1. Universität Stuttgart Institute of Parallel and Distributed Systems (IPVS) Universitätsstraße 38 D-70569 Stuttgart Voronoi Overlay Networks Pavel Skvortsov 21th August 2007

2. 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

3. 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

4. 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

5. 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

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

7. 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)

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

9. 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

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

11. 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? ?

12. 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=0.5+0.5+1.0+0.5+0.5=3 Solution: Recognize end by credit method (Kreditverfahren) ◦ Invariant: c = 3

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

14. 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

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

16. Universität Stuttgart IPVS Research Group Distributed Systems 16 Live Simulation

17. 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

18. 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