1. Know thy Neighbor’s Neighbor Better Routing for Skip Graphs and Small Worlds
Moni Naor
Udi Wieder

2. Properties of Greedy O(1) O(log2n) O(log n) O(log n)
Scheme
Degree
Greedy – path length
Kleinberg’s Small world
O(1)
O(log2n)
Randomized Chord
O(log n)
Skip Graph
O(log n)
Simple – to understand and to implement.
Local – If source and target are close, the path remains within a small area in the keyspace.
Howver, route not optimal with respect to the degree.

3. Can Greedy Routing be further shortened
Question
Can Greedy Routing be further shortened
Without compromising the good properties?
This paper examines this issue.

4. Optimal diameter For a graph in which each node has degree d,
The optimal diameter is O(logdn)
When d=log n, the optimal diameter is
O(log n/log log n)
Proof?

5. Neighbor of Neighbor (NoN) Routing
Each node has a list of its neighbor’s neighbors.The message is routed greedily to the closest neighbor of neighbor (2 hops). This means:
Let w1, w2, … wk be the neighbors of current node u
For each wi find zi, the closest neighbor to target t
Let j be such that zj is the closest to target t
Route the message from u via wj to zj
The first hop may not be a greedy choice.

6. Degree Optimal P2P Routing
Different routing schemes
Viceroy emulates the butterfly network
Constant degree
O(log n) hops for routing
Constructions emulating De-Bruijn graphs
Can achieve any degree/number of hops tradeoff
In particular degree O(log n) and O(log n/ log log n) hops
Routing is not greedy

7. Each node chooses a random string of bits.
Skip-Graphs
Each node (resource) has a name.
Nodes are arranged on a line sorted by name. a b c d e f 1
Each node chooses a random string of bits.
An edge is established if two nodes share a prefix which is not shared by the nodes between them.

8. Routing in Skip-Graphs
Greedy Routing – use longest edge possible.
Path length is (log n) w.h.p. 1
The NoN algorithm optimizes over two hops.
Theorem: Using the NoN algorithm, the expected path length of any lookup is

9. Skip Graphs – degree optimality= l o g d
X - # of two hop paths between d and
D - the event a message reached the node d. [ d l o g ; ]
Lemma: Prob [ ( X
> ) j D ] 1 2
Sufficiency of lemma:
Call a NoN 2-hop successful if it reduces the distance from d to log d
Need O(log log n) succesful 2-hops to get to distance 1.

10. The Cost/Performance of NoN
Cost of Neighbor of Neighbor lists:
Memory: O(log2n) - marginally higher.
Communication: Is it tantamount to squaring the degree?
Neighbor lists should be maintained (open connection, pinging, etc.)
NoN lists should only be kept up-to-date.
Lazy updates: Updates occur only when communication load is low – supported by simulations.
Networks of size 217 show 30-40% improvement

11. Conclusions
NoN Greedy seems like an almost free tweak that is a good idea in many settings.