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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Degree-Optimal Deterministic Routing for P2P Systems Meeting WEBMINDS 2005 Salerno, 20/21/22 giugno Università di Salerno - GL7 Dipartimento di Informatica e Applicazioni R.M. Capocelli Università di Salerno, 84081, Baronissi (SA) - Italy

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Outline P2P e DHTP2P e DHT DHT performance metricsDHT performance metrics Greedy Routing vs Non-Greedy RoutingGreedy Routing vs Non-Greedy Routing Neighbor of Neighbor Routing algorithmNeighbor of Neighbor Routing algorithm The Small World PhenomenaThe Small World Phenomena Our Proposal: H-NetworksOur Proposal: H-Networks Conclusions

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Distributed Hash Table (DHT) Distributed version of a hash table data structureDistributed version of a hash table data structure Stores (key, value) pairs Stores (key, value) pairs The key is like a filename The key is like a filename The value can be file contents The value can be file contents Goal: Efficiently insert/lookup/delete (key, value) pairsGoal: Efficiently insert/lookup/delete (key, value) pairs Each peer stores a subset of (key, value) pairs in the systemEach peer stores a subset of (key, value) pairs in the system Core operation: Find node responsible for a keyCore operation: Find node responsible for a key Map key to nodeMap key to node Efficiently route insert/lookup/delete requests to this nodeEfficiently route insert/lookup/delete requests to this node

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 DHT performance metrics Three performance metrics:Three performance metrics: Routing table size (degree)Routing table size (degree) Storage costStorage cost Measure the cost of self-stabilization for adapting to node joins/leavesMeasure the cost of self-stabilization for adapting to node joins/leaves Diameter and Average path lengthDiameter and Average path length Time costTime cost Fault toleranceFault tolerance

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 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 k 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 Chord

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 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

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 t Greedy routing: move to the neighbor that minimizes the distance to the target. s

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Greedy Routing in the Considered Networks Simple – to understand and to 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. Not optimal with respect to the degree.

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Greedy Routing in the Considered Networks Degree is (log n) Greedy routing needs (log n) hops. Lower bound is Ω(log n / log log n)

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Non Greedy Routing Viceroy Network Degree: O(1) Average Path Length: O(log N) De Bruijn graphs Degree: O(log N) Average Path Length: Routing is not local

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Neighbor of Neighbor (NoN) Greedy Routing Let d(x,y) be a metric for the nodes in the network. 1. Assume the message is currently at node u target. 2. Let N = {v 1, v 2, …, v k } be the neighbors of u. 3. For each 1 i k, let w i1, w i2, …, w ik be the neighbors of v i and let N'= { w ij 1 i, j k}. 4. Among these k 2 +k nodes, assume that z is the one closest to the target (with respect to metric d). 5. If z N route the message from u to z else z = w ij, for some i and j, and we route the message from u via v i to z. Manku, Naor, Wieder [2004] NoN-routing within O(log n / loglog n) hops in Small World Networks. Manku, Bawa, Ragahavan [2003]: a heuristic routing algorithm in Symphony – a Small World P2P network. Coppersmith, Gamarnik and Sviredenko [2002]: proved an upper bound on the diameter of a Small World graph.

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 s t Neighbor of Neighbor (NoN) Greedy Routing

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 The Small World Phenomena The six degree of separation experiment S. Milgram [M67]. The sociological experiment relied on social networks to transmit a letter from a person to unfamiliar targets by passing the letter only via acquaintances. Only a small number (around 6) of steps was needed. Problem: Locate a resource in a natural network based on partial information Question: How do people find short paths? Recent work [DRW03], shows that, in the first steps the message was forwarded to a person P by using a guess on who P knew or, in other words, on his/her neighbors.

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Small World Nodes points in a two dimensional grid Grid edge short range Each edge (x, y) appears independently with probability 1/d(x,y) 2 Degree of each node (log N)

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 R-Schemes R-Chord N=2 m [MNW04] For each 0 i < m, let r(i) denote an integer chosen uniformly at random from the interval [0,2 i ), node x is connected by edges to the nodes x+2 i +r(i); R-Hypercube [MNW04] For each 0 i m, node x is connected with y where y is defined as follows: the top i-1 bits of y are identical to those of x. The i th is flipped. The remaining m - i bits are chosen uniformly at random. x2i2i 2 i+1

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 [MNW04]DegreeGreedyNoN Greedy R-Chord, R-Hypercube, Skip Graphs Symphony k < log N NoN-Greedy Routing

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Neighbor of Neighbor (NoN): Degree Cost of Neighbor of Neighbor lists: Memory: O(log 2 n) Maintenance: O(log n) must be updated Neighbor lists should be maintained (open connection, pinging, etc.) SMLKKDB03In practice, a Chord ring will never be in a stable state; instead, joins and departures will occur continuously, interleaved with the stabilization algorithm. The ring will not have time to stabilize before new changes happen. [SMLKKDB03]

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 H-Chord Let N=2 m and H() denote a good hash function. For each 0 i m, node x is connected by edges to the nodes x+2 i + H(x) mod 2 i ; H-Hypercube Let H() denote a good hash function, for each 0 i m, node x is connected with y where y is defined as fallows: the top i-1 bits of y are identical to those of x. The i th is flipped. The remaining m - i bits are identical to those of H(x). x2i2i 2 i+1 H-Networks

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 H-NetworksDegreeGreedyNoN Greedy H-Chord, H-Hypercube, H-Skip-graphs H-Symphony k < log N H-Networks

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Lemma The average path length is O(log n / loglog n) hops for the NoN Greedy algorithm on H-Chord with n=2 m nodes. Proof Phase I : d < n 1/loglog n O(log n / loglog n) step to reach the destination n<2 m Chernoff Bound st d(s,t)=d H-Chord the distance decrease at last with a factor of ¾ for each step

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Phase II : d > n 1/loglog n | I |=d= d / log d Goal: The probability that s can reach the interval I in two hops is equal to a constant c st d(s,t)=d I H-Chord

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 s has at least p - 1 neighbors s 1,…,s p-1 Claim: The probability that s i can reach the interval I is at least d/2 p The probability that s can reach the interval I in two hops is equal to a constant 1-e -1 APL=O(log n / loglogn) s t d(s,t)=d I p - 1 neighborsH-Chord p= log d |I|=d= d / log d d > n 1/loglog n

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Skip – Graphs Each node (resource) has a name. Nodes are arranged on a line sorted by name. Each node x chooses a random string m(x) of bits. An edge is established if two nodes share a prefix which is not shared by the nodes between them. Allows prefix search. ??? Load balancing ??? 011100 11 111 0000 010 a b cf e d

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Routing in Skip – Graphs Greedy Routing – use longest edge possible. Path length and degree are (log N) w.h.p. 011100 11 111 0000 010

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Let H() denote a good hash function, H-Skip graphs are identical to Skip-graphs but with m(x)=H(x) 011100 11 111 0000 010 a b cf e d H-Skip-graphs

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 H-Skip-graphs A node in H-Skip-graphs, in spite of using a deterministic hash function, has no way of estimate its neighbors neighbors. Nevertheless, by using a deterministic hashing function the membership vector m(t) of the target become available to the source, and a more efficient search is now possible. d(x,y)=(y+2 m -x) mod 2 m 011100 11 111 0000 010 a b cf e d

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Chord: N IDs, N nodes Lower is better

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Chord: 2 32 IDs, N nodes Lower is better

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Skip-graphs: N nodes Lower is better

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Skip-graphs: N nodes Lower is better

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Conclusions H-Networks: Deterministic P2P networks No additional information is transmitted nor stored: Each node x, knowing y, can compute H(y) and then can estimate ys neighbors. Asymptotically optimal with respect to average path length and degree (No hidden constant) Allows a trade-off between efficiency and maintenance No overhead with respect to greedy routing system

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Bibliography "Degree-Optimal Deterministic Routing for P2P Systems. G.Cordasco, L.Gargano, M.Hammar, and V.Scarano. In Proc. of 10th IEEE Symposium on computers and communications (ISCC 2005) La Manga del Mar Menor, Cartagena, SPAIN June 27-30, 2005.

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Questions? Dipartimento di Informatica e Applicazioni R.M. Capocelli Università di Salerno, 84081, Baronissi (SA) - Italy

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Fine Diapositive

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Peer to Peer Systems (P2P) File sharing system; File storage system; Distributed file system; Redundant storage; Availability; Performance; Permanence; Anonymity; Scalability Motivation

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Uniform Routing Algorithm We consider a ring of N identifiers labeled from 0 to N-1We 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).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). AdvantagesAdvantages Easy to implementEasy to implement Greedy algorithm is optimalGreedy algorithm is optimal Simple – to understand and implementSimple – to understand and implement Local – routing occurs inside the portion of ring that is delimited by source and destinationLocal – routing occurs inside the portion of ring that is delimited by source and destination No node congestionNo node congestion Fast BootstrapFast Bootstrap Do not need to estimate nDo not need to estimate n DrawbackDrawback Less powerful (De Bruijn Graph and Neighbor of Neighbor Greedy routing are more powerful)Less powerful (De Bruijn Graph and Neighbor of Neighbor Greedy routing are more powerful) Routing is not greedy

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Asymptotic tradeoff curve 1 1 n -1 O(log n) LB O(log n/ log(log n)) Diameter Routing table size Chord et al. Ring Totally connected graph Uniform Routing algorithm Non-Uniform Routing algorithm

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 Classification…. Pure P2P Systems Uniform Systems Chord, CAN, Pastry, Tapestry… F-Chord Non Uniform Systems Non Greedy Routing Viceroy, De Bruijn graphs Greedy Routing Randomized Networks and Neighbor of Neighbor Routing

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 F-Chord( ) 2 5 13 34 89 F-Chord(1) F-Chord( ) Fib(2i), for i = 1,2, …, (1- )(m-2) Fib(i), for i = 2 (1- )(m-2) +2, …, m-1 Degree: F-Chord( ) use (m-2) jumps Diameter: For any value of, the diameter of F-Chord( ) is m/2 0.72021 log N Average Path Length: The average path length of the F-Chord( ) scheme is bounded by 0.39812 log N + (1- )0.24805 log N+1 1 3 8 21 55 even jumps all jumps [1/2,1] m=log n=1.44 log n

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 hops x log n Graphical results Lower is better

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Salerno, 20/21/22 giugno Meeting WEBMINDS 2005 H-Networks We denote by j 1, j 2, …, j d all the jumps of our schemes (ordered by their size); Let H() a good hash function that map an id on a sequence of m bits, for each 1 i d, node x is connected by edges to node x + j i + (H(x)/2 m )*(j i+1 - j i ) [0,1) ijiiji i+1 j i+1

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