1. Università di Salerno GL7 Distributed Adaptive Directory (DAD) F-Chord: Improved Uniform Routing on Chord Meeting Firb - Genova, 5-6 luglio 2004

2. Distributed Adaptive Directory (DAD) Sistema per il bookmark cooperativo Ambiente peer-to-peer permette di condividere i bookmark con gli utenti connessi Sistema adattivo DAD offre suggerimenti sulla base dei bookmark inseriti Sistema dinamico gli utenti possono fornire feedback sui bookmark di altri utenti modificando il peso di bookmark ed utenti Meeting Firb - Genova, 5-6 luglio 2004

3. Distributed Adaptive Directory (DAD) Meeting Firb - Genova, 5-6 luglio 2004 DAD CHILD Adaptivity Bookmark sharing Chord Bootstrap Authentication Kleinberg User Scores DHT dump MOM Graphical user interface Our extension to Kleinberg

4. Distributed Adaptive Directory (DAD) Meeting Firb - Genova, 5-6 luglio 2004 Suggeriti dal sistema Inseriti (o copiati) dallutente Trovati nel sistema (su un altro utente) Numero di occorrenze

5. F-Chord: Improved Uniform Routing on Chord Gennaro Cordasco, Luisa Gargano, Mikael Hammar, Alberto Negro, and Vittorio Scarano Summary Motivation to our work Peer to Peer Scalability Distributed Hash table F-Chord family The Idea Definition Our result Conclusions and Open Questions Meeting Firb - Genova, 5-6 luglio 2004

6. Motivation Peer to Peer Systems (P2P) File sharing system; File storage system; Distributed file system; Redundant storage; Availability; Performance; Permanence; Anonymity; Scalability Meeting Firb - Genova, 5-6 luglio 2004

7. 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 request to this node Meeting Firb - Genova, 5-6 luglio 2004

8. DHT performance metrics Three performance metric: 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 Meeting Firb - Genova, 5-6 luglio 2004

9. 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 No node congestion Drawback Less powerful (De Bruijn Graph and Neighbor of Neighbor Greedy routing are more powerful) Meeting Firb - Genova, 5-6 luglio 2004

10. 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 Meeting Firb - Genova, 5-6 luglio 2004 Totally connected graph

11. An Example: Chord Chord uses a one-dimensional circular key space (ring) of N=2 b 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 k nodes that immediately follows it in the key space A finger list of b = 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 Meeting Firb - Genova, 5-6 luglio 2004

12. An Example: Chord Meeting Firb - Genova, 5-6 luglio 2004 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

13. Previous Results The network diameter lower bound is when the routing table size is no more than Xu, Kumar, Yu (2003): The diameter lower bound for the network is if the degree is when we use an uniform routing algorithm. In particular, the diameter lower bound for the network is if the degree is when we use an uniform routing algorithm; Show an uniform routing algorithm with degree and diameter equals to Average path length is 0,6135 log N Meeting Firb - Genova, 5-6 luglio 2004

14. The Idea Meeting Firb - Genova, 5-6 luglio 2004 1-2x=0 x=1/21-x-x 2 =0 x=1/ x x2x2 S 1 =1 S i =(1/2) (i-1) … S d1/n d log 2 n S 1 =1 (1/ ) 2(i-1) S i (1/ ) (i-1) … S d1/n d log n Chord x

15. The Idea(2) Meeting Firb - Genova, 5-6 luglio 2004 We can use only the jumps x i s.t. i 1 mod 2 (x, x 3, x 5, x 7, …) 1 x2x2 x xx3x3 x3x3 x2x2 x2x2 x x2x2 x3x3 x2x2 d = (1/2)log n = (1/2)log n

16. The Idea(3) We construct an uniform routing algorithm using a novel number-theoretical technique, in particular our scheme is based on the Fibonacci number system. Fib(i) denote the i-th Fibonacci number. We recall that where is the golden ratio and [ ] represents the nearest integer function 12 4 8 16 32 64 Chord Meeting Firb - Genova, 5-6 luglio 2004

17. Fib-Chord 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 Meeting Firb - Genova, 5-6 luglio 2004

18. F-Chord( ) F a -Chord( ) Fib(2i), for i = 1,2, …, (1- )(m-2) Fib(i), for i = 2 (1- )(m-2) +2, …, m-1 F b -Chord( ) Fib(i), for i = 2, …,m-2 (1- )(m-2) Fib(2i), for i = (m-2 (1- )(m-2) )/2 +1, …, (m-1)/2 F a -Chord( ) and F b -Chord( ) use (m-2) jumps 2 5 13 34 89 Fib-Chord 1 3 8 21 55 even jumps all jumps even jumps [1/2,1] Meeting Firb - Genova, 5-6 luglio 2004

19. Property of F-Chord Degree: F-Chord( ) use (m-2) jumps Diameter: Theorem For any value of, the diameter of F-Chord( ) is m/2 0.72021 log N Average Path Length: Theorem The average path length of the F-Chord( ) scheme is bounded by 0.39812 log N + (1- )0.24805 log N Meeting Firb - Genova, 5-6 luglio 2004

20. F-Chord(1/2) Fib-Chord Diameter : Degree : F-Chord(1/2) = F a -Chord(1/2) = F b -Chord(1/2) Diameter : Degree : 2 5 13 34 89 Fib-Chord 1 3 8 21 55 F-Chord(1/2) Meeting Firb - Genova, 5-6 luglio 2004

21. The Lower Bound We provide a tradeoff of 1.44042 log N on the sum of the degree and the diameter in any P2P network using uniform routing on N identifiers. Theorem 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) F-Chord(1/2) is optimal Meeting Firb - Genova, 5-6 luglio 2004

22. Average path length Fib-Chord: 0.39812 log N F-Chord(1/2): 0.522145 log N Theorem For each [0.58929,0.69424] the F-Chord( ) schemes improve on Chord in all parameters (number of jumps, diameter, and average path length) Chord is better Meeting Firb - Genova, 5-6 luglio 2004

23. hops x log n Graphical results Meeting Firb - Genova, 5-6 luglio 2004 Lower is better

24. Congestion Our routing scheme is uniform, hence there is no node congestion [Xu, Kumar, Yu (2003)]. Theorem For each [1/2,1] the F-Chord( ) schemes is 1.38197-edge congestion free. A routing scheme is said to be c-edge congestion free if no edge is handling more than c times the average traffic per node Meeting Firb - Genova, 5-6 luglio 2004

25. Conclusions and Open Questions An optimal uniform routing algorithm with respect to diameter and degree A family of simple algorithms that improve uniform routing on Chord with respect to diameter, average path length and degree Open problem: Find a lower bound for the average path length on uniform routing algorithm Meeting Firb - Genova, 5-6 luglio 2004

26. Università di Salerno Dipartimento di Informatica ed Applicazioni R.M. Capocelli, 84081, Baronissi (SA) cordasco@dia.unisa.it Meeting Firb - Genova, 5-6 luglio 2004 GRAZIE