Geometric Routing without Geometry

Geometric Routing without Geometry Mirjam Wattenhofer Roger Wattenhofer Peter Widmayer

What are Ad-Hoc/Sensor Networks? Radio Power Processor Sensors Memory http://www.greatduckisland.net

Ad-Hoc/Sensor Networks: Abstraction • Unit Disk Graph: Link if and only if Euclidean distance at most 1.

Overview • Introduction • What are sensor networks? • Abstraction • Routing in Ad-hoc networks • With/Without position information • The nuts and bolts of obtaining position information • Pseudo geometric routing • The grid: naming and routing • The UDG: naming and routing • Summary of results & related work

Routing in Ad-Hoc Networks: with/without position information • Without position information: • Flooding does not scale • Distance Vector Routingdoes not scale • Source Routing • increased per-packet overhead • no theoretical results, only simulation • With position information: • Greedy Routing may fail: message may get stuck in a “dead end” • Geometric Routing  It is assumed that each node knows its position

Overview • Introduction • What are sensor networks? • Abstraction • Routing in Ad-hoc networks • With/Without position information • The nuts and bolts of obtaining position information • Pseudo geometric routing • The grid: naming and routing • The UDG: naming and routing • The hypercube: naming and routing • Summary of results & related work

Obtaining Position Information • Attach GPS to each sensor node • Often undesirable or impossible • GPS receivers clumsy, expensive, and energy-inefficient • Equip only a few designated nodes with a GPS • Anchor (landmark) nodes have GPS • Non-anchors derive their position through communication (e.g., count number of hops to different anchors) Anchor density determines quality of solution

What about no GPS at all? • In absence of GPS-equipped anchors...  ...nodes are clueless about real coordinates. • For many applications, real coordinates are not necessary  Virtual coordinates are sufficient 90 44' 58" East 470 30' 19" North 90 44' 56" East 470 30' 19" North (2,1) (1,0) 90 44' 57" East 470 30' 19" North 90 44' 55" East 470 30' 19" North (0,0) (1,1) vs. real coordinates virtual coordinates

What are „good“ virtual coordinates? • Given the connectivity information for each node and knowing the underlying graph is a UDG find virtual coordinates in the plane such that all connectivity requirements are fulfilled, i.e. find a realization (embedding) of a UDG: • each edge has length at most 1 • between non-neighbored nodes the distance is more than 1 • Finding a realization of a UDG from connectivity information only is NP-hard... • [Breu, Kirkpatrick, Comp.Geom.Theory 1998] • ...and also hard to approximate • [Kuhn,Moscibroda, Wattenhofer, DIALM 2004]

(1) (3) (2) (0) Geometric Routing without Geometry • For many applications, like routing, finding a realization of a UDG is not mandatory • Virtual coordinates merely as infrastructure for geometric routing Pseudo geometric coordinates: • Select some nodes as anchors:a1,a2, ..., ak • Coordinate of each node u is its hop-distance to all anchors: (d(u,a1),d(u,a2),..., d(u,ak)) • Requirements: • each node uniquely identified: Naming Problem • routing based on (pseudo geometric) coordinates possible: Routing Problem (4)

(4,4) (4,2) (4,6) (4,8) (4,10) Pseudo geometric routing in the grid: Naming Anchor 1 Anchor 2 (4) (4) (4) (4) Lemma: The naming problem in the grid can be solved with two anchors. (4) [R.A. Melter and I. Tomescu, Comput. Vision, Graphics. Image Process., 1984]: landmarks in graphs

Pseudo geometric routing in the grid: Routing Anchor 1 Anchor 2 Rule: pass message to neighbor which is closest to destination (5,5) (6,4) (3,9) (4,8) (5,7) (6,6) (4,10) (5,9) (6,8) (7,7) (5,11) (6,10) (7,9)

Pseudo geometric routing in the grid: Routing Anchor 1 Anchor 2 Lemma: The routing problem in the grid can be solved with two anchors.

Pseudo geometric routing in the UDT: Naming • recursive construction of a unit dist tree (UDT) which needs (n) anchors: k

Pseudo geometric routing in the UDT: Naming # nodes in distance k from root: L(k) L(k) = 4 L((k-3)/2)  L(k) = (k+3)2/8 # nodes in tree of depth k: N(k) N(k) < 4 N((k-3)/2)+2k  N(k) < 9k2 k Lemma: in a unit disk tree on n nodes there are (n) leaf-siblings leaf-siblings

Pseudo geometric routing in the UDT: Naming • Leaf-siblings can only be distinguished if one of them is an anchor: Anchor 1..Anchor k (a,b,c,...) (a+1,b+1,c+1,...) (a+1,b+1,c+1,...) Anchor k+1

Pseudo geometric routing in the UDT: Naming • Leaf-siblings can only be distinguished if one of them is an anchor: Anchor 1..Anchor k (a,b,c,..,1) (a+1,b+1,c+1,...,0) (a+1,b+1,c+1,...,2) Anchor k+1 Corollary: we need (n) anchors to solve the naming problem in the UDT

Pseudo geometric routing in the UDT: Routing Invariant: coordinate of destination t is smaller than coordinate of node u  t in at least one position ci (2,2,2,2) Rule: pass message to neighbor which decreases ci ( 1,1,3,3) ( 3,3,1,1) (0,2,4,4) (2,0,4,4) Anchor 1 A. 2 A. 2 A. 3 [Khuller, Raghavachari, Rosenfeld, Discrete Applied Mathematics 1996]: optimal anchor selection algorithm for trees

Pseudo geometric routing in the Ad-hoc networks • Naming and routing in grid quite good, in previous UDT example very bad • Real-world Ad-hoc networks are very probable neither perfect grids nor naughty unit disk trees Truth is somewhere in between... • GPS Free Coordinate Assignment and Routing in Wireless Sensor Networks [Caruso et al., Infocom 2005] • GLIDER: Gradient Landmark-Based Distributed Routing for Sensor Networks [Fang et al., Infocom 2005] • practical approach to pseudo geometric routing in Ad-hoc networks • unsolved problems: algorithmic anchor selection, partly very high density required, nodes have no unique name

Overview • Introduction • What are sensor networks? • Abstraction • Routing in Ad-hoc networks • With/Without position information • The nuts and bolts of obtaining position information • Pseudo geometric routing • The grid: naming and routing • The UDG: naming and routing • The hypercube: namig and routing • Summary of results & related work

Pseudo geometric routing in the hypercube: Naming Lemma: To solve the naming problem in a d-dimensional hypercube at least log n/log log n anchors are needed. Proof: each anchor can subdivide the nodes in at most d classes  k anchors can differentiate between at most dk nodes to differentiate between all n nodes it must hold that dk >= n  k >= log n/ log log n [Beerliova et al., WG 2005]: network discovery and verification

Pseudo geometric routing in the hypercube: Routing Lemma: The routing problem in the hypercube can be solved with log n anchors. Proof: choose all nodes with distance one to the origin as anchors  each node in the hypercube can obtain its (real) coordinate as a function of its pseudo coordinate  rule: pass message to neighbor with smallest Hamming distance to destination

Geometric Routing without Geometry: Facts & Figures

Pseudo geometric routing in general graphs: Related Work • Landmarks in Graphs [Khuller, Raghavachari, Rosenfeld, Discrete Applied Mathematics 1996] • results (a.o.): • optimal anchor selection algorithm for trees (also routing feasible with this selection) • NP-hardness proof for optimal anchor selection in general graphs • O(log n)-approximation algorithm for anchor selection in general graphs • Network Discovery and Verification [Beerliova et al., WG 2005] • results (a.o.): • for general graphs there is no o(log n)-approximation algorithm for anchor selection unless P = NP • there must be a selection of log n/log log n anchors solving the naming problem in the hypercube, yet proof not constructive

Pseudo geometric routing in general graphs: Related Work • GPS Free Coordinate Assignment and Routing in Wireless Sensor Networks [Caruso et al., Infocom 2005], GLIDER: Gradient Landmark-Based Distributed Routing for Sensor Networks [Fang et al., Infocom 2005] • practical approach to pseudo geometric routing in Ad-hoc networks • unsolved problems: algorithmic anchor selection, partly very high density required, nodes have no unique name

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Geometric Routing without Geometry