Universität Stuttgart Institute of Parallel and Distributed Systems (IPVS) Universitätsstraße 38 D Stuttgart Voronoi Overlay Networks Pavel Skvortsov 21th August 2007

Universität Stuttgart IPVS Research Group Distributed Systems 2 Overview Motivation Voronoi & Delaunay conditions Distributed Voronoi algorithms ◦ Routing ◦ Node insertion ◦ Node removal ◦ Credit method ◦ Conflicting insertions Live simulation Related work Conclusion

Universität Stuttgart IPVS Research Group Distributed Systems 3 Motivation: Problems in CAN Content Addressable Network (CAN) Ratnasamy et al, 2001 ◦ Hierarchic partitioning of service area ◦ Assumes uniform data distribution  Results in approximately regular grid Problems with non-uniform distributions (non-regular grid) ◦ High varying number of neighbours ◦ Data load on node insertions and removals refers to one other node only Nodes without sibling can not be removed in one step

Universität Stuttgart IPVS Research Group Distributed Systems 4 Motivation: Advantages of Voronoi Overlay Network-related advantages ◦ The number of neighbours of a node depends not on the size of network and equals 6 at the average ◦ Insertion and removal algorithms are uniform for any node ◦ The size of network is geometrically unbounded Data-related advantages ◦ The data distributing firstly depends on the distance to node ◦ Data load on node insertions and removals distributes between neighbours uniformly

Universität Stuttgart IPVS Research Group Distributed Systems 5 Voronoi & Delaunay Conditions Voronoi Minimum Distance condition: ◦ Each point belongs to closest node Delaunay Circumcircle condition: ◦ Empty circle around each triangle of nodes

Universität Stuttgart IPVS Research Group Distributed Systems 6 Distributed Voronoi Algorithms: Routing Destination node Initiator node “route!” Compass routing

Universität Stuttgart IPVS Research Group Distributed Systems 7 Distributed Voronoi Algorithms: Node Insertion (1) Boot node 1st neighbour 2nd neighbour 3rd neighbour New node 4th neighbour 5th neighbour “find triangle!” “find neighbour!” “add me!” The number of messages: M ins = r + (2n – 1)

Universität Stuttgart IPVS Research Group Distributed Systems 8 Distributed Voronoi Algorithms: Node Insertion (2) The number of messages: M ins = r + (2n – 1)

Universität Stuttgart IPVS Research Group Distributed Systems 9 Distributed Voronoi Algorithms: Node Removal (1) Node to remove “Remove me!” The number of messages: M rem = n

Universität Stuttgart IPVS Research Group Distributed Systems 10 Distributed Voronoi Algorithms: Node Removal (2) The number of messages: M rem = n

Universität Stuttgart IPVS Research Group Distributed Systems 11 Distributed Voronoi Algorithms: Conflict Example “find triangle!” New node 1 New node 2 Idea: To protect insertion process by distributed transaction ◦ Mark involved in previous insertion process nodes as “busy” ◦ Problem: How can involved nodes recognize the end of previous insertion process? ?

Universität Stuttgart IPVS Research Group Distributed Systems 12 Distributed Voronoi Algorithms: Credit Method (1) Boot node busy New node busy “find triangle!” “find triangle!” [c = 3] “find neighbour!” [c = 1] “find neighbour!” [c = 1] “find neighbour!” [c = 0.5] “add me!” c= =3 Solution: Recognize end by credit method (Kreditverfahren) ◦ Invariant: c = 3

Universität Stuttgart IPVS Research Group Distributed Systems 13 Distributed Voronoi Algorithms: Credit Method (2) “You are free” “You are free!” New node busy

Universität Stuttgart IPVS Research Group Distributed Systems 14 Distributed Voronoi Algorithms: Conflict (1) “find neighbour!” busy “roll back!” “back up!” New node 1 New node 2 busy

Universität Stuttgart IPVS Research Group Distributed Systems 15 Distributed Voronoi Algorithms: Conflict (2)

Universität Stuttgart IPVS Research Group Distributed Systems 16 Live Simulation

Universität Stuttgart IPVS Research Group Distributed Systems 17 Related Work Liebeherr et al, 2001: Application-Layer Multicasting with Delaunay Triangulation Overlays Compass routing o No distributed transaction but stabilization by periodic messages Kang et al, 2005: P2P Spatial Query Processing by Delaunay Triangulation Basics of node insertion algorithm o Non-uniform insertion algorithm o Conflicts during insertion are not considered Ohnishi et al, 2005: Incremental Construction of Delaunay Overlaid Network for Virtual Collaborative Space Approach of distributed transactions o Node insertion algorithm requires knowledge on all nodes of the overlay

Universität Stuttgart IPVS Research Group Distributed Systems 18 Conclusion Voronoi Overlay ◦ Voronoi Overlay Network has significant advantages in comparison with Context Addressable Network ◦ Compass method has been chosen because of convex hull exclusive situations ◦ With developed node insertion algorithm the optimal messages’ number has been achieved: N ins = r + (2n – 1) ◦ The represented node removal algorithm requires only the package of one-way directed messages during one time step ◦ Identification and reaction to conflict situation allows parallel execution of the represented algorithm Open questions ◦ Appearance of a new node right on the side of triangle ◦ Node failures and communication failures ◦ Many parallel node insertions and/or long message delays