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UNIVERSITY OF JYVÄSKYLÄ Optimal Resource Discovery Paths of Gnutella2 The IEEE 22nd International Conference on Advanced Information Networking and Applications (AINA 2008) Mikko Vapa, research student P2P Computing Group Department of Mathematical Information Technology

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UNIVERSITY OF JYVÄSKYLÄ Resource Discovery Problem In peer-to-peer (P2P) resource discovery problem any node in the network can possess resources and also query these resources from other nodes Node1: Where is ? Node 1 Node 2 Node 3 Node 4

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UNIVERSITY OF JYVÄSKYLÄ A Simple Solution for the Problem The most studied P2P network, Gnutella, for example used Breadth-First Search (BFS) flooding algorithm which sends query to all neighbors Problems: all resources in the network can be found, but network gets congested and there are lots of useless packets Node 1: Where is ? Node 1 Node 2 Node 3 Node 4 Query Node 4: I have it! Node 2: I have it! Node 4: Node 4 has it too! Reply

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UNIVERSITY OF JYVÄSKYLÄ Steiner Minimum Tree Problem Optimal paths for resource discovery can be found by using non- distributed algorithm which requires global knowledge of topology and resources Precisely, this problem can be formulated as a task of finding a Steiner Minimum Tree (SMT) from a graph:

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UNIVERSITY OF JYVÄSKYLÄ Steiner Minimum Tree Problem V = {Node 1, Node 2, Node 3, Node 4} R = {Node 1, Node 2, Node 4} min T = ({Node 1, Node 2, Node 4}, {1-2, 2-4}) min w(T) = 2 Node 1: Where is ? Node 1 Node 2 Node 3 Node 4 Query Node 4: I have it! Node 2: I have it! Node 4: Node 4 has it too! Reply

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UNIVERSITY OF JYVÄSKYLÄ Rooted k-Steiner Minimum Tree Problem SMT locates all resources in the network, but if only k instances of the matching resources need to be found the problem becomes k-Steiner Minimum Tree problem Also the problem is rooted to define which node starts the query

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UNIVERSITY OF JYVÄSKYLÄ MST k-Steiner Minimum Tree Algorithm MST k-Steiner Minimum Tree Algorithm was developed to find an approximation solution:

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UNIVERSITY OF JYVÄSKYLÄ MST k-Steiner Minimum Tree Algorithm Time Complexity: where E = number of edges in a graph G Worst-Case Approximation Ratio: where R = available resources

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UNIVERSITY OF JYVÄSKYLÄ Simulation Scenarios ScenarioPL10000N10000Gnutella2 DistributionPower-LawNormal- Nodes Edges Largest hub Resources Res. instances Queries100 Diameter81012

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UNIVERSITY OF JYVÄSKYLÄ Query Packets for Gnutella2 with ~75000 nodes MST k-Steiner Minimum Tree algorithm shows that current local search algorithms for peer-to-peer networks are far from optimal paths

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UNIVERSITY OF JYVÄSKYLÄ Hops for Gnutella2 with ~75000 nodes MST k-Steiner does not use the shortest paths to locate resources

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UNIVERSITY OF JYVÄSKYLÄ Highest Degree Search K-Steiner Minimum Tree K-Steiner Tree Algorithm locates 9 resource instances with 11 query packets. For this query the approximated solution is also the optimal solution. HDS uses almost twice as much query packets for this query.

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UNIVERSITY OF JYVÄSKYLÄ Future Work Conducting an extensive survey of related work in graph theory for k- Steiner Minimum Trees and modifying the problem to support multiple resource instances on a same node (Prize Collecting Steiner Tree problem with Quota) What makes the resource discovery problem hard in P2P networks is that only local information is available –It would be interesting to know how close to the optimum can algorithms get using local knowledge A record of the global network topology is used in Open Shortest Path First IP routing protocol and Dijkstras algorithm for computing the shortest paths –It might be possible that MST k-Steiner tree algorithm can be adapted to P2P networks –In this case, information about the resources needs to be at least partially cached in the nodes

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