Void Traversal for Guaranteed Delivery in Geometric Routing

Mikhail Nesterenko

Adnan Vora

Kent State University

MASSNovember 09, 2005

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Geometric Routing: Routing without Overhead

no tables: each node knows only neighborsno message overhead: message of constant sizeno flooding: only one message at a time per packetno 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?

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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

Face Routing [BMSU’99]

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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

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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?

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Outline

• memory requirement for traversal – intersection semi-closure

• traversal of voids of non-planar graphs

• simulation setup, examples, results

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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| 2/3 it follows that (u,w) N(u)

modest requirements on surrounding topology ensure intersection semi-closure

Intersection Semi-Closure

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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 currentedge segment

• sends edge_selection message 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

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Simulation Setup and Memory Usage

• implemented FACE and VOID traversal in Java and Matlab• uniform distribution random graphs

fixed area of 22 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

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FACE vs. VOID: Example Routes

• 50-node graph, fade factor is 2• FACE: 13 hops• VOID: 11 hops

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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

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Number of nodes in graph

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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?

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Void Traversal for Guaranteed Delivery in

Geometric Routing

Mikhail Nesterenko

Adnan Vora

thank you