Wireless Sensor Networks
7. Geometric Routing
Christian Schindelhauer
Technische Fakultät
Rechnernetze und Telematik
Albert-Ludwigs-Universität Freiburg
Version 30.05.2016
1
Literature - Surveys
 Stefan Rührup: Theory and Practice of Geographic
Routing. In: Hai Liu, Xiaowen Chu, and Yiu-Wing
Leung (Editors), Ad Hoc and Sensor Wireless
Networks: Architectures, Algorithms and Protocols,
Bentham Science, 2009
 Al-Karaki, Jamal N., and Ahmed E. Kamal. Routing
techniques in wireless sensor networks: a survey.
Wireless communications, IEEE 11.6 (2004): 6-28.
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Geometric Routing
 Routing target:
- geometric position
 Idea
- send message to the neighbor closest to the target node (greedy strategy)
 Advantagements
- only local decisions
- no routing tables
- scalable
(4,2)
13,5
s
(2,5)
(5,7)
(13,5)
t
(0,8)
(3,9)
3
Greedy forwarding and recovery
 Right-hand rule needs
planar topology
- otherwise endless recovery
cycles can occur
 Therefor the graph needs
to be made planar
- erase crossing edges
 Problem
- needs communication
between nodes
- must be done careful in order
to prevent graph from
becoming disconnected
4
Face Routing and AFR work on planar graphs. We use
the term planar graph for a specific embedding of a planar
graph, i.e. we consider Euclidean planar graphs. In this
case, the nodes and edges of a planar graph G partition the
Euclidean planeinto contiguous regions called the f faces of
G (see Figure 2 as an illustration). Note that we get f − 1
finite faces in the interior of G and one infinite face around
G.
The main idea of the Face Routing algorithm is to walk
along the faces which are intersected by the line segment st
between thesources and thedestination t. For completeness
we describe the algorithm in detail (see Figure 3).
Problems of Recovery
 Recovery strategy can
produce large detours
 Solutions
- Follow recovery strategy until
the situation has absolutely
improved
• e.g. until the target is closer
- Follow a thread
• Face Routing strategy, GOAFR
s
t
• Kuhn, Wattenhover, Zollinger,
Asymptotically Optimal
Geometric Mobile Ad-Hoc
Routing, DIAL-M 2002
Figure 3: T he Face Routing algorit
hm
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GOAFR: Adaptive Face Routing
 Adaptive Face Routing
 Faces are traversed completely while the search
area is restricted by a bounding ellipse
 Recovery strategy + greedy forwarding
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Adaptive Face Routing
 Spanning ratio/stretch factor
- max{shortest path(u,v)/
geometric distance(u,v)}
 Gabriel graphs
 Relative Neighborhood Graph
 Delaunay Triangulation
- but possibly long edges
- because the convex hull is always
a sub-graph of the DT
 A lot of better techniques
studied in literature
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s suggestions on how to construct a planar subPr oof . The theorem directly follows from Lemma 3.
he unit disk graph in a distributed way. Often
Lemma 4.3, and Lemma 4.4.
ection between the UDG and the Relative NeighGraph (RNG [24]) or the Gabriel Graph (GG [8]),
5.
LOWER
BOUND
y, have been proposed. In the RNG an edge beIn this section we give a constructive lower bound for ge
es u and v is present iff no other node w is closer
ometric ad-hoc routing algorithms.
o v than u is to v. In the Gabriel Graph an edge
 Kuhn, Wattenhover, Zollinger, Asymptotically Optimal Geometric Mobile Ad-Hoc
and v is present iff no other node w is inside or
Routing, DIAL-M 2002
w
le with diameter uv. The Relative Neighborhood
theGabriel Graph areeasily constructed in a dismanner. There have been other suggestions, such
ersection between the Delaunay triangulation and
sk graph [17]. All mentioned graphsareconnected
hat the unit disk graph is connected as well. We
abriel Graph, since it meets all requirements as
he following dlemma.
= length of shortest path
Lower Bound for Geometric Routing
4.4. In the Ω(1)-model the shortest path for any
time(Euclide
= #hops,
traffic = link
#messages
sidered metrics
an distance,
distance,
y) on the Gabriel Graph intersected with the unit
is only by a constant longer than the shortest path
t disk graph for the respective
2 metric.
t
Time: Ω(d )
u
w
e’
s
e’’
e
v
Figure 8: Lower bound graph
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Lower Bound for Greedy Routing
 J.Gao,L.J.Guibas,J.E.Hershberger,L.Zhang, A.Zhu,“Geometric spanner for routing
in mobile networks,” in 2nd ACM Int. Symposium on Mobile Ad Hoc Networking &
Computing (MobiHoc), 2001, pp. 45–55.
Time: Ω(d2)
d = length of shortest path
time = #hops, traffic = #messages
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A Virtual Cell Structure
nodes exchange beacon messages
node v knows positions of ist neighbors
v
Rührup et al. Online Multi-Path Routing in a Maze, ISAAC 2006
transmission radius
(Unit Disk Graph)
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A Virtual Cell Structure
each node classifies the cells
in ist transmission range
v
node cell
link cell
barrier cell
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Routing based on the Cell Structure
 Routing based on the cell
structure uses cell paths
cell path
- = sequence of orthogonally
neighboring cells
 Paths
- in the unit disk graph and cell
paths are equivalent up to a
constant factor
 no planarization strategy
needed
- required for recovery using the
right-hand rule
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Routing based on the Cell Structure
virtual forwarding using cells
w
v
physical forwarding from v to w,
if visibility range is exceeded
node cell
link cell
barrier cell
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Performance Measures
 competitive ratio:
solution of the algorithm
optimal offline solution
 competitive time ratio of a routing algorithm
- h = length of shortest barrier-free path
- algorithm needs T rounds to deliver a message
T
h
single-path
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Comparative Ratios
 optimal (offline) solution for traffic:
- h messages (length of shortest path)
 Unfair, because
- offline algorithm knows the barriers
h+
p
- but every online algorithm has to pay
exploration costs
 exploration costs
- sum of perimeters of all barriers (p)
 comparative traffic ratio
M = # messages used
h = length of shortest path
p = sum of perimeters
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