1. Stateless and Guaranteed Geometric Routing on Virtual Coordinate Systems Ke Liu and Nael Abu-Ghazaleh Dept. of CS, Binghamton University

2. Outlines Background and Motivation Virtual Coordinates System (VCS) Geometric Routing on VCS Contributions Dimensional Degradation Spanning Path Virtual Coordinate System Conclusion

3. Geographic Routing (GPSR) Proposed by B. Karp (MobiCom 2000), know as Greedy and Perimeter Stateless Routing (GPSR) A similar one proposed by Hannes Frey, know as Greedy and Face routing (GFG) Stateless: no path information, no (traditional) routing table. Only locations of neighborhood is used.

4. Geographic Routing Limitation Accurate Location GPS is expensive Indoor application Localization Algorithm is not Accurate: 40% localization error is common Perimeter Routing is not efficient (Possible hundred) times longer than greedy forward. Fail facing Localization error

5. Virtual Coordinates System (VCS) Reference (anchor) nodes are served as bases of VCS Each node sets up its VC as hop counts to reference nodes As localization algorithm at first, later independently used, replacing the physical coordinate system (GeoCS or PCS) Only based on communication connectivity Physical voids are avoided -- mostly Virtual voids arise, NOT with physical voids

6. VCS Variants VariantDimensionsDistanceBacktracking VCap3EuclideanRandom Walk LCR4EuclideanUniversal Record BVRN (>10, typically 80) ManhattanScoped Flooding GSpringDynamicsEuclideanVC Upgrading

7. Virtual Anomaly: Broken Naming Uniqueness

8. Important Definitions Given a graph G(V, E) Component: C(V’,E’), |V’| >= 2 Node cut Vc: |Vc| >=2, and {Vc == V’, or removing Vc would disconnect the rest of C(V’, E’) from G(V, E)} Network connectivity: the minimal size of any component Determinant Component: some anchor node in Vc Indeterminate Component Uniqueness Degree Ud: number of all unique virtual coordinate values for all nodes in network

9. Dimensional Degradation: D d Maximal number of virtual dimensions (virtual anchors) which can increase the naming uniqueness (U d ) if the U d of a n-dimensional virtual coordinatesystem on a network is x, and the U d of a (n+1)-dimensionalvirtual coordinate system is also x, we say the D d of this network is n.

10. Theorem 1: The D d of a 1-connected graph is 1 (High dimensional VCS does not increase naming uniqueness) A node cut Vc contains only this node, separate the network into 2 parts, one is determinant component, another is indeterminate component Increasing the virtual dimension means select one more node in the determinant component as new anchor Values for the new virtual dimension do not increase the naming uniqueness

11. Theorem 1: Proof

12. Lemma 2:

13. Theorem 3 Only (N-1)-Dimensional VCS maximize the naming uniqueness of a complete graph of N nodes If using the current VCS set up procedure, then complete graph suffers most It convergences to shortest path routing.

14. Spanning-Path VCS and Routing Why not use ONLY VCS – no localization at all Impossible? Possible? Yes, it is impossible if using the same VCS setting up (multi-dimension, hop-count based virtual coordinates) No, it is possible – if somehow we give each node unique name, with simple gradient between any pair of nodes Current VCS setting up breaks the naming uniqueness of coordinate system Giving each node a unique ID (VC value) globally and dynamically

15. Related Work Blind Searching: VCap, LCR VCap: Random detour LCR: each node records each packet forwarded Data Flooding: BVR Send the packet to the closest anchor node Anchor node scope floods the packet VCS Upgrading: GSpring Elect one more node as a new anchor

16. Motivation: Spanning-Tree GEM: Using spanning-tree structure (VPCS), as localization alogrithm GDSTR: Spanning-Tree structure: Hull Tree Convex Hull: aggregate all descendent nodes as a convex hull – a polygon covers the area of descendent nodes Negative false: failed to confirm some node in convex hull – routing failure Although those Spanning-tree structure based solution fail, we still believe it is a solution

17. Spanning-Path VCS One node is elected as anchor node DFS algorithm to set up a spanning-tree structure Each node is assigned a unique ID (SPVC) Maximal Range: After all descendent nodes are assigned SPVCs, the maximal SPVC is assigned to the root as its max range

18. Spanning-Path VCS Example

19. Spanning-Path Geometric Routing Descendent Range: node’s SPVC  node’s max range Forwarding candidates: any node whose descendent range contains the destination’s SPVC Using the one with the smallest descendent range as next hop

20. Aligned Greedy and Spanning Path (AGSP) Routing Greedy forwarding mostly based on our previous work (aligned Virtual coordinate system – MASS 2006)– greedy forwarding succeeds 98%+ on VCS If Greedy fowarding fails, using Spanning Path to route the data packets. It is delivery guaranteed, stateless, no localization algorithm used.

21. AGSP Evaluation: Path stretch Better than almost all other GR, both on VCS and GeoCS Approaching the optimal performance, as shortest path routing Deep alignment may not benefit much in high density

22. AGSP Evaluation: Odd deployment LCR provides similar performance – it benefits from less choice during blind searching AGSP is even better than random deployment

23. Conclusion Geometric Routing on VCS previously Geographic Routing was impractical GR on VCS was not even good routing Contribution Increasing Stateless delivery guaranteed GR on VCS Performance is not good as Greedy fowarding Easily to be used with any greedy forwarding, providing the best performance.

24. Thank you Questions ?