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Università di Salerno GL7 Distributed Adaptive Directory (DAD)

Università di Salerno GL7 Distributed Adaptive Directory (DAD) F-Chord: Improved Uniform Routing on Chord. Meeting Firb - Genova, 5-6 luglio 2004. Distributed Adaptive Directory (DAD). Sistema per il bookmark cooperativo Ambiente peer-to-peer

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Università di Salerno GL7 Distributed Adaptive Directory (DAD)

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

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

  5. F-Chord: Improved Uniform Routing on ChordGennaro 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. Ring Chord et al. Totally connected graph Asymptotic tradeoff curve Diameter Uniform Routing algorithm N -1 O(log N) Non-Uniform Routing algorithm 1 1 O(log N) N -1 Routing table size Meeting Firb - Genova, 5-6 luglio 2004

  11. 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 Meeting Firb - Genova, 5-6 luglio 2004

  12. ID indice Resp. Nodo 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 Nodo 1 m=6 Meeting Firb - Genova, 5-6 luglio 2004

  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. Chord x2 x x The Idea 1-2x=0  x=1/2 1-x-x2=0  x=1/ S1=1 Si=(1/2)(i-1) … Sd≤1/n  d ≤ log2 n S1=1 (1/)2(i-1) ≤ Si ≤ (1/ )(i-1) … Sd≤1/n  d ≤ log n Meeting Firb - Genova, 5-6 luglio 2004

  15.  = (1/2)log n The Idea(2) • We can use only the jumps xi s.t. i  1 mod 2 (x, x3, x5, x7, …) x2 x2 x 1 x3 x x3 d = (1/2)log n x2 x x2 x3 x2 Meeting Firb - Genova, 5-6 luglio 2004

  16. The Idea(3) 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 Meeting Firb - Genova, 5-6 luglio 2004

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

  18. 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 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. 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 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 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) Meeting Firb - Genova, 5-6 luglio 2004

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

  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. GRAZIE 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

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