Fine-Grained Localization in Sensor and Ad-Hoc Networks

1
Fine-Grained Localization in Sensor and Ad-Hoc
Networks
Ph.D. Dissertation Defense

• David Goldenberg

Dissertation Advisor Y. Richard Yang Committee
Members Jim Aspnes, A. Stephen Morse,
Avi Silberschatz, Nitin Vaidya (UIUC)
2
Overview

• This dissertation provides a theoretical basis
  for the localization problem, demonstrating
  conditions for its solvability and defining its
  computational complexity.
• We apply our fundamental results on localization
  to identify conditions under which the problem is
  efficiently solvable and to develop localization
  algorithms for a broader class of networks than
  previous approaches could localize.

3
Collaborators (2003-2006)

• Brian D.O. Anderson (Australia National
  University and NICTA)
• James Aspnes
• P.N. Belhumeur (Columbia University)
• Pascal Bihler
• Ming Cao
• Tolga Eren
• Jia Fang
• Arvind Krishnamurthy
• Jie (Archer) Lin
• Wesley Maness
• A. Stephen Morse
• Brad Rosen
• Andreas Savvides
• Walter Whiteley (York University)
• Y. Richard Yang
• Anthony Young

4
Outline

• Introduction to Localization
• Conditions for Unique Localization
• Computational Complexity of Localization
• Localization in Sparse Networks

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Why are Locations Important?

• Wireless ad-hoc networks are an important
  emerging technology
• Small, low-cost, low-power, multi-functional
  sensors will soon be a reality.
• Accurate locations of individual sensors are
  useful for many applications
• Sensing data without knowing the sensor location
  is meaningless. IEEE Computer, Vol. 33, 2000
• New applications enabled by availability of
  sensor locations.
• Location-aware computing
• Resource selection (server, printer, etc.).
• Location aware information services (web-search,
  advertisement, etc.).
• Sensor network applications
• Inventory management, intruder detection, traffic
  monitoring, emergency crew coordination,
  air/water quality monitoring, military/intelligenc
  e apps.

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Example Great Duck Island Sensor Network

• Monitoring breeding of Leachs Storm Petrels
  without human presence.
• 15 minute human visit leads to 20 offspring
  mortality.
• Sensors need to be small to avoid disrupting bird
  behavior.

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Great Duck Island Deployment Goals

• Occupancy pattern of nests?
• Environmental changes around the nests over time?
• Environmental variation across nests?
• Correlation with breeding success?
• Light, temperature, infrared, and humidity
  sensors installed.
• Infrared sensors detect presence of birds in
  nests.

• Sensor locations critical to interpreting data.
• Locations determined by manual configuration, but
  this will not be possible in the general case.

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Example ZebraNet Sensor Network

• Biologists want to track animals to study
• Interactions between individuals.
• Interactions between species.
• Impact of human development.
• Current tracking technology VHF collar
  transmitters
• Wishlist
• 24/7 position, data, and interaction logs.
• Wireless connectivity for mobility.
• Data storage to tolerate an intermittent base
  station.
• ZebraNet
• Mobile sensor net with intermittent base station.
• Records position using GPS every 3 minutes.
• Records Sun/shade info.
• Detailed movement information (speed, movement
  signature) 3 minutes each hour.
• Future head up/head down, body temperature,
  heart rate, camera.
• Goal, full ecosystem monitoring (zebras, hyenas,
  lions).

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Military Applications

• Intelligence gathering (troop movements, events
  of interest).
• Detection and localization of chemical,
  biological, radiological, nuclear, and explosive
  materials.
• Sniper localization.
• Signal jamming over a specific area.
• Visions for sensor network deployment
• Dropped in large numbers from UAV.
• Mortar-Launched.

!
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Why is Localization a Non-Trivial Problem?

• Manual configuration
• Unscalable and sometimes impossible.
• Why not use GPS to localize?
• Hardware requirements vs. small sensors.
• Obstructions to GPS satellites common.
• GPS satellites not necessarily overhead.
• Doesnt work indoors or underground.
• GPS jammed by sophisticated adversaries.
• GPS accuracy (10-20 feet) poor for short range
  sensors.

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Fine-Grained Localization (Savvides, 2001)

• Physically
• Network of n nodes, m of which have known
  location, existing in space at locations
  x1xm,xm1,,xn.
• Set of some pair-wise inter-node distance
  measurements.
• Usually between proximal nodes (iff d lt r in unit
  disk networks).
• Abstraction
• Given Graph GN, x1,...,xm, and d, the edge
  weight function.
• Find Realization of the graph.

x1,x2,x3
3
1
4
2
d14, d24, d25, d35, d45
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3
5
2
4
x4, x5
1
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Ranging Systems

• TDoA Time Difference of Arrival
• Uses ultrasound and radio signals to determine
  distance.
• Range of meters, cm accuracy.
• Possible to increase sensing range by increasing
  transmission power.

MIT cricket mote
UCLA medusa mote 2 (2002)
Yale ENALAB XYZ Motes
UCLA medusa mote (2001)
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Our Contributions

• Graph-theoretic conditions for the unique
  solvability of the localization problem in the
  plane.
• Proof that the problem is NP-complete even for
  the idealized case of unit-disk networks.
• Constructive characterization of classes of
  uniquely localizable and easily localizable
  networks for the plane and 3D.
• A localization algorithm that localizes a wider
  class of networks than was possible with existing
  approaches.
• In-depth study of the localizability properties
  of random networks
• New adaptive localizability-optimizing deployment
  strategies.
• Impact of non-uniquely localizable nodes on
  network performance.

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Outline

• Introduction to Localization
• Conditions for Unique Localization
• Computational Complexity of Localization
• Localization in Sparse Networks

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Unique Localizability

• Network is uniquely localizable if there is
  exactly one set of points xm1,,xn consistent
  with GN, x1,,xm and dE to R.
• Can we determine localizability by graph
  properties alone? (as opposed to the properties
  of d).
• In the plane, yes (more or less). Properties of
  the graph determine solvability in the generic
  case.
• Probability 1 for randomly generated node
  locations.

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Degenerate Cases Fool Abstraction
2
x1, x2, x3
2
d14, d24, d34
4
1
4
1
3
3
In general, this network is uniquely localizable.
probability 1 case
first case x4 second case ???
4
1
2
?
1
3
2
3
?
probability 0 case
In degenerate case, it is not The constraints
are redundant.
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Continuous Non-Uniqueness

• Continuous non-uniqueness
• Can move points from one configuration to another
  while respecting constraints.
• Excess degrees of freedom present in
  configuration.
• A formation is RIGID if it cannot be continuously
  deformed.

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Condition for Rigidity

• Purely combinatorial characterization of generic
  rigidity in the plane.
• 2n-3 edges necessary for rigidity, and

Lamans condition A graph G with 2n-3 edges is
rigid in two dimensions if and only if no
subgraph G has more than 2n-3 edges. where
n is the number of vertices in G
Lamans condition is a statement that any rigid
graph with n vertices must have a set of 2n-3
well-distributed edges.
Rigid!
not rigid!
Not enough edges Enough edges but not well
distributed Just right
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Discontinuous Non-Uniqueness in Rigid Graphs
2
3
1
Flip Ambiguities
0
4

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6
1 config
24 configs
1
4
5
Discontinuous Flex Ambiguities
2
3
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Unique Graph Realization
Solution
G must be rigid.
G must be 3-connected.
b
c
b
G must be redundantly rigid It must remain rigid
upon removal of any single edge.
d
f
a
e
e
c
a
d
f
A graph has a unique realization in the plane iff
it is redundantly rigid and 3-connected (globally
rigid). Hendrickson, 94
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Is The Network Uniquely Localizable?

• Problem By looking only at the physical
  connectivity structure, we would neglect our a
  priori knowledge of beacon positions.
• Solution The distances between beacons are
  implicitly known!
• By adding all edges between beacons to GN, we get
  the Grounded Graph of the network, whose
  properties determine network localizability.
• Theorem A network is generically uniquely
  localizable iff its grounded graph is globally
  rigid and it contains at least three beacons.
• By augmenting graph structure in this way, we
  fully express all available constraint
  information in a graph.

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Examples of GR graphs - Constructions

• Every globally rigid graph has a spanning
  subgraph that is minimally globally rigid.
• Every minimally globally rigid graph can be
  constructed inductively starting from K4 by a
  series of extensions (Berg-Jordan 01)
• New node w and edges uw and vw replace edge uv.
• Edge wx added for some node x distinct from u, v.
• Minimal globally rigid graphs have 2n-2 edges.

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Light edges are those subdivided by the extension
operation.
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Examples of Global Rigidity
Globally rigid components in green.
Random network avg node degree 6.
Regularized random network avg node degree 4.5.
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Construction Using Trilateration

• A position is uniquely determined by three
  distances to three non-collinear references.
• Minimal trilateration graphs formed by
  trilateration extension
• New node w and edges uw, vw, xw added, for u, v,
  x distinct.
• Minimal trilateration graphs are globally rigid.
• Minimal trilateration graphs have 3n-6 edges.

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Light edges are those removed in extension for
minimally GR graph but not in trilateration.
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Trilateration Graphs

• A trilateration graph G is one with an
  trilaterative ordering an ordering of the
  vertices 1,...,n such that the complete graph on
  the initial 3 vertices is in G and from every
  vertex j gt 3, there are at least 3 edges to
  vertices earlier in the sequence.
• Trilateration graphs are globally rigid.

Hand-made trilateration avg degree 6.
Trilateration graph from mobile network avg
degree 9.
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Tripled Connected Graphs are Trilateration
Graphs

• Theorem
• Let G (V,E) be a connected graph.
• Let G3 (V,E? E2 ? E3) be the graph formed from
  G by adding an edge between any two vertices
  connected by paths of 2 or 3 edges in E.
• Then G3 is a trilateration graph.

Example where G is a path.
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Doubled 2-connected Graphs are Globally
Rigid in 2D

• Theorem
• Let G be a 2-connected graph.
• Then G2 is globally rigid.

One gets G2 by doubling sensing radius or
measuring angles between adjacent edges.
Minimally GR graph by extension
Doubled cycles always have two edges more than a
minimally GR graph, so they are globally rigid.
Doubled cycle
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Tripled Biconnected Graphs are Globally Rigid
in 3D

• There is no known generic characterization of
  global rigidity in 3D, but our result on doubled
  graphs extends to 3D.
• Theorem
• Let G be a 2-connected graph.
• Then G3 is globally rigid in 3D.

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Summary of Constructive Characterization of
Globally Rigid Graphs

• 2D
• 3-connectivity necessary for GR.
• G2 GR if G 2-connected.
• G3 GR if G connected.
• 3D
• G3 GR in 3D if G 2-connected.
• G4 GR in 3D if G connected.
• Unique localizability by increasing sensing
  range, given initial connectivity.
• Conditions under which additional information can
  help.

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Outline

• Introduction to Localization
• Conditions for Unique Localization
• Computational Complexity of Localization
• Localization in Sparse Networks

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Localization
3
5
Search problem
Decision problem
2
4
Grounded graph
1
x1,x2,x3
1
4
1
4
2
d14, d24, d25, d35, d45
2
3
5
This graph has a unique realization. What is it?
3
5
Does this have a unique realization?
This problem is in general NP-hard.
???
Rigidity theory
x4,x5
Yes/No
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Computational Complexity

• Intuitively, reflection possibilities are linked
  with computational complexity

Suppose all edge distances known for small
triangles. Localization goes working out from any
beacon. Triangle reflection possibilities grow
exponentially.
and reflection possibilities are only sorted out
when one gets to another beacon.
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Complexity of GR Graph Realization

• If a network is localizable, how does one go
  about localizing it?
• It is NP-hard to localize a network in R2 even
  when it is known to be uniquely localizable.
• We will use two tools in our argument
• The NP-hard set-partition problem.
• The globally rigid wheel graph Wn.

The set partition problem Input A set of
numbers S. Output Can S be partitioned into two
subsets A and S-A such that the sum
of numbers in each subset is equal?
W6
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NP-hardness of Realization
Theorem Realization of globally rigid weighted
graphs that are realizable is NP-hard
Proof sketch Assume we have algorithm X that
takes as input a realizable globally rigid
weighted graph and outputs its unique
realization. We will find the set-partition of
the partitionable set S scaled w.l.o.g so that
the sum of elements in S is less than p/2 by
using calls to X. Suppose we have Ss1,s2,s3,s4
with a set-partition. Construct a graph G along
with its edge weights for X
Even without Set Partition, we have the edge
weights of G di,i12sin(si/2) that
uniquely determine the realization . When G is
realized, we obtain the picture on the left, from
which we obtain set partition!
sin(s1/2)
1
sin(s1/2)
2
3
s1s4s2s3
s4
s2
rights lefts
s1
4
s3
This is a realization of W5!
0
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Localization Complexity for Sparse Networks

• Problem with previous result is that edges exist
  arbitrarily.
• Graphs used in previous proof unlikely to arise
  in practice.
• In realistic networks, edges are more likely to
  exist between close nodes, and do not exist
  between distant nodes.
• Unit Disk Graphs edge present if distance
  between nodes less than parameter r.
• Therefore if edge absent, distance between nodes
  is greater than r.
• Does this information help us solve the
  localization problem?

1
2
3
Red edge would exist in unit disk graph, so unit
disk graph localization would not solve Set
Partition.
4
0
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Complexity of Localizing Unit Disk Graphs

• Theorem Localization for sparse sensor networks
  is NP-hard.
• Method
• Reduction from Circuit Satisfiability to Unit
  Disk Graph Reconstruction.
• Reduction is by construction of a family of
  graphs that represent Boolean circuits.
• Rigid bodies in the graph represent wires.
• Relative position of rigid bodies in the graph
  represent signals on wires.
• NOT and AND gates built out of constraints
  between these bodies expressed in the graph
  structure.
• There is a polynomial-time reduction from Circuit
  Satisfiability to Unit Disk Graph Reconstruction,
  in which there is a one-to-one correspondence
  between satisfying assignments to the circuit and
  solutions to the resulting localization problem.

Unit Disk Graph Reconstruction (decision
problem) Input Graph G along with a parameter
r, and the square of each edge length (luv)2 (to
avoid irrational edge lengths). Output YES iff
there exists a set of points in R2 such that
distance from u to v is luv if uv is an edge in G
and greater than r otherwise.
Circuit Satisfiability (NP-hard) Input A
boolean combinatorial circuit. Composed of AND,
OR, and NOT gates Output YES iff the circuit is
satisfiable.
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Localization in Trilateration Graphs

• As one adds more edges, localization becomes
  easier There are classes of globally rigid graph
  which are easy to localize.
• Trilateration graphs are localizable in
  polynomial time.
• Remember One gets a trilateration graph from a
  connected network by tripling the sensing radius.

• Algorithm
• If initial 3 vertices known, localize vertices
  one at a time until all vertices localized.
• Else starting with each triangle in the graph,
  proceed as above until all localized.
• O(V2) or O(V5).

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Connectivity in Random Networks
The following guarantees Gn(r) is k-connected
with high probability for some constant c large
enough and constant k Penrose, 99

• The random geometric graph Gn(r) is the random
  graph associated with formations with n vertices
  with all links of length less than r, where the
  vertices are points in 0,12 generated by a two
  dimensional Poisson point process of intensity n.

• Note Need nr2/(log n) gt c, for some c, to
  guarantee even connectivity.
• Theorem If nr2/(log n) gt 8, with high
  probability, Gn(r) is a trilateration graph.

• This identifies conditions under which a simple
  iterated trilateration algorithm will succeed in
  localization.

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Trilateration in Random Networks

• Sensors have 2 modes.
• Sensors determine distance from heard
  transmitter.
• All sensors are pre-placed and plugged in

But how fast?
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Asymptotics of Trilateration in Random Networks
Beacons Sensing radius Etloc



Running times to complete localization using
trilateration for different beacon densities.
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NP-hardness of Localization

• Fine-grained localization is NP-hard due to
  NP-hardness of realizing globally rigid graphs.
• This means that localization of networks in
  complete generality is unlikely to be efficiently
  solvable.
• Motivates search for reasonable special cases and
  heuristics. Explains hit-or-miss character of
  previous approaches.
• Changing sensing radius can predictably convert
  connectedness to global rigidity and
  trilateration.

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Outline

• Introduction to Localization
• Conditions for Unique Localization
• Computational Complexity of Localization
• Localization in Sparse Networks

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Motivation

• Being able to precisely localize only
  trilateration networks is unsatisfying.
• Trilateration networks contain significantly more
  constraints than necessary for unique
  localizability.
• Can we localize networks with closer to the
  minimal number of constraints?

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Red edges unnecessary for unique localizability.
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Trilateration graph
Globally rigid subgraph
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Bilateration Graphs

• A bilateration graph G is one with a bilateration
  ordering an ordering of the vertices 1,...,n
  such that the complete graph on the initial 3
  vertices is in G and from every vertex j gt 3,
  there are at least 2 edges to vertices earlier in
  the sequence.
• Theorem Bilateration graphs are rigid (but not
  globally rigid).
• Theorem Let G (V,E) be a connected graph.
• Then G2 is a bilateration graph.
• Bilateration graphs are finitely localizable in
  O(2V) time.

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6

• Algorithm
• If initial 3 vertices known, finitely localize
  vertices one at a time by computing all possible
  positions consistent with neighbor positions
  until all vertices finitely localized.
• Else starting with each triangle in the graph,
  proceed as above until all finitely localized.

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0
1
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4
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Localization in Doubled Cycles

• Based on finite localization of bilateration
  graphs, localization is uniquely computable for
  globally rigid doubled cycles.
• Completes in O(2V) time.
• Assumes nodes in general position.

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6

• Sweep Algorithm
• Fix the position of three vertices.
• Until no progress made
• Finitely localize each vertex connected to two
  finitely localized vertices.
• Remove possibilities with no consistent
  descendants.

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Localization in Doubled 2-connected Graphs

• 2-connected graphs are a union of cycles (they
  have an Ear Decomposition).
• The ear decomposition gives a ordering in which
  cycles may be localized using previous algorithm.
• Note This means if we have angles, we can
  localize 2-connected networks.

Biconnected network with its ear decomposition.
Doubled biconnected network.
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Localization on General Sparse Networks

• Worst-case exponential time algorithm for
  localization in sparse networks

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• For which types of network does sweep
  localization work?
• Theorem Shell sweep finitely localizes
  bilateration networks.
• Theorem Shell sweep uniquely localizes globally
  rigid bilateration networks.
• If G is connected, when run on G2, shell sweep
  produces all possible positions for each node.
  If G2 globally rigid, gives the unique positions.
• Question How many globally rigid networks are
  also bilaterations?

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Shell Sweep on Random Network

• Typical random graph.
• Starting nodes randomly chosen.
• Shell sweep uniquely localizes localizable
  portion.
• Also non-uniquely localizes nodes rigidly
  connected to localized region.

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Performance on Large Network

• 500 node graph with considerable anisotropy and
  4.5 average degree.
• Shell sweep computes in lt5 seconds with no
  intermediate position set exceeding 128.

As a JAVA applet on a zoo node with a dual
2.8GHz CPU and 2GB RAM
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Failing Case

• Globally rigid network.
• Connection between clusters unbridgeable by
  bilateration.

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Extent of Sweep Localization

• Sweeps localizes more nodes than trilateration,
  and almost all localizable nodes!
• In regular networks, sweeps localizes
  significantly more nodes than trilateration.
• Most incremental localization algorithms are
  trilateration based.
• Key point Many globally rigid random geometric
  graphs are bilateration graphs.

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Summary of Localization Density Spectrum

• Localization is NP-hard in general, but there are
  classes of graphs that are easy to localize.
• Complete graphs.
• Trilateration graphs.
• Graphs that we know how to localize in worst-case
  exponential time
• Doubled biconnected graphs.
• Basic idea more edges make localization easier.
• Goal to understand which networks can be
  localized and which are problematic.

Consider all possible networks on n sensors
Some networks can be localized in
O(V5) Trilateration graphs with unknown
ordering
Some networks can be localized in O(V2)
time Trilateration graphs with known ordering
Some networks can be localized in exponential
time Doubled biconnected graphs Globally rigid
bilateration graphs
Unlocalizable
53
When Does Localization Become Easy?
1
Dense
Easy
Complete Graph
3r1
Polynomial time
Trilateration Graph
Bilateration Graph
2r2
Exponential
Globally Rigid
NP-hard
3-connected
r3
Sparse
Unsolvable
0
Complexity of realization
Number of edges
Sensing radius in Gn(r)
54
Conclusion and Future Work

• Formalized the localization problem and its
  solvability.
• Showed that the problem is fundamentally
  computationally hard.
• Constructively characterized easily localizable
  networks.
• Provided algorithm that localizes more nodes than
  previous incremental algorithms.
• Next
• Localization using maps.
• Localization using angular order information.
• Localization in networks of mobile nodes.
• Localization in 3D or on 3D surfaces.
• Full system from deployment to localization.

55
Our Work in the Field

• Rigidity, Computation, and Randomization in
  Network Localization - Infocom 2004
• Conditions for unique fine-grained localization.
• Initial computational complexity results.
• On the Computational Complexity of Sensor
  Network Localization - Algosensors 2004
• Computational complexity results.
• A Theory of Network Localization -
  Transactions on Mobile Computing 2006
• Graphical Properties of Easily Localizable
  Sensor Networks - under review
• Characterizing easily localizable ad-hoc
  networks.
• Precise Localization in Sparse Sensor Networks
  - Accepted to Mobicom 2006
• Algorithm for localization in sparse ad-hoc
  networks.
• Localization in Partially Localizable Networks
  - Infocom 2005
• Investigation of partially localizable networks.
• Localizability-aware network deployment.
• Towards Mobility as a Network Control Primitive
  - Mobihoc 2004
• Location-aware controlled node-mobility algorithm
  for sensor network optimization.

56
Acknowledgements

• I would like to thank all my collaborators,
  without whom this work would not have been
  possible.
• Brian D.O. Anderson (Australia National
  University and NICTA)
• James Aspnes
• P.N. Belhumeur (Columbia University)
• Pascal Bihler
• Ming Cao
• Tolga Eren
• Jia Fang
• Arvind Krishnamurthy
• Jie (Archer) Lin
• Wesley Maness
• A. Stephen Morse
• Brad Rosen
• Andreas Savvides
• Walter Whiteley (York University)
• Y. Richard Yang
• Anthony Young

THANK YOU FOR LISTENING ANY QUESTIONS?