Void Traversal for Guaranteed Delivery in Geometric Routing

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

2. d2 ? 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 todestination • problem: local minimum • what if no closer neighbor? s d1

3. F4 Face Routing [BMSU’99] 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 d2 s F1 F3 d1 F2

4. 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

5. 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 ideaof traversal to voids innon-planar graphs? s V1 V3 d1 V2

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

7. 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-intersectionsemi-closed if for every node u and everyedge (w,x) such that |u,w| < 1 and |u,x| 2/3it follows that (u,w)N(u) • modest requirements on surrounding topology ensure intersection semi-closure x path(u,x)<d 1 u v w

8. s V1 V3 d1 V2 VOID Traversal Algorithm edge_selection follows segment of the edges that borders the void two parts • edge_changemessage sent to node adjacent to next segment edge, node selects beginning of next segment (next intersecting edge)the selection minimizes the currentedge segment • sends edge_selectionmessage to the other adjacent node to confirm selection and forward message to node adjacent tonext segment edge GVG – void traversal joinedwith greedy routing similarto GFG a c b g e f h d traversaldirection edge_change edge_change i void k j

9. 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 degreed • VOID – proportional tod f 1 probability f=1 f=2 f=3 u 2u 3u distance

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

11. 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

12. 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 v u w

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