1. Geometric Routing without Geometry
Mirjam Wattenhofer
Roger Wattenhofer
Peter Widmayer

2. What are Ad-Hoc/Sensor Networks?
Radio
Power
Processor
Sensors
Memory
In the spring of 2002, the Intel Research Laboratory at Berkeley initiated a collaboration with the College of the Atlantic in Bar Harbor and the University of California at Berkeley to deploy wireless sensor networks on Great Duck Island, Maine. These networks monitor the microclimates in and around nesting burrows used by the Leach's Storm Petrel. Our goal is to develop a habitat monitoring kit that enables researchers worldwide to engage in the non-intrusive and non-disruptive monitoring of sensitive wildlife and habitats. At the end of the field season in November 2002, well over 1 million readings had been logged from 32 motes deployed on the island. Each mote has a microcontroller, a low-power radio, memory and batteries. For habitat monitoring , we added sensors for temperature, humidity, barometric pressure, and mid-range infrared. Motes periodically sample and relay their sensor readings to computer base stations on the island. These in turn feed into a satellite link that allows researchers to access real-time environmental data over the Internet. In June 2003, we deployed a second generation network with 56 nodes. The network was augmented in July 2003 with 49 additional nodes and again in August 2003 with over 60 more burrow nodes and 25 new weather station nodes. These nodes form a multihop network transferring their data back "bucket brigade" style through dense forest. Some nodes are more than 1000 feet deep in the forest providing data through a low power wireless transceiver.

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

4. Overview Introduction Routing in Ad-hoc networks
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

5. Routing in Ad-Hoc Networks: with/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
source routing has the disadvantage of increased
per-packet overhead. With source routing, the size of each
packet is increased in order to carry the source route of hops
through which the packet is to be forwarded. Since the source
route is always present in the packet, the extra network overhead
caused by the presence of the source route is incurred
not only when the packet is originated, but also each time
it is forwarded to the next hop. This extra network overhead
decreases the bandwidth available for transmission of
data, increases the transmission latency of each packet, and
consumes extra battery power in the network transmitter and
receiver hardware.

6. Overview Introduction Routing in Ad-hoc networks
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

7. 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

8. 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' 56" East 
470 30' 19" North
90 44' 58" East 
470 30' 19" North
(2,1)
(1,0)
90 44' 55" East 
470 30' 19" North
90 44' 57" East 
470 30' 19" North
(0,0)
(1,1)
vs.
real coordinates
virtual coordinates

9. 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]

10. 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)

11. Overview Introduction Routing in Ad-hoc networks
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

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

13. 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)

14. Pseudo geometric routing in the grid: Routing
Anchor 1
Anchor 2
Lemma: The routing problem in the grid can be solved with two anchors.
real ad hoc networks not so regular, more holes, some places denser -> next example of a real bad unit disk graph

15. Overview Introduction Routing in Ad-hoc networks
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

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

17. 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

18. 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

19. 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

20. 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
we checked that with the such chosen anchors also routing is possible
[Khuller, Raghavachari, Rosenfeld, Discrete Applied Mathematics 1996]: optimal anchor selection algorithm for trees

21. 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

22. Overview Introduction Routing in Ad-hoc networks
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

23. 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
grid a special unit disk graph…
[Beerliova et al., WG 2005]:
network discovery and verification

24. 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
grid a special unit disk graph…

25. Overview Introduction Routing in Ad-hoc networks
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

26. Geometric Routing without Geometry: Facts & Figures
min
any
local
shortest path
Line 1 2
Ring 3
Grid
(n)
Tree
mc(T)
UDG
Butterfly
n/ log n
Hypercube
log n/log log n

27. 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]
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

28. 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

29. Questions? Comments?