1. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Non-uniform deterministic routing on F-Chord(  ) Gennaro Cordasco, Luisa Gargano, Mikael Hammar, Alberto Negro, Vittorio Scarano Dipartimento di Informatica e Applicazioni “R.M. Capocelli” Università di Salerno, 84081, Baronissi (SA) - Italy

2. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004Outline  P2P e DHT  DHT performance metrics  Uniform Routing Algorithm vs Non-uniform Routing Algorithm  F-Chord  F-Chord [CGHNS04]  Neighbor of Neighbor routing algorithm  Neighbor of Neighbor routing algorithm [MNW04]  Our Proposal  H-schemes  H-F-Chord

3. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Distributed Hash Table (DHT)  Distributed version of a hash table data structure  Stores (key, value) pairs  The key is like a filename  The value can be file contents  Goal: Efficiently insert/lookup/delete (key, value) pairs  Each peer stores a subset of (key, value) pairs in the system  Core operation: Find node responsible for a key  Map key to node  Efficiently route insert/lookup/delete requests to this node

4. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 DHT performance metrics  Three performance metrics:  Routing table size (degree)  Storage cost  Measure the cost of self-stabilization for adapting to node joins/leaves  Diameter and Average path length  Time cost  Fault tolerance

5. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Uniform Routing Algorithm  We consider a ring of N identifiers labeled from 0 to N-1  A routing algorithm is uniform if for each identifier x, x is connected to y iff x+z is connected to y+z (i.e. : all the connection are symmetric).  Advantages  Easy to implement  Greedy algorithm is optimal  Simple – to understand and implement  Local – routing occurs inside the portion of ring that is delimited by source and destination  No node congestion  Drawback  Less powerful (De Bruijn Graph and Neighbor of Neighbor Greedy routing are more powerful) Routing is not greedy

6. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Asymptotic tradeoff curve Routing table size 1 1 N -1 O(log N) Chord et al. Ring O(log N) Diameter Uniform Routing algorithm Non-Uniform Routing algorithm Totally connected graph

7. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 F-Chord(  ) [CGHNS04] 2 5 13 34 89 F-Chord(1) 1 3 8 21 55 even jumps all jumps  F-Chord(  ) Fib(2i), for i = 1,2, …,  (1-  )(m-2)  Fib(i), for i = 2  (1-  )(m-2)  +2, …, m-1  Degree: F-Chord(  ) use  (m-2)  jumps  Diameter: For any value of , the diameter of F-Chord(  ) is  m/2   0.72021 log N  Average Path Length: The average path length of the F-Chord(  ) scheme is bounded by 0.39812 log N + (1-  )0.24805 log N+1 The average path length of the F-Chord(  ) scheme is bounded by 0.39812 log N + (1-  )0.24805 log N+1  [1/2,1]

8. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004  Let N( ,d) denote the maximum number of consecutive identifiers obtainable trough a uniform algorithm using up to  jumps (i.e. degree  ) and diameter d. For any  0, d  0, it holds that N( ,d)  Fib(  +d+1)  For each  [0.58929,0.69424] the F-Chord(  ) schemes improve on Chord in all parameters (number of jumps, diameter, and average path length) F-Chord(1/2) is optimal F-Chord(  ) [CGHNS04]

9. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004  hops x log n Graphical results Lower is better

10. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 The Small World Phenomena  The “six degree of separation” experiment S. Milgram [M67].  The sociological experiment relied on social networks to transmit a letter from a person to unfamiliar targets by passing the letter only via acquaintances.  Only a small number (around 6) of steps was needed.  Recent work [DRW03], shows that, in the first steps the message was forwarded to a person P by using a guess on who P knew or, in other words, on his/her neighbors.

11. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Small World Nodes  points in a two dimensional grid Grid edge  short range Each edge (x, y) appears independently with probability 1/d(x,y) 2 Degree of each node  (log N)

12. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004R-Schemes  R-Chord N=2 m [MNW04]  For each 0 ≤ i < m, let r(i) denote an integer chosen uniformly at random from the interval [0,2 i ), node x is connected by edges to the nodes x+2 i +r(i);  R-Hypercube [MNW04]  For each 0 ≤ i ≤ m, node x is connected with y where y is defined as follows: the top i-1 bits of y are identical to those of x. The i th is flipped. The remaining m - i bits are chosen uniformly at random. x2i2i 2 i+1

13. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Neighbor of Neighbor (NoN) Greedy Routing Neighbor of Neighbor (NoN) Greedy Routing [MNW04] Let d(x,y) be a metric for the nodes in the network. 1. Assume the message is currently at node u ≠ target. 2. Let N = {v 1, v 2, …, v k } be the neighbors of u. 3. For each 1 ≤ i ≤ k, let w i1, w i2, …, w ik be the neighbors of v i and let N'= { w ij  1 ≤ i, j ≤ k}. 4. Among these k 2 +k nodes, assume that z is the one closest to the target (with respect to metric d). 5. If z  N route the message from u to z else z = w ij, for some i and j, and we route the message from u via v i to z.

14. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 [MNW04]DegreeGreedy NoN Greedy R-Chord,R-Hypercube, Small World Percolation, Skip Graphs Symphony k < log N k < log N NoN-Greedy Routing

15. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Neighbor of Neighbor (NoN) Degree Cost of Neighbor of Neighbor lists: Memory: O(log 2 n) Maintenance: O(log n) must be updated Neighbor lists should be maintained (open connection, pinging, etc.) SMLKKDB03 “In practice, a Chord ring will never be in a stable state; instead, joins and departures will occur continuously, interleaved with the stabilization algorithm. The ring will not have time to stabilize before new changes happen.” [SMLKKDB03]

16. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004H-Schemes  We denote by j 1, j 2, …, j d all the jumps of our schemes (ordered by their size);  Let H() a good hash function that map an id on a sequence of m bits, for each 1 ≤ i ≤ d, node x is connected by edges to node x + j i +  (H(x)/2 m )*(j i+1 - j i )  x + j i +  (H(x)/2 m )*(j i+1 - j i )  [0,1) ijiiji i+1 j i+1

17. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004H-Schemes  Deterministic network  No additional information is transmitted nor stored:  Each node x, knowing y, can compute H(y) and then can estimate y’s neighbors.

18. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 H-SchemesDegreeGreedy NoN Greedy H-Chord,H-Hypercube, H-Symphony k < log N k < log N NoN-Greedy Routing

19. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 H-F-Chord(  ) Theorem 1: The average path length is O(log N/ log log N) hops for the NoN Greedy algorithm on H-F-Chord(  ) in a ring of size Fib(m) where the number of nodes alive is N < Fib(m). We show, by simulation, that H-F-Chord(  ) is more efficient than the corresponding F-Chord(  ) by a percentage that goes from 10% to 25% even for small N.

20. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 H-F-Chord(  ) – APL optimality Case 1: d=d(s,t)<O(2 log n / loglog n ) we can use greedy routing d=d(s,t)<O(2 log n / loglog n ) we can use greedy routing Case 2: d’=d loglog n / log n Let v  [s, t] and let jv i a jump of v Pr[  i st jv i  ] > loglog n / log n Pr[  i st jv i  I ] > loglog n / log n s t s t d’ I d

21. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 H-F-Chord(  ) – APL optimality s t d’ I Pr[s can reach I in two hops when d ≥O(2 log n / loglog n )] ≥ 1 – e -1 Hence after O(log d / loglog n) hops the distance is decreased to O(log n / loglog n) and we have reduced case 2 to case 1.

22. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Lower is better

23. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Lower is better

24. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Lower is better

25. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Lower is better

26. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004 Higher is better

27. Hot Topics in Peer-to-Peer Computing (HOT-P2P 2004) Volendam 08 October 2004Conclusions  H-F-Chord(  ):  Deterministic P2P network  Asymptotically optimal with respect to average path length and degree (No hidden constant)  Allows a trade-off between efficiency and maintenance  No overhead with respect to greedy routing system