1. On Graphs Supporting Greedy Forwarding for Directional Wireless Networks W. Si, B. Scholz, G. Mao, R. Boreli, et al. University of Western Sydney National ICT Australia University of Sydney 1

2. Outline Background The problem we consider Some propositions Optimal solution by an Integer Linear Program A polynomial-time approximation algorithm Evaluation Conclusions 2

3. Background Networks are modelled by geometric graphs. Geographic routing algorithms use the position information of nodes to route packets in the networks. If a geographic routing algorithm A can route a packet from any source s to any destination t in a geometric graph G, A is said to succeed on G. Dst Src 3

4. Background (cont’d) The greedy forwarding algorithm  When a node forwards a packet with destination Dst, it sends this packet to its neighbor that is closest to Dst. Greedy forwarding does not succeed on a graph that contains the void scenario Dst 4

5. Background (cont’d) As a result, it is desirable to design a network without the void scenario. Many research efforts assign virtual coordinates to the nodes to achieve a greedy embedding.  Considering omni- directional antennas  Links are known, how to change the coordinates? 5

6. The problem we consider We also study the problem of obtaining graphs that support greedy forwarding. To ease the discussion, we call such graphs greedy forwarding graph (GFG) later. We consider a different setting in establishing GFGs.  Directional antennas  Coordinates are known, how to change the links?  The goal is to minimize the number of links, thus the network installation cost. 6

7. The problem we consider (cont’d) Given an arbitrary set of nodes with known coordinates, how can we add minimum number of point-to-point links among the nodes, such that the resulting geometric graph is a GFG. 7

8. Propositions on GFGs Proposition 1: Any GFG is connected. This implies that any GFG(V) has a spanning tree as its subgraph. Since the number of edges in a spanning tree on V is n-1, where n is the number of nodes in V, a lower bound for |E min (V)| is n-1. We use GFG(V) to denote a GFG on a node set V, and GFG min (V) to denote the GFG(V) with the minimum number of links, and E min (V) to denote the link set in GFG min (V). 8

9. Propositions on GFGs (cont’d) Proposition 2: There exists a node set V such that GFG min (V) has no planar instances. 9

10. Propositions on GFGs (cont’d) Proposition 3: Given an arbitrary node set V, the DT(V) is a GFG(V). Since a DT(V) has 3n-k-3 edges, where k is the number of convex hull edges on V, an upper bound for |E min (V)| is 3n-k-3. A Delaunay triangulation on a node set V (denoted by DT(V)) is a triangulation graph on V such that no nodes are contained in the interior of the circumcircle of any of its triangles. 10

11. Propositions on GFGs (cont’d) Proposition 4: There exists a node set V such that the EMST(V), RNG(V), and GG(V) are not a GFG(V). The Euclidean Minimum Spanning Tree on V ( EMST(V) ), the Relative Neighborhood Graph on V ( RNG(V) ), the Gabriel Graph on V ( GG(V) ), and DT(V) have the following relationship: EMST(V) ≤ RNG(V) ≤ GG(V) ≤ DT(V) 11

12. Propositions on GFGs (cont’d) Proposition 5: Given an arbitrary node set V, the NNG(V) is a subgraph of any GFG(V). Unfortunately, EMST(V), RNG(V), and GG(V) are not always subgraphs of a GFG(V). A nearest neighbor graph on a node set V (denoted by NNG(V)) is obtained by connecting each node in V with its nearest neighbor. 12

13. Propositions on GFGs (cont’d) Definition 1. Given a geometric graph G, the closer region of a node p in G is defined as the intersection of all h(p, u)’s, where u is a neighbor of p in G. In a geometric graph, for an edge ab with two end nodes a and b, the bisector of a and b is defined as the perpendicular bisector of the edge ab. For the two open half planes delimited by this bisector, the one containing a is denoted h(a, b), and the other one containing b is denoted h(b, a). 13

14. Propositions on GFGs (cont’d) The closer region concept differs from the Voronoi region concept in the following aspects: The closer regions are defined on a geometric graph, while the Voronoi regions are defined on a set of nodes. The closer regions can overlap with each other, while the Voronoi regions do not overlap. Given a geometric graph G(V, E) and the Voronoi diagram on V, the closer region of a node p regarding G contains the Voronoi region of p regarding V. 14

15. Propositions on GFGs (cont’d) Proposition 6: A necessary and sufficient condition for a geometric graph G to be a GFG is that for every node v in G, the closer region of v and its boundary do not contain any other node in G. 15

16. Optimal Solution by an Integer Linear Program To obtain optimal solution to the minimum GFG problem, we formulate it into an Integer Linear Program (ILP). For each link (u,v)  V  V, we introduce a binary variable x(u,v) in the ILP that indicates whether link (u,v) exists in E. 16

17. Optimal Solution by an ILP (cont’d) An exemplar GFG min (V) obtained by solving the previous ILP on a 100-node set V. 17

18. A Polynomial-time Approximation Algorithm As established in the previous propositions, given a set of nodes V, the DT(V) is a GFG(V) and the NNG(V) is a subgraph of any GFG(V). The basic idea of our algorithm is to calculate the DT(V) and the NNG(V) first, and then attempt to remove the edges that are in DT(V) but not in NNG(V), i.e., DT(V) - NNG(V). 18

19. A Polynomial-time Approximation Algorithm 1.Calculate DT(V). 2.Based on DT(V), calculate NNG(V) and determine the list of edges that are in DT(V) - NNG(V). 3.Go through the list of edges in DT(V) - NNG(V). For each edge uv (u and v are the two end nodes of this edge), test whether it can be removed such that the remaining graph is still a GFG(V). 4.The remaining graph is the output GFG(V). Input: a set of nodes V with known (x, y)-coordinates Output: a GFG that approximates the minimum GFG 19

20. A Polynomial-time Approximation Algorithm The DT(V) on a 100-node set V. 20

21. A Polynomial-time Approximation Algorithm The NNG(V) on a 100-node set V. 21

22. A Polynomial-time Approximation Algorithm The GFG(V) obtained by our algorithm on this 100-node set V. 22

23. A Polynomial-time Approximation Algorithm Result 1. The above algorithm has a complexity of O(n 2 ), where n is the number of given nodes. Result 2. The above algorithm is a 3- approximation algorithm in terms of the number of edges. We have the following results about this algorithm. 23

24. Evaluation We compare our polynomial-time algorithm with the optimal solution on the number of edges generated to construct the GFGs. We also evaluate the path quality of the GFGs generated by our polynomial-time algorithm for the greedy forwarding algorithm.  absolute stretch  relative stretch 24

25. On the number of edges 25

26. On the path quality – absolute stretch 26

27. On the path quality – relative stretch 27

28. Conclusions This paper considers the problem of generating minimum GFGs for the wireless networks with directional antennas. We first presented six propositions regarding GFGs.  Revealing the properties of GFGs and the relationships between GFGs and other well-known graphs. We then give an optimal solution by ILP as well as a polynomial time 3-approximation algorithm for the minimum GFG problem. The experiments show that the polynomial-time algorithm  can actually produce within 1.1 times the number of links found by the optimal solution  can provide paths with small stretches for the greedy forwarding algorithm 28

29. Thank you! Questions and Suggestions? 29