Geometric Routing without Geometry

Geometric Routing without Geometry

Mirjam Wattenhofer

Roger Wattenhofer

Peter Widmayer

What are Ad-Hoc/Sensor Networks?

Power

Processor

Radio

SensorsMemory

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




– 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 sufficient90 44' 55" East 470 30' 19" North

90 44' 56" East 470 30' 19" North

90 44' 57" East 470 30' 19" North

90 44' 58" East 470 30' 19" North

(0,0)

(1,0)

(1,1)

(2,1)

real coordinates virtual coordinates

vs.

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]

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

(0) (1) (2) (3) (4)

Overview





Pseudo geometric routing in the grid: Naming(4)

(4)

(4)

(4)

(4)

Anchor 1 Anchor 2

(4,4)

(4,2)

(4,6)

(4,8)

(4,10)

Lemma: The naming problem in the grid can be solved with two anchors.

[R.A. Melter and I. Tomescu, Comput. Vision, Graphics. Image Process., 1984]:

landmarks in graphs

Pseudo geometric routing in the grid: Routing

(4,10)

Anchor 1 Anchor 2

(6,4)

(5,11)

(3,9)

(5,9) (6,8)

(5,7)

(7,7)

(6,6)

(5,5)

(6,10)

(4,8)

(7,9)

Rule: pass message to neighbor which is closest to destination

Pseudo geometric routing in the grid: Routing

Anchor 1 Anchor 2

Lemma: The routing problem in the grid can be solved with two anchors.

Overview





Pseudo geometric routing in the UDT: Naming

k

• recursive construction of a unit dist tree (UDT) which needs (n) anchors:


k

# 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

Lemma: in a unit disk tree on n nodes there are (n) leaf-siblings

leaf-siblings


• Leaf-siblings can only be distinguished if one of them is an anchor:

(a,b,c,...)

(a+1,b+1,c+1,...)(a+1,b+1,c+1,...)

Anchor k+1

Anchor 1..Anchor k


• Leaf-siblings can only be distinguished if one of them is an anchor:

(a,b,c,..,1)

(a+1,b+1,c+1,...,2)(a+1,b+1,c+1,...,0)

Corollary: we need (n) anchors to solve the naming problem in the UDT

Anchor k+1

Anchor 1..Anchor k

Pseudo geometric routing in the UDT: Routing

Rule: pass message to neighbor which decreases ci

Invariant: coordinate of destination t is smaller than coordinate of node u t in at least one position ci

Anchor 1 A. 2 A. 2 A. 3

( 1,1,3,3) ( 3,3,1,1)

(2,2,2,2)

(0,2,4,4) (2,0,4,4)

[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 networksunsolved problems: algorithmic anchor selection, partly very high density required, nodes have no unique name

Overview




– 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

[Beerliova et al., WG 2005]: network discovery and

verification

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

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

Overview





Geometric Routing without Geometry: Facts & Figuresmin any local shorte

st path

Line 1 2

Ring 2 3

Grid 2 n)

Tree mc(T) n)

UDG n) n)

Butterfly

n/ log n n)

Hypercube

log n/log log n

n)

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

Questions?Comments?

Geometric Routing without Geometry

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