SKIP GRAPHS

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SKIP GRAPHS
James Aspnes Gauri Shah SODA 2003
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P2P system
Key

• Bunch of peers.
• Store resources identified by keys.
• Peers subject to crash failures.
• Goal locate resources efficiently.

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Properties of ideal network

• Data availability
• Decentralization
• Fault-tolerance
• Scalability
• Load balancing
• Maintaining the network
• Dynamic node addition/deletion
• Self-stabilization
• Efficient searching
• Incorporating geography
• Incorporating locality temporal, spatial

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Distributed Hash Tables
Virtual Route
v4
Nodes
Keys
v2
v1
HASH
Physical Link
v3
Virtual Link
v1 v2 v3 v4
Actual Route
PHYSICAL NETWORK
VIRTUAL OVERLAY NETWORK
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Advantages
Disadvantages

• Load balancing.
• Decentralization.
• O(log n) space and search time.
• O(log2n) insert and delete time search for (log
  n) neighbors.
• Tolerance of random faults.

• No locality properties.
• No tolerance to adversarial faults.
• No self-stabilization.
• No optimization wrt. geography.

SKIP GRAPHS
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Skip List Pugh 90
Data structure based on a linked list.
TAIL
HEAD
Level 0
A
G
J
M
R
W
Each node linked at higher level with probability
1/2.
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Searching in a skip list
Search for key R
HEAD
TAIL
success
failure
J
Level 2
A
J
M
Level 1
Level 0
A
G
J
M
R
W
-?
?
Time for search O(log n) on average. On average,
constant number of pointers per node.
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Skip lists for P2P?
Advantages

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

Disadvantages

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

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A Skip Graph
W
G
Level 2
R
A
J
M
101
100
000
001
011
110
G
100
W
R
Level 1
A
J
M
101
110
001
001
011
Membership vectors
A
J
M
R
W
Level 0
G
001
001
011
100
110
101
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. Searching.
2. Node insertions.
3. Independence from system size.
4. Locality and range queries.

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Searching avg. O (log n)
Restricting to the lists containing the starting
element of the search, we get a skip list.
G
W
Level 2
R
A
J
M
G
W
R
Level 1
A
J
M
Level 0
A
G
J
M
R
W
Same performance as DHTs.
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Node Insertion 1
buddy
new node
W
G
Level 2
R
M
A
101
100
000
011
110
R
G
W
Level 1
A
M
101
110
100
001
011
A
R
G
M
W
Level 0
001
011
100
110
101
Starting at buddy node, find nearest key at level
0. Basically a range query looking for key
closest to new key. Takes O(log n) on average.
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Node Insertion - 2
At each level i, find nearest node with matching
prefix of membership vector of length i1.
W
G
Level 2
R
M
A
101
100
000
011
110
G
W
R
Level 1
A
M
101
110
100
001
011
A
R
G
M
W
Level 0
001
011
100
110
101
Total time for insertion O(log n) DHTs take
O(log2n)
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Independent of system size
No need to know size of keyspace or number of
nodes.
E
Z
Level 1
Z
E
Level 0
1
0
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 lt F, gt F.
• Find largest key lt x.
• Find least key gt x.
• Find all keys in interval D..O.
• Initial node insertion at level 0.

D
F
A
I
D
F
A
I
O
S
L
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Applications of locality
Version Control
e.g. find latest news from yesterday. find
largest key lt news10/29.
Level 0
news10/29
news10/27
news10/28
news10/26
news10/25
Data Replication
e.g. find any copy of some Britney Spears song.
Level 0
britney05
britney03
britney04
britney02
britney01
DHTs cannot do this easily as hashing destroys
locality.
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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.
?
?
?
?
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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 lt 2/(d1).
where V nodes v u lt v lt t and V d1.
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Skip list restriction
s
Level 2
Level 1
Level 0
Node u is on the search path from s to t only if
it is in the skip list formed from the lists of s
at each level.
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Tallest nodes
s
u is not on path.
u is on path.
?
u
u
t
Node u is on the search path from s to t only if
it is in T the set of k tallest nodes in u..t.
Heights independent of position, so distances are
symmetric.
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Load on node u
Start with n nodes. Each node goes to next set
with prob. 1/2. We want expected size of T last
non-empty set.
We show that ET lt 2.
Asymptotically ET 1/(ln 2) ? 2x10-5 ?
1.4427 Trie analysis
Average load on a node is inversely proportional
to the distance from the destination.
We also show that the distribution of average
load declines exponentially beyond this point.
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Experimental result
Load on node
Node location
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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
131072 nodes
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Searches with random failures
131072 nodes 10000 messages
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Adversarial faults
dA nodes adjacent to A but not in
A. Expansion ratio min dA/A,
1 lt A lt n/2.
A
dA
Theorem A skip graph with n nodes has
expansion ratio (1/log n).
f failures can isolate only O(flog n ) nodes.
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Proof intuition
Consider neighbors of set A at level 0.
A
Level 0
1. Clumpy sets
dA
Low probability of clumpy sets.
A
A
2. Non-clumpy sets
Level 0
Non-clumpy sets have many neighbors at level
0. Gives high expansion ratio.
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Expansion ratio
All sets have low probability of few neighbors at
level h. And there are not too many clumpy
sets. Low probability that any set A has few
neighbors at level 0 or h. This gives
expansion ratio (1/log n).
Same analysis applicable to DHTs?
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Need for repair mechanism
G
W
Level 2
R
A
J
M
W
R
G
Level 1
A
J
M
A
G
J
R
W
M
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.
If xLi, xRi exist
xLi lt x lt xRi. xLiRi xRiLi x.
Invariant
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Basic repair
If a node detects a missing neighbor, it tries
to patch the link using other levels.
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3
Also relink at other lower levels.
Successor constraints may be violated by node
arrivals or failures.
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Constraint violation
Neighbor at level i not present at level (i-1).
Level i
x
x
Level i-1
..00..
..01..
..01..
..01..
..00..
..01..
..01..
..01..
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Self-stabilization
zOp(B)
zOp(E)
zOp(I)
A
C
D
F
J
zipperOp message
Level i
B
E
G
H
I
zOp(D)
zOp(A)
zOp(F)
Eventually want each connected component of the
skip graph to reorganize itself into an ideal
skip graph.
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Conclusions
Similarities with DHTs

• Decentralization.
• O(log n) space at each node.
• O(log n) search time.
• Load balancing properties.
• Tolerant of random faults.

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Differences
Property DHTs Skip Graphs
Insert/Delete time O(log2n) O(log n)
Locality No Yes
Repair mechanism ? Partial
Tolerance of adversarial faults ? Yes
Keyspace size Reqd. Not reqd.
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Open Problems

• Design efficient repair mechanism.

• Incorporate geographical proximity.

• Study multi-dimensional skip graphs.

• Evaluate performance in practice.

• Study effect of byzantine failures.