Two-step Routing and Apex Angle Routing for Delaunay Triangulations

1. Two-step Routing and Apex Angle Routing for Delaunay Triangulations PhD Candidate: Weisheng Si Supervisor: Prof. Albert Y. Zomaya School of Information Technologies

2. Outline • Background • Related work • Our work • The Two-step Routing algorithm • The Apex Angle Routing algorithm • Evaluation • Open problems • Conclusion

3. Background • Online routing • Our evaluation metrics for online routing • Delaunay triangulations

4. Online Routing • In some networking scenarios, a packet only has local information to find out its routes. Routing algorithms designed for such scenarios are called online routing algorithms. • We consider online routing in the same settings as those described in “Online routing in triangulations”: • The environment is modeled by a geometric graph G(V, E), where V is the set of nodes with known (x, y) coordinates and Eis the set of links connecting the nodes. • When a packet travels from a source node s to a destination node t, it can remember the coordinates of s and t, and at each node v being visited, can learn the coordinates of the nodes in N(v), where N(v) denotes the set of v’s neighbors.

5. Online routing (cont’d) • If an online routing algorithm A can route a packet from any source s to any destination tin G, A is said to work for G. • If at each node v traversed by a packet,A makes the routing decision for this packet only according to the coordinates of v, t, and the nodes in N(v), A is said to be memoryless or oblivious. • ‘memoryless’ means that a packet records no information learned during the traversal of a graph. • Because the memoryless online routing (MOR) algorithms have low complexity in both space and time for nodes and packets, they have received wide attention.

6. Our evaluation metrics for online routing • For a source/destination pair (s, t) in G, we define the detour ratioof(s, t) by a routing algorithm A as the length of the path found by A from s to t versus the length of the shortest path from s to t. • For a graph G, we define the detour ratio of G by A as the average detour ratio of all (s, t) pairs in G. • In practice, the path length generally has two metrics: • link distance • Euclidean distance • Correspondingly, we obtain two types of detour ratios called the link detour ratio and the Euclidean detour ratio.

7. Our evaluation metrics (cont’d) • The detour ratio concept is different from the c-competitiveconcept • A routing algorithm is c-competitive for a graph G, if for all (s, t) pairs in G, their detour ratios are not greater than a constant c. • The detour ratio concept concerns the average performance of a routing algorithm on a graph, while the c-competitive concept concerns the worst-case performance of a routing algorithm on a graph. • The detour ratio concept is different from the dilation concept and the stretch factor concept • Both of them are defined to measure the path quality of a subgraph with respect to the complete graph. • Both of them are not used to evaluate routing algorithms.

8. Delaunay Triangulations • A Delaunay triangulation (DT) is a triangulation graph in which no node lies in the interior of the circumcircle of any its triangle. • DTs have the following properties in favor of routing: • Let n denotes the number of nodes, e the number of edges, k the number of convex hull edges, e = 3n – 3 – k holds for any triangulation graph. Therefore, the total number of links in a DT is less than 3n and the average node degree is less than 6, thus simplifying the operation of routing. • In a DT, the Euclidean length of the shortest path between any two nodes u and v is less than C times the Euclidean distance between u and v, where C is proved to be between 1.5846 and 2.42. As an aside, determining C exactly is one of the most challenging problems in the area of computational geometry. • DTs are planar graphs.

9. Delaunay Triangulations (cont’d) • Therefore, DTs have been widely used as the network topologies. • In light of the above, this paper particularly focuses on the MOR algorithms for DTs.

10. Related work • Three well-known MOR algorithms are proved to work for DTs • The Compass Routing algorithm • The Greedy Routing algorithm • The Greedy Compass algorithm Hereafter, we will use t to denote the destination node of a packet P, v to denote the current processing node of P, d(a, b) to denote the Euclidean distance between node a and node b, and to denote the angle between the linkva and the link vb.

11. The Compass Routing algorithm • The node v always moves P to the node win N(v) that minimizes the angle . c t v a b

12. The Greedy Routing algorithm • The node v always moves P to the node win N(v) that minimizes d(w, t). c t v a b

13. The Compass Greedy algorithm • The node v first decides the two nodes cw(v) and ccw(v), where cw(v) denotes the node w that has the smallest clockwise angle from the line vt, and ccw(v) denotes the node w that has the smallest counterclockwise angle from the line vt. • Then, P is moved to one of cw(v) and ccw(v), whichever has a smaller Euclidean distance to t. c ccw(v) t v cw(v) a b

14. Our work • The Two-step Routing algorithm • The Apex Angle Routing algorithm

15. The Two-step Routing Algorithm • The basic idea is to to minimize the sum of d(v, w) and d(w, t). • Two-step Routing essentially restricts its searching area of a w in N(v) to the area enclosed by an ellipse with v and t as the two foci, i.e., it increases this area gradually until the first win N(v) is encountered.

16. The Two-step Routing (cont’d) • Unfortunately, if the Two-step Routing algorithm only tries to minimize the d(v, w) + d(w, t), it fails for some DTs.

17. The Two-step Routing (cont’d) • To fix this problem, we add a checking condition that only when aw in N(v) has a smaller d(w, t) than d(v, t), minimizing the d(v, w) + d(w, t) is used as the criteria to select the next hop. 1 for each w in N(v) { 2 if ( w has a smaller d(w, t) than d(v, t) ) { 3 sum = d(v, w) + d(w, t); 4if ( w has a smaller sum than previous ) { 5 next(v) = w; 6 } 7 } 8 }

18. The Two-step Routing (cont’d) • For the Two-step Routing algorithm, we have the following theorem. Theorem 1: The Two-step Routing algorithm works for DTs. Proof: As mentioned previously, the Two-step Routing algorithm moves a packet strictly closer to the destination in each routing step, so a packet visits a different node in each move. Since there is limited number of nodes in a DT, the packet must reach the destination eventually; otherwise, there should be infinite number of nodes in a DT. 

19. The Apex Angle Routing Algorithm • We call the apex angle for the current processing node v, aw in N(v), and the destination t. The basic idea of Apex Angle Routing is to maximize this apex angle in each routing step. • Apex Angle Routing essentially restricts its searching area of a w in N(v) to the area enclosed by the two arcs with vt as the chord, i.e., it increases this area gradually until the first win N(v) is encountered.

20. The Apex Angle Routing (cont’d) • As to be proved later, simply maximizing the apex angle guarantees a packet reaching the destination in DTs. Thus, the Apex Angle Routing algorithm can be described as follows. 1 for each w in N(v) { 2 if ( w has a larger than previous ) { 3 next(v) = w; 4 } 5 }

21. The Apex Angle Routing (cont’d) Theorem 2: The Apex Angle Routing algorithm works for DTs. Proof: We prove this theorem by showing that in each routing step, a packet gets strictly closer tot. This proof also exploits the fact that a DT is the dual graph of a Voronoi diagram. a w v t o i D

22. Evaluations • Euclidean and link detour ratios of the five MOR algorithms: • Compass Routing • Greedy Routing • Compass Greedy • Two-step Routing • Apex Angle Routing • The possibility for the shortest paths in DTs under Euclidean metric and link metric to be discrepant.

23. Euclidean and link detour ratios • We develop a computer program that implements the above five MOR algorithms and calculates their Euclidean detour ratios and link detour ratios. • We totally conduct experiments on 1000 DTs of 100 nodes. For each DT, the positions of its 100 nodes are randomly uniformly distributed in a square area.

24. Euclidean detour ratios • In terms of both average and 99th percentile Euclidean detour ratios, MinArc and MinElp perform better than the other three algorithms. • In terms of both average and 99th percentile Euclidean detour ratios, GdyRtg performs the worst.

25. Link detour ratios • In terms of both average and 99th percentile link detour ratios, MinArc and MinElp perform worse than the other three algorithms. • In terms of both average and 99th percentile link detour ratios, GdyRtg performs the best.

26. Possibility for the shortest paths under Euclidean and link metrics to be discrepant • Our conjecture: in a DT with random node placement, the shortest paths in the Euclidean distance metric are discrepant with those in the link distance metric in a large proportion. • Evaluation with discrepancy ratio: the number of the (s, t) pairs for which the shortest paths under the Euclidean and link metrics are discrepant versus the total number of (s, t) pairs considered in a DT.

27. Discrepancy Ratio • The discrepancy ratios are significantly large for the link distances 2, 3, …, 7, ranging from 0.153 to 0.737, which verifies our conjecture. • The discrepancy ratio increases with the link distance of the paths, reflecting that the longer the path, the larger the possibility is for the shortest paths in these two metrics to be discrepant.

28. Real-life meaning • Driver A: I can drive through only two streets to reach the airport. • Driver B: I can drive through ten streets, but travel less kilometers than you. • Driver A: Are you kidding? • Driver B: No, if we are driving on Delaunay triangulations.

29. Open Problem 1 • It is known that there exist some DTs for which both Compass Routing and Greedy Routing are not c-competitive in Euclidean metric. • As seen in our evaluations, both the Two-step Routing and the Apex Angle Routing perform better than Compass Routing and Greedy Routing in Euclidean metric. • Thus, we conjecture that the Two-step Routing and the Apex Angle Routing are c-competitive for all DTs. • If this conjecture is proved true, more meanings will be added to these two algorithms.

30. Open Problem 2 • Our evaluations show that in DTs, the shortest paths in the Euclidean metric are discrepant with those in the link metric in a large proportion. • Intuitively true by looking at the previous DT figure. • Pathological cases of DTs (e.g., figure below) can also exist.

31. Open Problem 2 (cont’d) • We see that DTs are not suitable if an application (e.g., routing in the wireless networks) desires that the path found in a graph be shortest in both Euclidean and link metrics. • So the following problem can be meaningful: in what kind of graphs the shortest paths in Euclidean metric and in link metric are identical for any pair of nodes? • To be more specific, can we give a low-complexity testing method for this problem?

32. Conclusions • Two new MOR algorithms are presented and proved to work for DTs • These two new algorithms are also evaluated and compared with other three well-known MOR algorithms. Our evaluations reveal that: • Both of the two new algorithms perform better than the other three algorithms in terms of Euclidean detour ratio, but worse in terms of link detour ratio. • DTs with random node placement have the property that the shortest paths in link metric and in Euclidean metric are discrepant in a large proportion. • Two open problems are posed, with their importance explained.

33. One question • There also exist other works using ellipses to specify areas for taking certain routing actions: • One uses a bounding ellipse to signal a packet to change its traversing direction during a face traversal • Another two use ellipses to specify regions in which nodes will relay packets in flooding mode. • Is it a good idea to rename our algorithms “Two-step Routing” and “Apex Angle Routing”? • These two names seem more straight-forward and easy to remember. Thank you!And other questions or comments?