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
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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 2006Genova, 27/28 ottobre
Università di Salerno - GL7
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
OutlineOutline
• 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
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
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
Overlay networkOverlay 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”
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
Randomized NetworksRandomized 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)
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
t
NoN Greedy routing NoN Greedy routing
s
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 RoutingNeighbor of Neighbor Routing
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
• Consider a deterministic network U having n nodes and degree
• Let j1, j2, . . . , 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+ji 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+ji+(ji+1− ji)H(v) mod n (where j+1 = n)
H-Networks (2005)H-Networks (2005)
jiiji+1i+1
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
0
1
2
3
4
5
6
7
8
9
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
log N
hops
Chord
R-ChordGreedyR-Chord NoN
R-Chord SP
H-ChordGreedyH-Chord NoN
H-Chord SP
ChordChordLower is
better
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
Overlay Network with classOverlay 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?
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 classOverlay Network with class
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
• Let U be uniform overlay network having N nodes and degree .
• Let j1, j2, . . . , j all the jumps of D (ordered by their size).
• Let c be any given positive integer in the interval [1, 2m] 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 Hc-U is obtained from U as follows:
• For each i = 0,…, -1, node v is connected by an edge to the nodes
v + ji + cv(ji+1- ji),
where cv = cH(v).
Overlay Network with classOverlay Network with class
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
• TheoremTheorem Let 1 < c Let 1 < c log n/ log log n, the average path log n/ log log n, the average path
length is O(loglength is O(logcc n) hops for the NoN Greedy n) hops for the NoN Greedy algorithm on Halgorithm on Hcc-Chord with n = 2-Chord with n = 2mm peers peers
• CorollaryCorollary When c = log n/ log log n, the average path When c = log n/ log log n, the average path
length islength is
O(log n/ log log n) hops and the number of hops O(log n/ log log n) hops and the number of hops for the completion of the join operation are for the completion of the join operation are O(log n log log n) (w.h.p.) O(log n log log n) (w.h.p.)
n<2m
Chernoff Bound
Overlay Network with classOverlay Network with class
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
• TheoremTheorem Both the diameter and the average Both the diameter and the average
(shortest) path length of an (shortest) path length of an Hc-Network with degree O(log n) and c classes are with degree O(log n) and c classes are (log(logcc n) n)
Hc-Chord is
asymptotically optimal
Overlay Network with classOverlay Network with class
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
• Extension to other networks:• Symphony −> Hc-Symphony
• Hypercube −> Hc-Hypercube
• Small-world -> …
How Much Independent Should
Individual Contacts be to Form a Small-
World?
Overlay Network with classOverlay Network with class
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
Simulation ResultsSimulation Results
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
• Hc-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 classOverlay Network with class
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006
ReferencesReferences• 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)
Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006