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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