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An Algorithmic Approach to Geographic Routingin Ad Hoc and Sensor Networks- IEEE/ACM Trans. on Networking, Vol 16, Number 1, February 2008
D94725004 許明宗R97725039 林世昌
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Authors
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Fabian Kuhn
Member, IEEE
Roger Wattenhofer
Aaron Zollinger
Member, IEEE
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Outline
•Introduction•Related Work•Models and Preliminaries•Geographic Routing•Conclusion
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Introduction (1/2)• Wireless Ad Hoc Networks
▫Emergency and rescue operations, disaster relief efforts
• Wireless Sensor Networks▫Monitoring space, things, and the interactions of
things with each other and the encompassing space
• Routing Challenges in Wireless Ad Hoc Networks▫Energy conservation▫Low link communication reliability▫Mobility
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Introduction (2/2)
•Geographic Routing (directional, location-based, position-based, geometric routing) ▫Each node knows its own position and
position of neighbors▫Source knows the position of the
destination•Why “Geographic Routing”?
▫No routing tables stored in nodes ▫Independence of remotely occurring
topology changesACNs 2009 Spring
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Related WorkKleinrock et al. 1975~ MFR et
al.Geographic routing proposed
Finn USC/ISI Report 1989
Greedy Routing
Greedy routing using the locations of nodes
Kranakis, Singh, Urrutia
CCCG 1999 Face Routing
First correct algorithm
Bose, Morin, Stojmenovic, Urrutia
DialM 1999 GFG First average-case efficient algorithm (simulation but no proof)
Karp, Kung MobiCom 2000
GPSR A new name for GFG
Kuhn, Wattenhofer, Zollinger
DialM 2002 AFR First worst-case analysis. Tight (c2) bound.
Kuhn, Wattenhofer, Zollinger
MobiHoc 2003
GOAFR Worst-case optimal and average-case efficient, percolation theory
Kuhn, Wattenhofer, Zhang, Zollinger
PODC 2003 GOAFR+ Improved GOAFR for average case, analysis of cost metrics
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Models and Preliminaries (1/3)•Definition 3.1: (Unit Disk Graph)
▫Let V ⊂ R2 be a set of points in the two-dimensional plane. The graph with edges between all nodes with distance at most 1 is called the unit disk graph of V.
•Definition 3.2: (Cost Function): ▫A cost function c:]0,1] R+ is a nondecreasing
function which maps any possible edge length d (0<d 1) to a positive real value ≦ c(d) such that d’ > d c(d’) ≧ c(d). For the cost of an edge e ∈ E we also use the shorter form c(e) := c(d(e)).
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Models and Preliminaries (2/3) •Definition 3.3: (Ω(1)-Model):
▫If the distance between any two nodes is bounded from below by a term of order Ω(1), i.e., there is a positive constant d0 such that d0 is a lower bound on the distance between any two nodes, this is referred to as the Ω(1)-model.
•For the routing algorithms in the paper, the network graph is required to be planar.▫In order to achieve planarity on the unit disk
graph , the Gabriel Graph is employed.ACNs 2009 Spring
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Models and Preliminaries (3/3)• Definition 3.4: (Geographic Ad Hoc Routing Algorithm)
▫ Let G =(V,E) be a Euclidean graph. The task of a geographic ad hoc routing algorithm A is to transmit a message from a source S ∈ V to a destination D ∈ V by sending packets over the edges of while complying with the following conditions: All nodes v ∈ V know their geographic positions as well as the
geographic positions of all their neighbors in G. The source S is informed about the position of the destination
D. The control information which can be stored in a packet is
limited by O(log n) bits. Except for the temporary storage of packets before
forwarding, a node is not allowed to maintain any information.ACNs 2009 Spring
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Geographic Routing
•Greedy Routing•Face Routing
▫Planar Graph•Greedy Other Adaptive Face Routing
(GOAFR)▫OFR, OBFR, and OAFR▫GOAFR+
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- Nodes learn 1-hop neighbors’ positions from beaconing- A node forwards packets to its neighbor closest to DA stateless and scalable routing for Wireless Ad Hoc (Sensor)
Networks
Greedy Routing (1/2)G.G. Finn ‘87 Lemma 4.1:
If GR reaches D, it does so with O(d2) cost, where d denotes the Euclidean distance between S and D.pf: the disk with center D and radius d contains at most O(d2) nodes with pairwise distance at least 1.
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Greedy Routing (2/2)
x is a local minimum (dead end) to D; w and y are far from D
Greedy Routing not always possible!
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Face Routing (1/2)
Well-known graph traversal: the right-hand rule• (1) Traverse a face• (2) Requires only neighbors’ positionsFails when there are cross links in the graph! planar graph, e.g., RNG, GG
E. Kranakis, H. Singh, and J. Urrutia ‘99
x
y z
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Face Routing (2/2)Face (Perimeter) traversal on a planar graph
S
D
F1
F2
F3
F4
With O(n) messages Many existing algorithms like GFG, GPSR, GOAFR+,
and etc. combine greedy routing with face routing.
Walking sequence: F1 -> F2 -> F3 -> F4
Two primitives: (1) the right-hand rule (2) face-changes
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Planar Graph (1/2) Given a radio graph, make a planar sub-graph
in which every cross-edge is eliminated.
u vw
GG (Gabriel Graph)
Gabriel Graph
u v
w
Relative Neighborhood Graph (RNG)
Relative Neighborhood Graph
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Planar Graph (2/2)
Full Radio Graph
GG Sub-graph
Important assumptions - Unit-disk graph & Accurate localization
How well do planarization techniques work in real-world?
RNG Sub-graph
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GOAFR - Other Face Routing
S
D
F1
F2
P1
P2
Lemma 5.1:OFR always terminates in O(n) steps, where n is the number of nodes. If S and D are connected, OFR reaches D; otherwise, disconnection will be detected.
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GOAFR – Other Bounded Face Routing (1/2)
DS
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GOAFR – Other Bounded Face Routing (2/2)•Lemma 5.2:
▫If the length of the major axis of ε is at least the length of a—with respect to the Euclidean metric—shortest path between S and D, OBFR reaches the destination. Otherwise OBFR reports failure to the source. In any case, OBFR expends cost at most .
c~
)~( 2cO
The shortest path between S and D
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GOAFR – Other Adaptive Face Routing (1/2)•OAFR ( Other Adaptive Face Routing )
0) Initialize to be the ellipse with foci and the length of whose major axis is .
1) Start OBFR with ε. 2) If the destination has not been reached,
double the length of ε’s major axis and go to step 1.
SD2
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GOAFR – Other Adaptive Face Routing (2/2)• Theorem 5.3
▫OAFR reaches the destination with cost O(c2(p*)), p* is an optimal path
• Theorem 6.1▫Any deterministic (randomized) geographic ad hoc
routing algorithm has (expected) cost Ω(c2)• Theorem 6.2
▫Let c be the cost of an optimal path on a unit disk graph. In the worst case, the cost for applying OAFR to find a route from the source to the destination is Θ(C2). This is asymptotically optimal.
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GOAFR
OAFR
greedy fails
After First Face Traversal
greedy works
Greedy Routing
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GPSR
Perimeter Routing
greedy fails
A location closer than where greedy routing failed
greedy failsgreedy works
Greedy Routing
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We could fall back to greedy routing as soon as we are closer to D than the local minimum
But:
Early Fallback to Greedy Routing?
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Greedy
Face
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GOAFR+
Counter p: closer to D than uCounter q: farther from D than uFall back to greedy routing if
p > q
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Performance
FR
OAFR
GFG/GPSR
GOAFR+
AFR
Network Connectivity
Greedy Success Rate
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Conclusion (1/2)
•GOAFR+
▫Combination of the greedy forwarding and face routing approaches Using greedy forwarding, the algorithm also
becomes efficient in average-case networks Average-case efficiency, correctness, and
asymptotic worst-case optimality▫Bounded searchable area and a counter
technique Proved to require at most O(c2) steps
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Conclusion (2/2)
Greedy Routing/MFR ()
Face Routing
GFG/GPSR
AFR
GOAFR/GOAFR+
Correct
Routing
Avg-Case
Efficient
Worst-Case
Optimal
Comprehensive
Simulation
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Thanks for Your Listening
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Discussion
•Lemma 3.3: ▫The shortest p ath for cost function
intersected with the unit disk graph is only longer than the shortest path on the unit disk graph for the respective metric.
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