Know thy neighbor’s neighbor: Better Routing for Skip-Graphs and Small Worlds Αθανασόπουλος Διονύσης Καμωνά Λαμπρινή Φωτιάδου Αικατερίνη Moni Nao, Udi Wieder

2 Introduction zAim: Propose an approach for routing in DTH’s which is better than greedy routing zGreedy routing: move to the node that minimizes the L 1 distance to the target. zExamples:Chord, Skip Nets, Skip Graphs,

3 Greedy Routing Advantages zSimplicity: Easy to understand and implement zFault Tolerance: as long as each node has some edge towards the target, it is guaranteed that the message will reach its destination zLocality in the key space: Message do not “wander” in the key space

4 Greedy Routing zWhy use something else? Not degree optimal yGreedy -> O(logn) != Optimal -> O(logn/loglogn) zNoN Greedy algorithm (Neighbor-of- Neighbor) Enjoys the advantages of greedy, while being degree optimal

5 Kleinberg’s model [2000] yPeople  points on a two dimensional grid yGrid edges (short range) yOne long range contact chosen with the Harmonic distribution  probability of ( u, v ) proportional to 1/ d ( u, v ) 2 x Degree of each node Θ(logn) yNaturally generalizes to q long range links

6 Small Worlds zd-dimensional grid zEach edge (u,v) is connected with probality ||u-v|| -d yDegree of each node Θ(logn) yOriginates from long range percolation model yShares structural properties with some popular randomized P2P networks: R-Chord, R-Hypercube, Skip Lists…

7 The NoN-Greedy Algorithm

8 zStep (2) is implemented by putting all z in a search tree. Search time = O(log(k 2 )) zK=logn => Search time = O(loglogn)

9 Greedy vs NoN-Greedy z2 24 nodes z150 executions for each size z34% improvement

10 The NoN-Greedy Algorithm zPhase1: the message is sent to a neighbor whose neighbor is close to the target zPhase2-greedy step: the message moves to the neighbor of the neighbor

11 Fault Tolerance – Optimistic Scenario zA node knows if its lists are updated zIf not updated performs a greedy step zP(NoN)= ½ zP(Greedy)= ½

12 Fault Tolerance – Pessimistic Scenario zNode is unaware that its list are up-to-date zWith probability ½ the edge (w,z) no longer exists zi) w performs a greedy step zii) w performs a NoN step

13 NoN - Chord zMake Chord resemble the Small World zEach node x is connected to logn nodes y 0,y 1,y 2 … zy i is a random point in [x+2 i, x+2 i+1 ] zPath length= O(logn/loglogn)