Know thy neighbor

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Know thy neighbors neighbor Better Routing for
Skip-Graphs and Small Worlds
Moni Nao, Udi Wieder

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Introduction

• Aim Propose an approach for routing in DTHs
  which is better than greedy routing
• Greedy routing move to the node that minimizes
  the L1 distance to the target.
• ExamplesChord, Skip Nets, Skip Graphs,

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Greedy Routing Advantages

• Simplicity
• Easy to understand and implement
• Fault Tolerance
• as long as each node has some edge towards the
  target, it is guaranteed that the message will
  reach its destination
• Locality in the key space
• Message do not wander in the key space

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Greedy Routing

• Why use something else?
• Not degree optimal
• Greedy -gt O(logn) ! Optimal -gt O(logn/loglogn)
• NoN Greedy algorithm (Neighbor-of-Neighbor)
• Enjoys the advantages of greedy, while
  being degree optimal

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Kleinbergs model 2000

• People ?? points on a two dimensional grid
• Grid edges (short range)
• One long range contact chosen with the Harmonic
  distribution
• probability of (u,v) proportional to 1/d(u,v)2
• Degree of each node T(logn)
• Naturally generalizes to q long range links

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Small Worlds

• d-dimensional grid
• Each edge (u,v) is connected with probality
  u-v-d
• Degree of each node T(logn)
• Originates from long range percolation model
• Shares structural properties with some popular
  randomized P2P networks R-Chord, R-Hypercube,
  Skip Lists

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The NoN-Greedy Algorithm
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The NoN-Greedy Algorithm

• Step (2) is implemented by putting all z in a
  search tree. Search time O(log(k2))
• Klogn gt Search time O(loglogn)

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Greedy vs NoN-Greedy

• 224 nodes
• 150 executions for each size
• 34 improvement

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The NoN-Greedy Algorithm

• Phase1 the message is sent to a neighbor whose
  neighbor is close to the target
• Phase2-greedy step the message moves to the
  neighbor of the neighbor

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Fault Tolerance Optimistic Scenario

• A node knows if its lists are updated
• If not updated performs a greedy step
• P(NoN) ½
• P(Greedy) ½

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Fault Tolerance Pessimistic Scenario

• Node is unaware that its list are up-to-date
• With probability ½ the edge (w,z) no longer
  exists
• i) w performs a greedy step
• ii) w performs a NoN step

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NoN - Chord

• Make Chord resemble the Small World
• Each node x is connected to logn nodes y0,y1,y2
• yi is a random point in x2i, x2i1
• Path length O(logn/loglogn)