Addressing and routing in hexagonal networks with applications in location update and c onnection rerouting in mobile ph

1. Addressing and routing in hexagonal networks with applications in location update and connection rerouting in mobile phone network Fabian Garcia Nocetti Ivan Stojmenovic Jingyuan Zhang

2. Hexagonal networks Honeycomb mesh -interconnection network Stojmenovic 1998 Cellular phone network Base stations at centers of hexagons Benzenoid hydrocarbons

3. Triangular networks Hexagonal mesh -interconnection network – Chen, Shin, Kandlur 1990 Centers of cells in cellular network and connections with neighbors

4. Dual triangular-hexagonal networks BS= base stations = centers of hexagons Cells in cellular network = hexagons

5. Location management in cellular networks Mobile phones update periodically current cell position with their home location server Time, movement and distance based methods Bar-Noy, Kessler, Sidi 1994 Distance from last updated BS=? BS mobile phone

6. Connection rerouting in cellular nets Wired network path modification or Wireless path extension How to shorten extended wireless path?

7. Addressing in ‘hexagonal’ networks (0,-2,1) i+j+k=0 i (a,b,c)= ai + bj + ck x k j z (0,0,0) (a’,b’,c’)= (a”,b”,c”)  a’i+b’j+c’k= a”i+b”j+c”k y (-1,0,1) (-1,0,0)=(0,1,1) (-1,0,2)

8. Shortest path representation Lemma: D-S=(a,b,c) shortest path from S to D  abc=0 Proof:assume abc0 , a>0, b>0 D-S= ai+bj+ck= ai+bj+ck –(i+j+k)= (a-1)i+ (b-1)k +(c-1)k, a=|a-1|+1, b=|b-1|+1, |a|+|b|+|c|= |a-1|+|b-1|+2+|c| > |a-1|+|b-1|+|c-1| contradiction.

9. Length of shortest path Corollary: c=0  ab 0. Assume ab>0 a>0 b>0ai+bj=ai+bj-(i+j+k)= (a-1)i+(b-1)j - k |a|+|b|=|a-1|+1+|b-1|+1> |a-1|+|b-1|+|-1| contradiction. Lemma: D-S= ai+bj  |D-S|=min(|a|+|b|, |a-b|+|b|, |a-b|+|a|) Proof: D-S=ai+bj=(a-b)i+bk=(b-a)j+ak

10. Routing and connection rerouting D-S= ai+bjandab 0. Route |a| times in direction sign(a)x (=ai) Route |b| times in direction sign(b)y (=bj) C(|a|+|b|,|a|) shortest paths (choice may depend on load) Distance determined by three integers (one is 0) Routing corresponds to GEDIR = distance based routing: Route to neighbor that is closest to destination

11. Fault tolerance in ‘hexagonal’ networks Hexagonal network is planar graph Apply GFG = GEDIR-FACE-GEDIR for guaranteed delivery GFG also works in other planar graphs: square mesh, honeycomb mesh, semi-regular tessellations GEDIR works for other networks e.g. hypercubes

12. Extension – BS at arbitrary positions BS determine Delaunay triangulation = dual Voronoi diagram GEDIR guarantees delivery in Delaunay triangulations (Bose, Morin, 2000) Faulty BSs: apply GFG (DT is planar graph)