1. Searching via Your Neighbor’s Neighbor: The Power of Lookahead in P2P Networks Moni Naor Udi Wieder The Weizmann Institute of Science Gurmeet Manku Stanford

2. The Small World Phenomena a very brief history Folklore – People are connected via short chains – The graph of social networks has small diameter. Barabasi: belief may have originated from a story by Frigyes Karinthy, 1929 Quantitative approach initiated by Milgram in the 1960 ’ s - “ The six degrees of separation ”. Mathematical modeling: Model a social network by some distribution on graphs. A precursor of P2P – need to locate a resource in a ‘ natural ’ network based on partial information. P2P = Peer-to-Peer = a highly dynamic network

3. Routing in a Small World Common question: do short paths exist? Kleinberg ’ s algorithmic question: assuming short paths exist, how do people find them?

4. Modeling Small Worlds Kleinberg ’ s 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.  Naturally generalizes to k long range links (Symphony [MBR03],[ADS02].).  Naturally generalizes to any dimension. Captures the intuitive notion that people know people who are close to them.

5. Modeling Small Worlds Small World Percolation: People  points on a two dimensional grid. Grid edges (short range). Each edge appears independently with probability = inverse of its distance squared.  Degree of each node.  Originates from long range percolation model. Shares structural properties with some popular randomized P2P networks: R-Chord, R-Hypercube, Skip Lists … £ ( l ogn )

6. Routing in Small Worlds Greedy algorithm: move to the node that minimizes the L 1 distance to the target. SchemeDegreeGreedy – path length Kleinberg ’ s Model P2P - [MBR03],[ADS02] Percolation Small World, R-Chord, R-Hypercube Skip Lists – Skip Nets [AS02],[HDJ+03] £ ( l ogn ) £ ( l ogn ) O ( l ogn ) O ( l ogn ) £ ( l og 2 n k ) k · l ogn

7. Properties of Greedy Simple – to understand and to implement. Local – If source and target are close, the path remains within a small area. In some cases – (Hypercube, Chord) – the best we can do. Not optimal with respect to the degree. SchemeDegreeGreedy – path length Kleinberg ’ s Model [MBR03],[ADS02] Percolation Small World, R-Chord, R-Hypercube Skip Lists – Skip Nets [AS02],[HDJ+03] Can Greedy Routing be shortened? Without compromising the good properties £ ( l ogn ) £ ( l og 2 n k ) £ ( l ogn ) O ( l ogn ) O ( l ogn ) k · l ogn

8. 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). Let w 1, w 2, … w k be the neighbors of current node u For each w i find z i, the closet neighbor to target t Let j be such that z j is the closest to target t Route the message from u via w j to z j Effectively it is Greedy routing on the squared graph. The first hop may not be a greedy choice. Previous incarnations of the approach: Coppersmith, Gamarnik and Sviridenko [2002]: proved an upper bound on the diameter of a small world graph.  No routing algorithm Manku, Bawa and Ragahavan [2003]: a heuristic routing algorithm in ‘ Symphony ’ - a Small-World like P2P network.

9. What can we show about Non Greedy PSW, R-Chord, R-Hypercube are degree optimal w.h.p. Skip Lists – degree optimal on expectation. Kleinberg ’ s model and P2P variations – improved. Lower bounds for algorithms based on neighbor lists only (Greedy is a special case). SchemeDegreeGreedy – path length NoN Greedy – path length Kleinberg ’ s Model P2P - [MBR03],[ADS02] Percolation Small World, R-Chord, R-Hypercube Skip Lists – Skip Nets [AS02],[HDJ+03] £ ( l og n l og l og n ) £ ( l og n l og l og n ) k £ ( l ogn ) £ ( l ogn ) £ ( l ogn ) £ ( l ogn ) £ ( l og 2 n k ) £ ( l og 2 n k l og k )

10. Degree Optimal P2P Routing Different routing schemes Viceroy [MNR02] : 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 Recent construction [AM] fixes that. Even if target and source are close in label space message might be routed away No (natural) prefix search Random keys are necessary.

11. Skip – Graphs [AS02],[HDJ+03] Each node (resource) has a name. Nodes are arranged on a line sorted by name. 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. Allows prefix search. 011100 11 111 0000 010 a b cf e d

12. Routing in Skip – Graphs Greedy Routing – use longest edge possible. Path length is  (log n) w.h.p. The NoN algorithm optimizes over two hops. 011100 11 111 0000 010 Theorem: Using the NoN algorithm, the expected path length of any lookup is.

13. Call a NoN 2-hop successful if it reduces the distance from d to. Need succesful 2-hops to get to distance 1. From Lemma, this would take in expectation. Skip Graphs – degree optimality d 0 X - # of two hop paths between d and D - the event a message reached the node d. Lemma: Prob O ( l ogn = l og l ogn ) O ( l ogn = l og l ogn ) Sufficiency of lemma: d = l og d [ d l og d ; 0 ] d = l og d [( X > 0 ) j D ] ¸ 1 2

14. A i, j - There exists an edge between i, j. Lemma: X - # of two hop paths between d and Want to show Prob. Ignore dependence on D. c 1 j i ¡ j j · P r [ A i ; j ] · c 2 j i ¡ j j Proof: For prefix of length k the probability of an edge is: Let k be log (| i - j |). 2 ¡ k ¢ ( 1 ¡ 2 ¡ k ) j i ¡ j j ¡ 1 Skip Graphs – degree optimality 0 d = l og d d [ d l og d ; 0 ] E [ X ] ¸ d l og d ¢ n X i = 1 c 1 i ( n ¡ i ) ¸ 5 Choice of constants [( X > 0 ) j D ] ¸ 1 2

15. 0 d ijxy Which implies: var [ X ] · E [ X ] + 1 2 E 2 [ X ] P r [ X = 0 ] · E [ X ] + 1 2 E 2 [ X ] E 2 [ X ] · 0 : 7 A i, j - There exists an edge between i, j. X - # of two hop paths between d and Skip Graphs – degree optimality [ d l og d ; 0 ] Careful calculation: deal with dependencies cov [ A d ; i ; x ; A d ; j ; y ] · 1 2 P r [ A d ; i ; x ] ¢ P r [ A d ; j ; y ]

16. The Cost/Performance of NoN Cost of Neighbor of Neighbor lists: Memory: O(log 2 n) - marginal. 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. Reduce communication by piggybacking updates on top of the maintenance protocol. Lazy updates: Updates occur only when communication load is low – supported by simulations. Networks of size 2 17 show 30-40% improvement

17. Simulation Results Small World - one dimensionSkip Graphs

18. Simulation Results 2-dimensional small world 1-dimensional Small World each edge fails with probability 1/2

19. A Case for Randomized Topology Average diameter of hypercube is. Average diameter of ‘ perfect ’ skip graph is. Average diameter of Chord is. Conclusion – The randomization of edges reduces the average path lengths. Common design rule – reduce randomization in topology. The long edges are just in the right density, so that NoN finds them without increasing the degree. Other advantages: Security, fault tolerance …. ­ ( l ogn ) ­ ( l ogn ) ­ ( l ogn )

20. Do People Use the NoN Algorithm? Experiment based on email [DRW03] About 25% sent the mail because: The recipient traveled to target ’ s geographical region. The recipient ’ s family originates from target ’ s geographical region.

21. Lower Bounds – A Probing Model Goal: Find a path between two nodes in an unknown graph. The algorithm may probe a node. If the probing reveals a neighborhood of radius k, then the algorithm is k – local. A lower bound on the number of probes implies a lower bound on the sequential running time of routing. The Greedy algorithm is 1-local. NoN is 2-local. Theorem: Every 1-local algorithm requires probes w.h.p, both in small worlds and in skip graphs. ­ ( l ogn ) Conclusion: Some extra information is necessary.

22. Greedy algorithm dominates1-local algorithms. Let A be a 1-local algorithm. Denote by the r.v. counting the number of probes it takes A (Greedy) to find a path between 0 and d. P r [ g d · k ] ¸ P r [ f d · k ] Lemma: For all k;d>0 ; If a probe finds node i, reveal all edges (prefixes) in [ d;i ]. Only increases. The ‘ best chance ’ of getting close to 0 is by probing the node closest to 0. 0d i revealed f d ( g d ) P r [ f d · k ]

23. Lower Bounds on Greedy Partition the nodes to balls B 0, B 1, …, B log d Define X i – the indicator of the event : “ Greedy probed a node in B i ” The probe complexity is at least. d 0 B0B0 B3B3 B2B2 B1B1 l og n X i = 0 X i Lemma: Both for skip graphs and small worlds, there exists a constant c such that: P r [ X i = 1 j X 0 = 1 ; X 1 = 1 ;:::; X i ¡ 1 = 1 ] ¸ c Azuma ’ s inequality: P r [ P X i · 1 2 c l ogn ] · n ¡ ² E [ P X i ] ¸ c l ogn

24. Lower Bounds on Greedy X i depends only on the last ball visited. When a ball is visited – skip to the last node. Assume X 0 =1, X 1 =0. The probability the dangling edge would skip over B 2 is at most. Lemma: Both for skip graphs and small worlds, there exists a constant c such that: P r [ X i = 1 j X 0 = 1 ; X 1 = 1 ;:::; X i ¡ 1 = 1 ] ¸ c d 0 B0B0 B3B3 B2B2 B1B1 1 ¡ c

25. Conclusions NoN Greedy seems like an almost free tweak that is a good idea in many settings. Do not be perfect (all the time) – randomization helps. What is more important Prefix search. Easy and ‘ natural ’ degree optimality. Better understanding of the ‘ small world ’ phenomena.