1. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Degree–Optimal Routing for P2P Systems Giovanni Chiola, Gennaro Cordasco, Luisa Gargano, Mikael Hammar, Alberto Negro, and Vittorio Scarano Meeting WEBMINDS 2006 Genova, 27/28 ottobre Università di Salerno - GL7

2. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Outline Peer-to-Peer Distribute Hash Table Overlay Network Uniform Networks vs. Randomized Networks Greedy Routing vs. Neighbor of Neighbor Routing Our Proposals F-Chord (2004) ----> Generalize Base-k (2006) H-Networks (2005) ----> Overlay network with class (2006) Conclusions

3. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 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 Distributed Hash Table (DHT)

4. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Overlay network [I. Stoica] “An overlay network is a virtual network of nodes and logical links that is built on top of an existing network with the purpose to implement a network service that is not available in the existing network. In particular an overlay network: adds an additional layer of indirection/virtualization changes properties in one or more areas of underlying network ”

5. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Overlay Network: Performance 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 Congestion Clustering

6. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Chord Chord uses a one-dimensional circular key space (ring) of N=2 m identifiers The node responsible for the key is the node whose identifier most closely follows the key Chord maintains two sets of neighbors: A successor list of r nodes that immediately follows it in the key space A finger list of m = log N nodes spaced exponentially around the key space Routing consists in forwarding to the node closest, but not past, the key Performance: Diameter: log N (O(log n) whp) where n denote the number of nodes present in the network Routing table size: log N (O(log n) whp) Average path length: ½ log N Routing correctness Routing efficiency

7. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 An Example: Chord IDResp. 8+1=914 8+2=1114 8+8=1621 8+16=2424 8+32=4042 8+4=1214 m=6 indiceNod o 114 221 432 538 642 324 Successors Predecessor Nodo 1

8. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 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 v, v is connected to u iff v+w is connected to u+w (i.e. : all the connection are symmetric). Advantages  Easy to implement  Greedy algorithm  Simple – to understand and implement  Local – routing occurs inside the portion of ring that is delimited by source and destination  In some cases – (Hypercube, Chord) – the best we can do  No node congestion  Quick Bootstrap Drawback –Less powerful (De Bruijn Graph and Neighbor of Neighbor Greedy routing are more powerful)

9. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 t Greedy Routing: move to the neighbor that minimizes the distance to the target. s Greedy Routing

10. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Fib-Chord (2004) Formally Let N  (Fib(m-1), Fib(m)]. The scheme uses m-2 jumps of size Fib(i) for i = 2,3, …, m-1 Fib-Chord Diameter : Degree : 123 5 8 13 21 34 55 89 Fib-Chord

11. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 F-Chord(  ) (2004) F-Chord(  ) Fib(2i), for i = 1,2, …,  (1-  )(m-2)  Fib(i), for i = 2  (1-  )(m-2)  +2, …, m-1 F-Chord(  ) uses  (m-2)  jumps 2 5 13 34 89 Fib-Chord 1 3 8 21 55 even index jumps all jumps  [1/2,1]

12. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Property of F-Chord Diameter: Theorem [CGHSN04] For any value of , the diameter of F-Chord(  ) is  m/2   0.72021 log n Average Path Length: Theorem [CGHSN04] The average path length of the F-Chord(  ) scheme is bounded by 0.39812 log n + (1-  )0.24805 log n Degree: F-Chord(  ) use  (m-2)  jumps

13. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006  hops x log n F-Chord (2004) Lower is better

14. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Generalized Base-k We generalize the property of F-Chord(1/2) to bases greater than 2 Let R(d) the size of the greatest ring with diameter d (using Greedy routing). Generalized Base-k: J(0)=1, R(0)=1, for each l  0 for each i=1,…,k-1 J((k-1)l+i)=J((k-1)l)+iR(l) R(l+1)= J((k-1)l)+kR(l)

15. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Generalized Base-k: An Example Generalized Base-k: J(0)=1, R(0)=1, for each l  0 for each i=1,…,k-1 J((k-1)l+i)=J((k-1)l)+iR(l) R(l+1)= J((k-1)l)+kR(l) Example k=3 and n=R(2)=15 J(0)=1, R(0)=1  J(1)=2, J(2)=3, R(1)=4  J(3)=7, J(4)=11, Generalized Base-3 1 2 3 7 11

16. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Generalized Base-k For k=2 the sequence of jump is 1,2,5,13,34,… (F-Chord(1/2) jumps) Generalized Base-2 = F-Chord(1/2) Generalized Base-k is the proper generalization of F- Chord(1/2) to base k 12 5 13 34 89

17. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Property of Generalized Base-k Diameter: Let d the diameter of Generalized Base-k scheme with n=R(d) nodes, d<log k+((k-1)/k) (n+1) Average Path Length: The average path length of Generalized Base-k scheme with n=R(d) nodes is bounded by ((k+2)/(k+3))d Degree: Let  the node degree of Generalized Base-k scheme with n=R(d) nodes,  = (k-1)d+1

18. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Simulation Results: Degree B-k: Base-k G-k: Gen. Base-k Fib: F-Chord F-k: Ext. Fib. G-(k+1)  B-k B-3G-3B-2

19. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Simulation Results: APL B-k: Base-k G-k: Gen. Base-k Fib: F-Chord F-k: Ext. Fib. G-k  B-k for k>2

20. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Randomized Networks Networks whose nodes are connected to each other randomly  Small Average Path Length (APL)  Small Clustering  The global structure of the network cannot be a global information  Routing is not local  Optimal trade-off can be reached only through a estimation of the number of participants in the system  The optimal trade-off will only occur for a range of n  ?Quick Bootstrap?  ?Congestion? Examples: Viceroy Network Degree: O(1) APL: O(log N) Mariposa Degree: O(  ) APL: O(log  N)

21. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 t NoN Greedy routing s

22. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Benefits Efficient Local Routing Estimation of n not required Drawbacks Neighbor lists should be maintained No quick bootstrap Neighbor of Neighbor Routing

23. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Consider a deterministic network U having n nodes and degree  Let j 1, j 2,..., j  all the jumps of U (ordered by their size). (i.e.: For each node v and for each i=1,…, , v is connected with v+j i mod n) Let H() denote a good hash function, that maps an id on the interval [0,1) The network H-U is obtained from U as follows: For each i=1,…, , node v is connected to the nodes v+j i +(j i+1 − j i )H(v) mod n (where j  +1 = n) H-Networks (2005) ijiiji i+1 j i+1

24. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Chord Lower is better

25. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 H-F-Chord vs. H-Chord Lower is better

26. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Overlay Network with class Uniform Networks Greedy routing is optimal: log N hops with log N fingers per node Greedy routing is easy to program Uniformity allows quick bootstrap for new peers joining an already established DHT Congstion-free and high Clustered Non-Uniform Networks Greedy routing is not optimal: other strategies such as “Neighbor of Neighbor” (NoN) yield log N / log log N hops with log N degree per node NoN routing is similar to Greedy routing in terms of implementation Non-Uniformity prevents quick bootstrap for new peers joining an already established DHT ?Congestion and Clustering?

27. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Split nodes into a c classes Routing tables are uniform within the same class Different classes follow different patterns Each peer chooses its class pseudo-randomly (based on Hash digest of its identifier) Quick bootstrap is obtained when finding a neighbor belonging to the same class Overlay Network with class

28. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Let U be uniform overlay network having N nodes and degree . Let j 1, j 2,..., j  all the jumps of D (ordered by their size). Let c be any given positive integer in the interval [1, 2 m ] and H() be a cryptographic hash function that maps an id on the interval [0, 1). Consider c real numbers 0, 1,…, c-1 in the interval [0, 1) such that 0 = 0 < 1 <…< c-1 < 1 The network H c -U is obtained from U as follows: For each i = 0,…,  -1, node v is connected by an edge to the nodes v +  j i + c v (j i+1 - j i ) , where c v =  cH(v) . Overlay Network with class

29. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 TheoremTheorem Let 1 < c  log n/ log log n, the average path length is O(log c n) hops for the NoN Greedy algorithm on H c - Chord with n = 2 m peers Let 1 < c  log n/ log log n, the average path length is O(log c n) hops for the NoN Greedy algorithm on H c - Chord with n = 2 m peers CorollaryCorollary When c = log n/ log log n, the average path length is When c = log n/ log log n, the average path length is O(log n/ log log n) hops and the number of hops for the completion of the join operation are O(log n log log n) (w.h.p.) O(log n/ log log n) hops and the number of hops for the completion of the join operation are O(log n log log n) (w.h.p.) n<2 m Chernoff Bound Overlay Network with class

30. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 TheoremTheorem Both the diameter and the average (shortest) path length of an with degree O(log n) and c classes are  (log c n) Both the diameter and the average (shortest) path length of an H c -Network with degree O(log n) and c classes are  (log c n) H c -Chord is asymptotically optimal Overlay Network with class

31. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Extension to other networks: Symphony −> H c -Symphony Hypercube −> H c -Hypercube Small-world -> … How Much Independent Should Individual Contacts be to Form a Small- World? Overlay Network with class

32. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Simulation Results

33. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 H c -Networks: NoN routing on few Classes gives almost the same performance as NoN routing on truly randomized systems Few Classes guarantee a quick bootstrap with very high probability 2 Classes are sufficient for DHTs with up to hundred thousand peers 4 Classes are sufficient for DHTs with billions of peers The result applies to other network topologies Overlay Network with class

34. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006References G. Cordasco, L. Gargano, M.Hammar, A. Negro and V. Scarano, “F- Chord: Improved Uniform Routing on Chord”. In Proc. of 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004. G. Cordasco, L. Gargano, M.Hammar and V. Scarano, “Degree- Optimal Deterministic Routing for P2P Systems”. In Proc. of 10th IEEE Symposium on computers and communications (ISCC 2005) La Manga del Mar Menor, Cartagena, SPAIN June 27-30, 2005. (see also Twenty-Third Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2004) (Brief Announcement) St. John's, Newfoundland, Canada July 25-28, 2004.) G. Cordasco, L. Gargano, M.Hammar, A. Negro and V. Scarano, “Non- uniform deterministic routing on F-Chord(α)”. In Proc. of First International Workshop on Hot Topics in Peer-to-Peer Systems (Hot-P2P04) Volendam (The Netherland) October 08, 2004. G. Chiola, G. Cordasco, L. Gargano, A. Negro and V. Scarano, “Overlay networks with class”. In Proc. of 8th International Symposium on Parallel Architectures, Algorithms, and Networks (I-SPAN 2005) Las Vegas, Nevada, USA. December 7-9, 2005. G. Chiola, G. Cordasco, L. Gargano, A. Negro and V. Scarano, “Optimizing the finger table in Chord-like DHTs”. In Proc. of Third International Workshop on Hot Topics in Peer-to-Peer Systems (HotP2P 2006) (Co-Located with IPDPS 2006) Rhodes Island, Greece. April 29, 2006. G. Cordasco and L. Gargano, “How Much Independent Should Individual Contacts be to Form a Small-World?”. In Proc. of The 17th International Symposium on Algorithms and Computation (ISAAC 2006) December 18-20, 2006 - Kolkata, India. (To Appear)

35. Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Thanks for your attention Questions?