1. 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

2. 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

3. 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

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 idea of traversal to voids in non-planar graphs? s V1 V2 V3 d1

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-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

8. 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

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 degree d
VOID – proportional to d 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 u v w

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