1. SKIP GRAPHS (continued)
Some slides adapted from the original slides by
James Aspnes
Gauri Shah

2. So far... Coming up...      Load balancing. Decentralization.
Tolerance to faults.
Self-stabilization.
Random faults.
Adversarial faults.
Decentralization.
Locality properties.
O(log n) neighbors per node.
O(log n) search, insert, and delete time.
Independent of system size.     

3. 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 < 2/(d+1).
where V = {nodes v: u <= v <= t} and |V| = d+1.

4. Skip list restriction u t s
Level 2
Node u
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.

5. Tallest nodes s t u 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 the path [u..t].
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.

6. Load on node u Average load on a node is inversely proportional
Start with n nodes. Each node goes to next set with prob. 1/2.
We want expected size of T = last non-empty set.
= T
We show that: E[|T|] < 2.
Asymptotically: E[|T|] = 1/(ln 2)  2x10-5  … [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.

7. Experimental result Load on node Node location Expected load
Actual load
Destination = 76542
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.0
Load on node
Node location

8. 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.

9. Random faults
nodes

10. Searches with random failures
nodes
10000 messages

11. Adversarial faults Theorem: A skip graph with n nodes has
dA = nodes adjacent to A but
not in A.
Expansion ratio = min |dA|/|A|,
1 <= |A| <= n/2. A dA
Theorem: A skip graph with n nodes has
expansion ratio =  (1/log n).
f failures can isolate only O(f•log n ) nodes.

12. Need for repair mechanism
Level 2 G W A J M R G R W
Level 1 A J M
Level 0 A G J M R W
Node failures can leave skip graph in inconsistent state.

13. Let xRi (xLi) be the right (left) neighbor of x
Ideal skip graph
Let xRi (xLi) be the right (left) neighbor of x
at level i.
If xLi, xRi exist: k
xRi = xRi-1.
xLi = xLi-1.
Successor
constraints x
Level i
Level i-1 i xR
i-1 1 2
..00..
..01..
xLi < x < xRi.
xLiRi = xRiLi = x.
Invariant

14. Basic repair
If a node detects a missing neighbor, it tries
to patch the link using other levels. 1 5 1 3 5 6 1 2 3 4 5 6
Also relink at other lower levels.
Successor constraints may be violated by node
arrivals or failures.

15. Constraint violation Neighbor at level i not present at level (i-1). x
..00..
..01.. x
Level i x x
Level i-1
..00..
..01..
..01..
..01..
..00..
..01..
..01..
..01..
zipper
Level i-1
Level i x

16. Self-stabilization
zOp(B)
zOp(E)
zOp(I) A C D F J
Level i
zipperOp
message B E G H I
zOp(A)
zOp(D)
zOp(F)
Eventually want each connected component of the skip
graph to reorganize itself into an ideal skip graph.

17. Conclusions Similarities with DHTs Decentralization.
O(log n) space at each node.
O(log n) search time.
Load balancing properties.
Tolerant of random faults.

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

19. Open Problems Design efficient repair mechanism.
Incorporate geographical proximity.
Study multi-dimensional skip graphs.
Evaluate performance in practice.