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F-Chord: Improved Uniform Routing on Chord

F-Chord: Improved Uniform Routing on Chord Gennaro Cordasco, Luisa Gargano, Mikael Hammar, Alberto Negro, and Vittorio Scarano. Dipartimento di Informatica ed Applicazioni ”R.M. Capocelli” Università di Salerno, 84081, Baronissi (SA) – Italy.

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F-Chord: Improved Uniform Routing on Chord

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  1. F-Chord: Improved Uniform Routing on Chord Gennaro Cordasco, Luisa Gargano, Mikael Hammar, Alberto Negro, and Vittorio Scarano Dipartimento di Informatica ed Applicazioni ”R.M. Capocelli” Università di Salerno, 84081, Baronissi (SA) – Italy 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  2. Summary • Motivation to our work • Peer to Peer • Scalability • Distributed Hash table • Previous results • F-Chord family • The Idea • Definition • Our result • Conclusions and Open Questions 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  3. Motivation • Peer to Peer Systems (P2P) • File sharing system; • File storage system; • Distributed file system; • Redundant storage; • Availability; • Performance; • Permanence; • Anonymity; • Scalability 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  4. 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 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  5. 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 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  6. 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) 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  7. Asymptotic tradeoff curve Diameter Ring Uniform Routing algorithm N -1 Chord et al. Totally connected graph O(log N) O(log N) Non-Uniform Routing algorithm 1 1 O(log N) N -1 Routing table size 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  8. An Example: Chord • Chord uses a one-dimensional circular key space (ring) of N=2b 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 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  9. ID index Resp. Node 8+1=9 1 14 14 2 8+2=11 21 14 3 8+4=12 14 24 4 8+8=16 21 32 8+16=24 5 24 38 8+32=40 6 42 42 An Example: Chord Successors Predecessor Node 1 b=6 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  10. 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  11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  11. The Idea Chord • 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 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  12. Fib-Chord 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 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  13. 2 5 13 34 89 F-Chord() Fib-Chord • Fa-Chord() Fib(2i), for i = 1,2, …,(1-)(m-2) Fib(i), for i = 2 (1-)(m-2) +2, …, m-1 • Fb-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 • Fa-Chord() and Fb-Chord() use (m-2) jumps 1 3 8 21 55 [1/2,1] even jumps even jumps all jumps all jumps 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  14. 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 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  15. 2 5 13 34 89 F-Chord(1/2) Fib-Chord F-Chord(1/2) • Fib-Chord • Diameter :  • Degree :  • F-Chord(1/2) = Fa-Chord(1/2) = Fb-Chord(1/2) • Diameter :  • Degree :  1 3 8 21 55 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  16. 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 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  17. Average path lenght Chord is better • 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) 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  18. 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 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

  19. 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 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004

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