1. SKIP GRAPHS
James Aspnes
Gauri Shah
SODA 2003

2. P2P system Bunch of peers. Store resources identified by keys.
Peers subject to crash failures.
Goal: locate resources efficiently.

3. 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]

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

5. Advantages Disadvantages SKIP GRAPHS 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

6. Skip List [Pugh ’90] Data structure based on a linked list. J A J M A
HEAD
TAIL J
Level 2 A J M
Level 1
Level 0 A G J M R W
Each node linked at higher level with probability 1/2.

7. Searching in a skip list
Search for key ‘R’
HEAD
success
TAIL
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.

8. 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.

9. A Skip Graph G W A J M R G R W A J M A G J M R W
Level 2 A J M R
101
100
000
001
011
110
100 G R W
Level 1 A J M
110
101
001
001
011
Membership vectors
Level 0 A G J M R W
001
100
001
011
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.

10. Properties of skip graphs
Searching.
Node insertions.
Independence from system size.
Locality and range queries.

11. Searching: avg. O (log n)
Restricting to the lists containing the starting
element of the search, we get a skip list.
Level 2 G W A J M R G R W
Level 1 A J M
Level 0 A G J M R W
Same performance as DHTs.

12. Node Insertion – 1
buddy
new node J
001 G W
Level 2 A M R
100
101
000
011
110 G R W
Level 1 A M
110
101
100
001
011
Level 0 A G M R W
001
100
011
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.

13. Node Insertion - 2 Total time for insertion: O(log n)
At each level i, find nearest node with matching
prefix of membership vector of length i+1. G W
Level 2 A J
001 M R
100
101
000
011
110 G R W
Level 1 A J
001 M
110
101
100
001
011 A
Level 0 G J
001 M R W
001
100
011
110
101
Total time for insertion: O(log n)
DHTs take: O(log2n)

14. Independent of system size
No need to know size of keyspace or number of nodes. E Z 1 J 00 01
Level 0
Level 1
Level 2 J
insert E Z
Level 1 E Z
Level 0 1
Old nodes extend membership vector as required with arrivals.
DHTs require knowledge of keyspace size initially.

15. 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. A D F I A D F I L O S

16. Applications of locality
Version Control
e.g. find latest news from yesterday.
find largest key < news:10/29.
Level 0
news:10/25
news:10/26
news:10/27
news:10/28
news:10/29
Data Replication
e.g. find any copy of some Britney Spears song.
Level 0
britney01
britney02
britney03
britney04
britney05
DHTs cannot do this easily as hashing destroys locality.

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

18. 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.

19. Skip list restriction s
Level 2
Nodes u
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.

20. 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 [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.

21. 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.
We show that: E[|T|] < 2.
= T
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.

22. 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

23. 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.

24. Random faults
nodes

25. Searches with random failures
nodes
10000 messages

26. 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.

27. Proof intuition Consider neighbors of set A at level 0.
1. Clumpy sets
Level 0
Low probability of clumpy sets. dA A A
2. Non-clumpy sets
Level 0
Non-clumpy sets have many neighbors at level 0.
Gives high expansion ratio.

28. This gives expansion ratio = (1/log n).
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?

29. Need for repair mechanismG W
Level 2 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.

30. 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

31. 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.

32. 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

33. Self-stabilization
zOp(B)
zOp(E)
zOp(I) A C D F J
zipperOp
message
Level i 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.

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

35. 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.

36. Open Problems Design efficient repair mechanism.
Incorporate geographical proximity.
Study multi-dimensional skip graphs.
Evaluate performance in practice.
Study effect of byzantine failures. ?