Void Traversal for Guaranteed Delivery in Geometric Routing Mikhail Nesterenko Adnan Vora Kent State University MASS November 09, 2005 s d2 d1 V1 V2 V3

Geometric Routing: Routing without Overhead no tables: each node knows only neighbors no message overhead: message of constant size no flooding: only one message at a time per packet no memory: no info is kept at node after message is routed no global knowledge static nodes each node knows its global coordinates sender knows coords of receiver simplest approach: greedy routing message carries coords of dest. each node forwards to neighbor closer to destination problem: local minimum what if no closer neighbor? d2 ? s d1

Face Routing [BMSU’99] HOWEVER d2 s F1 F3 d1 F2 F4 face – continuous area of planar graph not intersected by edges observation – finite number of faces intersect source-destination line idea - traverse each face intersecting sd-line, switch to next face when encountered to traverse a face select to be outgoing the next edge after incoming counter-clockwise optimization (GFG/GPSR) – use greedy, switch to face to leave local minimum, switch back to greedy after approach destination closer than the local minimum, proceed iteratively to use GFG need planar graph unit-disk graph – each vertex pair is connected if distance is less than fixed unit assume – approximates radio model can locally construct Gabriel or Relative Local Neighborhood planar subgraph -- guaranteed connectivity -- no extra communication required HOWEVER F4 d2 s F1 F3 d1 F2

Radio Networks are Not Unit-Disk [David Culler, UCB] non-isotropic large variation in affinity asymmetric links long, stable high quality links short bad ones THUS

What to Do with Non-Planar Graphs? planarization removes edges useful for routing irregular signal propagation forces conservative estimates of edge length increases route size requires greater node deployment density void – continuous area in (not necessarily planar) graph not intersected by edges if unit-disk based planar graphs are inadequate is it possible to apply the idea of traversal to voids in non-planar graphs? s V1 V2 V3 d1

Outline memory requirement for traversal – intersection semi-closure traversal of voids of non-planar graphs simulation setup, examples, results

Intersection Semi-Closure to traverse voids nodes need to have more information about surrounding topology Definition: neighbor relation N over graph G is d-incident edge intersection semi-closed if for every two intersecting edges (u,v) and (w,x) either (w,x)  N(u) and there exist path(u,w)  N(u) and path(u,x)  N(u) neither one is more than d hops; or (w,x)  N(v) and there exist path(v,w)  N(v) and path(v,x)  N(v) neither one is more than d hops Lemma: in a unit-disk a neighborhood relation is 2-intersection semi-closed if for every node u and every edge (w,x) such that |u,w| < 1 and |u,x|  2/3 it follows that (u,w)  N(u) modest requirements on surrounding topology ensure intersection semi-closure x path(u,x)<d 1 u v w

VOID Traversal Algorithm follows segment of the edges that borders the void two parts edge_change message sent to node adjacent to next segment edge, node selects beginning of next segment (next intersecting edge) the selection minimizes the current edge segment sends edge_selection message to the other adjacent node to confirm selection and forward message to node adjacent to next segment edge GVG – void traversal joined with greedy routing similar to GFG edge_selection a c b g e f d h traversal direction edge_change edge_change i void k j s d1 V1 V2 V3

Simulation Setup and Memory Usage implemented FACE and VOID traversal in Java and Matlab uniform distribution random graphs fixed area of 22 units 50, 100, 200 nodes connectivity unit 0.3, 0.25, 0.2 respectively fading factors of 1, 2 and 3 generated graphs and computed unit-disk subgraphs only 1 out of 350 generated had a connected subgraph for factors 2 and 3 generated connected unit-disk graphs and added extra edges according to fading factor memory usage FACE – proportional to average node degree d VOID – proportional to d f 1 probability f=1 f=2 f=3 u 2u 3u distance

FACE vs. VOID: Example Routes 50-node graph, fade factor is 2 FACE: 13 hops VOID: 11 hops

VOID vs. FACE: Average Route Length randomly generated 10 pairs of nodes for each graph used paired comparison to estimate route length improvement comparison based on (HopCountFACE - HopCountVOID)/HopCountFACE

Future Work for degenerate graphs, to establish the neighborhood, the node has to explore sizable portion of the network what are the practical criteria for limiting graph exploration? how certain are we that all intersecting edges are discovered? what are the adverse effects of missed edges on VOID? x u v w

Void Traversal for Guaranteed Delivery in Geometric Routing Mikhail Nesterenko Adnan Vora thank you