An Algorithmic Approach to Geographic Routing in Ad Hoc and Sensor Networks - IEEE/ACM Trans. on Networking, Vol 16, Number 1, February 2008 D 許明宗 R 林世昌

Authors ACNs 2009 Spring 2 Fabian Kuhn Member, IEEE Roger WattenhoferAaron Zollinger Member, IEEE

Outline Introduction Related Work Models and Preliminaries Geographic Routing Conclusion 3 ACNs 2009 Spring

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 4 ACNs 2009 Spring

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 changes 5 ACNs 2009 Spring

Related Work Kleinrock et al.1975~MFR et al.Geographic routing proposed FinnUSC/ISI Report 1989 Greedy Routing Greedy routing using the locations of nodes Kranakis, Singh, Urrutia CCCG 1999Face Routing First correct algorithm Bose, Morin, Stojmenovic, Urrutia DialM 1999GFGFirst average-case efficient algorithm (simulation but no proof) Karp, KungMobiCom 2000 GPSRA new name for GFG Kuhn, Wattenhofer, Zollinger DialM 2002AFR First worst-case analysis. Tight  (c 2 ) bound. Kuhn, Wattenhofer, Zollinger MobiHoc 2003 GOAFRWorst-case optimal and average-case efficient, percolation theory Kuhn, Wattenhofer, Zhang, Zollinger PODC 2003GOAFR+Improved GOAFR for average case, analysis of cost metrics 6 ACNs 2009 Spring

Models and Preliminaries (1/3) Definition 3.1: (Unit Disk Graph) ▫Let V ⊂ R 2 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  c(d’) ≧ c(d). For the cost of an edge e ∈ E we also use the shorter form c(e) := c(d(e)). 7 ACNs 2009 Spring

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 d 0 such that d 0 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. 8 ACNs 2009 Spring

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. 9 ACNs 2009 Spring

Geographic Routing Greedy Routing Face Routing ▫Planar Graph Greedy Other Adaptive Face Routing (GOAFR) ▫OFR, OBFR, and OAFR ▫GOAFR+ 10 ACNs 2009 Spring

-Nodes learn 1-hop neighbors’ positions from beaconing -A node forwards packets to its neighbor closest to D  A stateless and scalable routing for Wireless Ad Hoc (Sensor) Networks Greedy Routing (1/2) G.G. Finn ‘87 11 Lemma 4.1: If GR reaches D, it does so with O(d 2 ) cost, where d denotes the Euclidean distance between S and D. pf: the disk with center D and radius d contains at most O(d 2 ) nodes with pairwise distance at least 1. ACNs 2009 Spring

Greedy Routing (2/2) 12 x is a local minimum (dead end) to D; w and y are far from D Greedy Routing not always possible! ACNs 2009 Spring

Face Routing (1/2) Well-known graph traversal: the right-hand rule (1) Traverse a face (2) Requires only neighbors’ positions Fails when there are cross links in the graph!  planar graph, e.g., RNG, GG 13 E. Kranakis, H. Singh, and J. Urrutia ‘99 x y z ACNs 2009 Spring

Face Routing (2/2) 14 Face (Perimeter) traversal on a planar graph S D F1F1 F2F2 F3F3 F4F4 With O(n) messages  Many existing algorithms like GFG, GPSR, GOAFR+, and etc. combine greedy routing with face routing. Walking sequence: F 1 -> F 2 -> F 3 -> F 4 Two primitives: (1) the right-hand rule (2) face-changes ACNs 2009 Spring a

Planar Graph (1/2) 15  Given a radio graph, make a planar sub-graph in which every cross-edge is eliminated. u v w GG (Gabriel Graph) Gabriel Graph u v w Relative Neighborhood Graph (RNG) Relative Neighborhood Graph ACNs 2009 Spring

Planar Graph (2/2) 16 Full Radio GraphGG Sub-graph  Important assumptions - Unit-disk graph & Accurate localization  How well do planarization techniques work in real-world? RNG Sub-graph ACNs 2009 Spring

GOAFR - Other Face Routing 17 S D F1F1 F2F2 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. ACNs 2009 Spring

GOAFR – Other Bounded Face Routing (1/2) 18 D S ACNs 2009 Spring

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. 19 The shortest path between S and D ACNs 2009 Spring

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 ACNs 2009 Spring

GOAFR – Other Adaptive Face Routing (2/2) Theorem 5.3 ▫OAFR reaches the destination with cost O(c 2 (p*)), p* is an optimal path Theorem 6.1 ▫Any deterministic (randomized) geographic ad hoc routing algorithm has (expected) cost Ω(c 2 ) 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(C 2 ). This is asymptotically optimal. 21 ACNs 2009 Spring

GOAFR 22 OAFR greedy fails After First Face Traversal greedy works Greedy Routing ACNs 2009 Spring

GPSR 23 Perimeter Routing greedy fails A location closer than where greedy routing failed greedy fails greedy works Greedy Routing ACNs 2009 Spring

24 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? ACNs 2009 Spring Greedy Face

GOAFR+ 25 Counter p: closer to D than u Counter q: farther from D than u Fall back to greedy routing if p >  q ACNs 2009 Spring

Performance FR OAFR GFG/GPSR GOAFR+ AFR Network Connectivity Greedy Success Rate 26 ACNs 2009 Spring

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(c 2 ) steps 27 ACNs 2009 Spring

Conclusion (2/2) Greedy Routing/MFR ( ) Face Routing GFG/GPSR AFR GOAFR/GOAFR+ 28 Correct Routing Avg-Case Efficient Worst-Case Optimal Comprehensive Simulation ACNs 2009 Spring

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Discussion Lemma 3.3: ▫The shortest path 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. ACNs 2009 Spring 30

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