Post on 01-Feb-2016
description
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
• 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 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
• 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
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
• 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
Pseudo geometric routing in the UDT: Naming
k
• recursive construction of a unit dist tree (UDT) which needs (n) anchors:
Pseudo geometric routing in the UDT: Naming
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
Pseudo geometric routing in the UDT: Naming
• 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
Pseudo geometric routing in the UDT: Naming
• 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
• 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
[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
• 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
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?