Asymptotic Critical Transmission Radius for Greedy Forward Routing in Wireless Ad Hoc Networks Chih-Wei Yi Submitted to INFOCOM 2006

2 Wireless Ad Hoc Networks

3 Greedy Forward Routing What is greedy forward routing? –Packets are discarded if there is no neighbor which is nearer to the destination node than the current node; otherwise, packets are forwarded to the neighbor which is nearest to the destination node. –Each node needs to know the locations of itself, its 1-hop neighbors and destination node. Pros: easy implement Cons: deliverability

4 Examples u w4w4 v w1w1 w2w2 w3w3 w5w5 w6w6 w 6 is a local minimum w.r.t. v

5 Each node has an omni-directional antenna, and all nodes have the same transmission radii. CTR for GFR: Critical Transmission Radius for Greedy Forward Routing vu w

6 Random Deployment Deterministic deployment at some regular pattern is prohibited due to –Large network size –Harsh environment –Mobility Random deployment –Nodes are independently and uniformly distributed in the deployment region

7 r-Disk Graphs D : deployment region of unit-area V n : a random point process with rate n over D r: transmission radius (a function of n) G r ( V n ): r-disk graph over V n

8 Relative Works: Critical Transmission Radius for Connectivity D is a unit-area square or disk. V n is a uniform point process or Poisson point process.

9 Relative Works: the Longest Edge of the Gabriel Graph D is a unit-area disk. V n is a Poisson point process. Let A Gabriel edge is called long if its length is larger than r n. The number of long Gabriel edges is asymptotically Poisson with mean 2e -ξ. The probability of the event that the length of the longest edge is less than r n is asymptotically equal to exp(-2e -ξ ).

10 Main Results D is a convex unit-area region. V n is a Poisson point process with rate n over D, denoted by P n. Let Suppose n  r n 2 = (  +o(1))lnn for some . –If  >  0, then  ( P n ) ≤ r n is a.a.s.. –If  <  0, then  ( P n )  r n is a.a.s.. Let L uv denote the lune area B(u,||u-v||)  B(v,||u-v||). –If ||u-v|| = (1/  ) 1/2, |L uv | = 1/  0. –|L uv | = (  ||u-v|| 2 )/  0.

11 A Sufficient Condition v uw If some node exists on L uv, packets can be forwarded from u toward v. Assume u needs to forward packets to v. Let w denote the intersection point of the ray uv and circle  B(u,r).

12 Minimum Scan Statistic Minimum scan statistic –D : the deployment region –C : the scanning set –V  D : a point set –The minimum scan statistic for V (with scanning set C) is the smallest number of points of V covered by a copy of C. Assume C n =B(o,r n ) and n  r n 2 =  lnn. Let S m ( V n,C n ) = min x  D | V n ∩(C n +x)|.  =1 is a threshold. –If  >1, S m ( V n,C n )>0 is a.a.s.. –If  <1, S m ( V n,C n )=0 is a.a.s..

13 Upper Bounds of the CTR For any  >  0 and ||u-w|| = r n, we have Since  /  0 > 1, according to minimum scan statistics, there almost surely exist nodes on L uw. Therefore, u can forward packets toward v.

14 A Necessary Condition uv Assume ||u-v||>r. If u can forward packets to v, there must exit nodes in L uv. If we can find a pair of nodes u and v such that there doesn’t exist node in L uv, it implies ρ( P n )  r n.

15 Lower Bounds of the CTR u v For any  <  0, we can find a pair of nodes u and v whose distance is larger than r n such that there is no other node on the lune L uv.

16 Conclusion Threshold of critical transmission radius for greedy forward routing Future works –Critical transmission radius for other geographic routing heuristics –Relation between the length of the longest edge of the relative neighbor graph and the critical transmission radius for the greedy forward routing