SKIP GRAPHS (continued)

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SKIP GRAPHS(continued)
Some slides adapted from the original slides
by James Aspnes Gauri Shah
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So far...
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Decentralization. Locality properties.
O(log n) neighbors 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. how
many searches from source s to destination
t 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
u
Level 0
t
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.
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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 the
path 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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Need for repair mechanism
G
W
Level 2
R
A
J
M
W
R
G
Level 1
A
J
M
Level 0
A
G
J
R
W
M
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.