Scalable Routing in 3D High Genus Sensor Networks Using Graph Embedding

1. Scalable Routing in 3D High Genus Sensor Networks Using Graph Embedding Xiaokang Yu1, Xiaotian Yin2, Wei Han2, Jie Gao3, Xianfeng David Gu3 1Shandong University, PRC 2Harvard University 3Stony Brook University

2. Routing in a high genus 3D network • Sensors monitoring underground tunnels • water, sewer, gas system; coal mine [Li, Liu 2009] • Non-planar, 3D network • Sparsewith complex shape

3. Geographical routing in 2D • Greedy routing: next hop closer to dest • Face routing: tour around a planar face towards dest t S

4. Geographical routing in 3D • 2D geographical routing • Greedy routing works in any dimension • Face routing is unique to 2D • Geo-routing in 3D is strictly harder. • Analogy of “faces” is “voids” in 3D; how to get around a void is not obvious. • In 3D unit ball graph, no deterministic algorithm can guarantee delivery with O(1) state. [DKN08]

5. Geographical routing in 3D • Ideas to do geo-routing in 3D: • Handle special cases of dense networks [XYW+11]. • Project to 2D. [KFO05, LCW+08] • Use hull tree [ZCL+10] • Random walk [FW08] • Face routing on a general graph [ZLJ+07]. • Our idea: • Embed a graph on a general surface • Apply greedy routing with guaranteed delivery

6. Embedding a graph on a surface • Every graph can be embedded on a (high genus) surface without crossing edges • Genus: # handles. • # times one can cut the surface without disconnecting it genus=1 genus=2 genus=3

7. Special case: planar graphs • Combinatorial: no K5, or K3, 3 minor. • K3, 3 cannot be embedded in the plane without crossing edges.

8. Special case: planar graphs • Combinatorial: no K5, or K3, 3 minor. • K3, 3 can be embedded on a torus

9. Special case: planar graphs • Planar graphs can always be embedded on a sphere (genus 0), e.g., in Thuston’s embedding Point of infinity Red: planar graph; Blue: dual graph

10. Example: embedding on a torus Planar illustration: Left & right boundaries are identified; Top & bottom boundaries are identified.

11. Example: embedding on a torus Red: planar graph; Blue: dual graph

12. Algorithm • Define planar faces for a general graph • In a combinatorial sense. • Same idea used in FaceTracing [ZLJ+07] • Embed these faces on a general surface • Use the embedding for greedy routing

13. Define faces on a general graph • Rotation system: around each vertex, define a (arbitrary) cyclic order of the edges. • Trace edges of a face: come in from edge i of a vertex v, leave at edge i+1 at v. • Each edge only belongs to two faces, once in each direction. 3 2 1 4 5

14. Genus of the embedding surface • Determining the embedding surface with minimum genus is NP-hard. [T89] • We tested a heuristic algorithm: • Swap the orders of two adjacent edges. • Keep it if the genus goes down. • In practice we can bring down the genus to about 1/3 the original.

15. Compute the embedding • Determine the edge lengths for such a surface embedding • Introduce the dual graph • Introduce a vertex for intersection of a primal & its dual edge • This defines a partitioning into 4-gons, each with one primal vertex, one dual vertex, two edge vertices.

16. Compute the embedding • Use circle packing metric • Red circles: primal graph, tangent to neighbors. • Green circles: dual graph, tangent to neighbors • Red circle orthogonal to adjacent green circle.

17. Compute the embedding • Existence of such a metric is guaranteed in theory. • Genus=0, spherical metric, curvature > 0 • Genus=1, Euclidean metric, curvature = 0 • Genus=2, hyperbolic metric, curvature < 0 • Use Ricci flow (curvature flow) to compute the final metric • Distributed algorithm introduced in [SYG+08] • Convergence is guaranteed by theory.

18. Algorithm outline • Define planar faces for a general graph • Embed these faces on a general surface • Use the embedding for greedy routing • Define virtual coordinates.

19. Route on the embedded surface • Decompose into |2-2 genus| “pants” (genus 0 with 3 boundaries)

20. Route on the embedded surface • Each pant can be partitioned into two hexagonal pieces (half pants). • With the computed embedding, each piece is convex inside which greedy routing guarantees delivery! Embedding in hyperbolic space

21. Route on the embedded surface • Routing in two stages • Global planning: • Find the sequence of hexagonal pieces to visit • Each node stores the adjacency of the half pants. • Storage = O(genus of the embedding) • Local routing: • Perform greedy routing inside each piece • Each node stores virtual coordinate on its half pants

22. Example: pants decomposition

23. Example: global planning phase

24. Example: local routing phase • Greedy routing to the closest point on the boundary to the next piece to visit.

25. Example: entire path

26. Performance comparison • Compare with • Face tracing [ZLJ+08]: Use rotation system to define faces, do DFS on faces • Shortest path routing • Greedy routing using geographical coordinates • Assume the ideal setting of no transmission failures.

27. Performance comparison • Sensors densely deployed on two surfaces. genus=4

28. Simulation results • Delivery rate • Average path lengths

29. Path length histogram Global guidance is helpful in finding paths in the correct direction.

30. Conclusion and open questions • A general solution for routing on an arbitrary graph. • Storage size of each node: • adjacency of pants decomposition • virtual coordinates in each half pants • O(genus of the embedding). • Open: approximation algorithm for determining the embedding surface with minimum genus?

31. Questions and Comments? • http://www.cs.sunysb.edu/~jgao • jgao@cs.sunysb.edu