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d 2. s. V 1. V 3. d 1. V 2. Void Traversal for Guaranteed Delivery in Geometric Routing. Mikhail Nesterenko Adnan Vora Kent State University MASS November 09, 2005. d 2. ?. Geometric Routing: Routing without Overhead. no tables : each node knows only neighbors - PowerPoint PPT Presentation
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Void Traversal for Guaranteed Delivery in Geometric Routing
Mikhail Nesterenko
Adnan Vora
Kent State University
MASSNovember 09, 2005
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211/9/2005 MASS
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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311/9/2005 MASS
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
511/9/2005 MASS
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
711/9/2005 MASS
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
u
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1
path(u,x)<d
811/9/2005 MASS
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
void
traversaldirection
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edge_change
edge_selection
edge_change
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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
prob
abili
ty
1
distanceu 2u 3u
f=1 f=2 f=3
1011/9/2005 MASS
FACE vs. VOID: Example Routes
• 50-node graph, fade factor is 2• FACE: 13 hops• VOID: 11 hops
1111/9/2005 MASS
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
0
20
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50 100 200
Number of nodes in graph
Ho
p c
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imp
rove
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t, %
Factor of 1
Factor of 2
Factor of 3
1211/9/2005 MASS
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?
u v
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Void Traversal for Guaranteed Delivery in
Geometric Routing
Mikhail Nesterenko
Adnan Vora
thank you
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