to failures.• Lacks redundancy.

Disadvantages

Advantages• O(log n) expected search time.• Retains locality.• Dynamic node additions/deletions.

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A Skip Graph

A001

J001

M011

G100 W101

R110

Level 1

G

R

W

A J M000 001 011

101

110

100Level 2

A G J M R W001 001 011100 110 101Level 0

Membership vectors

Link at level i to nodes with matching prefix of length i.Think of a tree of skip lists that share lower layers.

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Properties of skip graphs

1. Efficient Searching.2. Efficient node insertions & deletions.3. Independence from system size.4. Locality and range queries.

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Searching: avg. O (log n)

Same performance as DHTs.

A J MG WR

Leve

l 1

GR

WA J MLe

vel 2

A G J M R W

Leve

l 0

Restricting to the lists containing the starting element of the search, we get a skip list.

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Node Insertion – 1

A

001

M

011

G

100

W101

R110Le

vel 1

G

R

W

A M

000 011

101

110

100Leve

l 2

A G M R W

001 011100 110 101

Leve

l 0J

001

Starting at buddy node, find nearest key at level 0.Takes O(log n) time on average.

buddy new node

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Node Insertion - 2At each level i, find nearest node with matching

prefix of membership vector of length i.

A001

M011

G

100

W101

R110Le

vel 1

G

R

W

A M

000 011

101

110

100Leve

l 2

A G M R W

001 011100 110 101Leve

l 0 J

001

J

001

J

001

Total time for insertion: O(log n)DHTs take: O(log2n)

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Independent of system sizeNo need to know size of keyspace or number of nodes.

E Z1 0

E ZJ

insert

Level 0

Level 1

E Z1 0

E Z

J0

J00 01

E ZJ

Level 0

Level 1

Level 2

Old nodes extend membership vector as required with arrivals.DHTs require knowledge of keyspace size initially.

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Locality and range queries

• Find key < F, > F.• Find largest key < x.• Find least key > x.

• Find all keys in interval [D..O].

• Initial node insertion at level 0.

D F A I

D F A I L O S

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Applications of locality

news:02/13

e.g. find latest news from yesterday. find largest key < news: 02/13.

news:02/11 news:02/12news:02/10news:02/09Level 0

DHTs cannot do this easily as hashing destroys locality.

e.g. find any copy of some Britney Spears song.

britney05britney03 britney04britney02britney01Level 0

Data Replication

Version Control

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So far...

Decentralization.

Locality properties.

O(log n) space per node.

O(log n) search, insert, and delete time.

Independent of system size.

Coming up...

• Load balancing.

•Tolerance to faults.

• Self-stabilization.

• Random faults.• Adversarial faults.

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Load balancing

Interested in average load on a node u.i.e. the number of searches from source s to destination t that use node u.

Theorem: Let dist (u, t) = d. Then the probability that a search from s to t passes through u is < 2/(d+1).

where V = {nodes v: u <= v <= t} and |V| = d+1.

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Nodes u

Observation

Level 0

Level 1

Level 2

Node u is on the search path from s to t only if it is inthe skip list formed from the lists of s at each level.

s

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Tallest nodes

Node u is on the search path from s to t only if it isin T = the set of k tallest nodes in [u..t].

u

u t

s u is not on path.

tu

u

s

u

u is on path.

Pr [u T] = Pr[|T|=k] • k/(d+1) = E[|T|]/(d+1).k=1

d+1

Heights independent of position, so distances are symmetric.

∈

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Load on node uStart with n nodes. Each node goes to next set with prob. 1/2.We want expected size of T = last non-empty set.

= T

Average load on a node is inversely proportional to the distance from the destination.

We show that: E[|T|] < 2.

Asymptotically: E[|T|] = 1/(ln 2) ± 2x10-5 ≃ 1.4427… [Trie analysis]

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Expected loadActual loadDestination = 76542

76400 76450 76500 76550 76600 76650

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0.0

Node location

Load

on

nod

eExperimental result

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Fault tolerance

How do node failures affect skip graph performance?

Random failures: Randomly chosen nodes fail. Experimental results.

Adversarial failures: Adversary carefully chooses nodes that fail. Bound on expansion ratio.

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Random faults

Size of largest connected component

as fraction of live nodes

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.0

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

0.3

0

0.3

5

0.4

0

0.4

5

0.5

0

0.5

5

0.6

0

0.6

5

0.7

0

0.7

5

0.8

0

0.8

5

0.9

0

0.9

5

Probability of node failure

Siz

e

131072 nodes

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Searches with random failures

Fraction of f ailed searches

0.00

0.05

0.10

0.15

0.20

0.25

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Probability of node f ailure

Fai

led s

earc

hes 131072 nodes

10000 messages

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Adversarial faults

Theorem: A skip graph with n nodes has expansion ratio = (1/log n).Ω

A

dA

dA = nodes adjacent to A but not in A. To disconnect A all nodes in dA must be removed

Expansion ratio = min |dA|/|A|, 1 <= |A| <= n/2.

f failures can isolate only O(f•log n ) nodes.

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Need for repair mechanism

A J MG WR

Level 1

GR

WA J MLe

vel 2

A G J M R W

Level 0

Node failures can leave skip graph in inconsistent state.

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Ideal skip graph

Let xRi (xLi) be the right (left) neighbor of x at level i.

xLi < x < xRi.xLiRi = xRiLi = x.

Invariant

kxRi = xRi-1.

kxLi = xLi-1. Successor

constraints

x

Level i

Level i-1

ixR

i-1xR1

i-1xR2

x

..00..

..01.. ..00..

If xLi, xRi exist:

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Basic repairIf a node detects a missing neighbor, it tries

to patch the link using other levels.

1 2 4 5 63

31 5 6

1 5

Also relink at other lower levels.

Successor constraints may be violated by node arrivals or failures.

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Constraint violationNeighbor at level i not present at level (i-1).

x

x Level i-1

Level i

Level i-1

Level ix

x

zipper..00.. ..01.. ..01.. ..01.. ..00.. ..01.. ..01....01.. ..00.. ..01..

..01.. ..00.. ..01..

x

x..01..

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Conclusions

• Decentralization.

• O(log n) space at each node.

• O(log n) search time.

• Load balancing properties.

• Tolerant of random faults.

Similarities with DHTs

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Property DHTs Skip Graphs

Insert/Delete time

O(log2n) O(log n)

Locality No Yes

Repair mechanism

? Partial

Tolerance ofadversarial faults

? Yes

Keyspace size Reqd. Not reqd.

Differences

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Open Problems

• Evaluate performance in practice.

• Incorporate geographical proximity.

• Study multi-dimensional skip graphs.

• Study effect of byzantine failures.

?

• Design efficient repair mechanism.