Panos Pardalos
Distinguished Professor
Center for Applied Optimization
Dept. of Industrial and
Systems Engineering,
University of Florida
DIMACS/DyDAn Workshop: Approximation Algorithms
in Wireless Ad Hoc and Sensor Networks
April 22 - 24, 2009
Sensors Everywhere
 Introduction
 Data Fusion
 Sensor Network Design
 Sensor Network Localization
 Sensor Scheduling
 Network Interdiction
Sensors Everywhere
 Introduction
 Data Fusion
 Sensor Network Design
 Sensor Network Localization
 Sensor Scheduling
 Network Interdiction
What are sensors?
 A sensor is a device that responds to a physical stimulus (e.g.
heat, light, sound, pressure, magnetism, or motion). It collects
and measures data regarding some property of a phenomenon,
object, or material. Typical sensors are cameras, radiometers and
scanners, lasers, radio frequency receivers, radar systems, sonar,
thermal devices, seismographs, magnetometers, gravimeters,
and scintillometers.
 The term "Remote Sensing" indicates that the measuring device
is not physically in close proximity with the phenomenon being
observed.
 Each
year hundreds millions of sensors are
manufactured. They are in domestic appliances,
medical equipment, industrial control systems, airconditioning systems, aircraft, satellites and toys.
 Sensors are becoming smarter, more accurate and
cheaper. They will play an ever increasing role in just
about every field imaginable.
 How can nanotechnology improve the performance of
sensors?
 The application of nanotechnology to sensors should
allow improvements in functionality. In particular,
new biosensor technology combined with micro and
nanofabrication technology can deliver a huge range of
applications. They should also lead to much decreased
size, enabling the integration of nanosensors into
many other devices.
Sensor Networks
 A sensor network is a collection of some (sometimes
even hundreds & thousands) smart sensor nodes
which collaborate among themselves to form a sensing
network.
Sensor Applications
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Homeland security
Radiation detection and standards
X-ray detectors and imaging
Integrated System Health Management (ISHM)
Multisensor Data Fusion
Nondestructive Evaluation and Remote Sensing
Commercial Development
Environmental sensing
Medical/healthcare sensing
Robotic and remote sensing Tomography
Domestic electronics and smart homes
Crime prevention
Automotive and aerospace
Leisure industry and toys
Food and agriculture
Marine
Energy and Power
Sensors Everywhere
 Introduction
 Data Fusion
 Sensor Network Design
 Sensor Network Localization
 Sensor Scheduling
 Network Interdiction
Data Fusion
 Combine information from many sensors to have a better
picture than the sensors were used individually
 More accurate, more complete, more reliable
 Sensors fusion algorithms use machine learning
techniques:
 Statistical inference and forecasting
 Kalman Filter
 Bayesian Networks
 Neural Networks
 Fuzzy Logic
 Dempster-Shaffer
Sensor Network Design
 Finding optimal network topology accounts the
following characteristics:
 Fault tolerance
 The ability to sustain sensor network functionalities
without any interruption due to sensor node failures
 Scalability
 A well designed architecture must be able to work with
large number of nodes
 Costs constraints
 Deployment, Maintenance, etc
 Hardware constraints
 Size, Weight, Transmitting, etc
Sensors Everywhere
 Introduction
 Data Fusion
 Sensor Network Design
 Sensor Network Localization
 Sensor Scheduling
 Network Interdiction
Sensor Network Localization
 Network topology identification:
 Ad hoc and dynamics networks;
 Sensor’s parameters can depend on it’s location:
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Transmission characteristics;
Energy consumption;
Reliability.
 Installing GPS receivers in every sensor – too expensive;
 Mathematical programming techniques often allow to
find efficient solutions.
Ad hoc positioning system using
angle of arrival
 Typically, a few nodes of the network know their
location - landmarks (equipped with GPS);
 The rest of the nodes can communicate to other nodes;
 Every node has a capability to determine the angle of
the arriving signal;
 Every node in the network has fixed main axis to
measure all angles against it.
 Every node can only communicate with its neighbors
within the radio range (they may not know their
location).
Ad hoc positioning system using
angle of arrival
Nodes with angle of arrival (AOA) capability
Ad hoc positioning system using
angle of arrival
 Nodes immediately adjacent to a landmark get their
angle directly from the landmark.
 If a node has some neighbors with orientation for a
landmark, it will be able to compute its own
orientation with respect to that landmark, and forward
it further into the network.
 Knowledge of orientation to two other nodes (which
are not on one line) allows to calculate location of the
node by triangulation.
Ad hoc positioning system using
angle of arrival
Node A computes its orientation to remote node L
using information from B and C
Ad hoc positioning system using
angle of arrival
Probability for a node to satisfy conditions
necessary for orientation forwarding
Ad hoc positioning system using
angle of arrival
 The proposed method calculates position of nodes in
Ad hoc network where nodes can measure angle of
arriving signal;
 All nodes have AOA capability and only a fraction have
self position capability
 Simulations showed that resulted positions have an
accuracy comparable to the radio range between
nodes, and resulted orientations are usable for
navigational or tracking purposes.
Localization via Semidefinite
Programming
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Tomorrow (April, 23)
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Semidefinite Programming, Graph Realization, and
Sensor Network Localization. Yinyu Ye, Stanford
University
Reduction to Semidefinite Programming
Solution existence
Statistical interpretation of the formulation (distance
values are random with normally distributed
measurement errors)
Reference
 Sorokin, A.; Boyko N.; Boginski V.; Uryasev S.; Pardalos P.
Mathematical Programming Techniques for Sensor
Networks. Algoritms, 2009, p. 565-581
Sensors Everywhere
 Introduction
 Data Fusion
 Sensor Network Design
 Sensor Network Localization
 Sensor Scheduling
 Network Interdiction
Sensor Scheduling
 Scheduling problem – m sensors, n sites to observe,
n>m. The problem is to find the schedule that reduces
potential loss of limited observations.
 Single sensor scheduling
 Multiple sensor schedule using percentile type
constrains
Sensor Scheduling
 Scheduling problem – m sensors, n sites to observe,
n>m. The problem is to find the schedule that reduces
potential loss of limited observations.
 Single sensor scheduling
 Multiple sensor schedule using percentile type
constrains
Single Sensor Scheduling
 The simplest case is to model one sensor that observes
a group of sites at discreet time point
 Time for changing a site being observed is negligibly
small
 Assume we need to observe n sites during T time
periods
 During every period a sensor is allowed to watch only
at one of n sites
Single Sensor Scheduling
Decision variable:
Penalty for not observing site i at time t:
ai - fixed penalty;
bi ,t - variable penalty;
yi ,t - time of last observing site i before time moment t;
yi ,t is set to t if and only if the sensor is observing site i at time t
otherwise it remains constant
- only one site may be observed at a time
Single Sensor Scheduling
Problem Formulation:
Reference
 This problem was first formulated for one sensor in
 M. Yavuz and D.E. Jeffcoat. Single sensor
scheduling for multi-site surveillance. Technical
report, Air Force Research Laboratory, 2007.
Sensor Scheduling
 Single sensor scheduling
 Multiple sensor schedule using percentile type
constrains
Multiple Sensor Scheduling
 Next talk: Vladimir Boginski will present sensors
scheduling problem
 Multiple sensors
 Stochastic Setup
 Robust formulation using Conditional Value at Risk
(CVaR)
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Joint work with N. Boyko, T. Turko, D.E. Jeffcoat, S. Uryasev,
P.M. Pardalos
Sensors Everywhere
 Introduction
 Data Fusion
 Sensor Network Design
 Sensor Network Localization
 Sensor Scheduling
 Network Interdiction
Network Interdiction
 An important issue in military applications is to
neutralize the communication in the sensors network
of the enemy
 Given a graph whose arcs represent the
communication links in the graph.
 (Offense) Select at most k nodes to target whose
removal creates the maximum network disruption.
 (Defense) Determine which of your nodes to protect
from enemy disruptions.
Problem Definition
 Decision Version: K-CNP
 Input: Undirected graph G = (V,E) and integer k
 Question: Is there a set M, where M is the set of
all maximal connected components of G
obtained by deleting k nodes or less, such that

iM
 i ( i  1)
2
K
where σi is the cardinality of component i, for all i
in M?
Theoretical Results
 Lemma 1: Let M be a partition of G = (V,E) in to L
components obtained by deleting a set D, where
 | V | k 
|D| = k Then the objective
(| V | k function
)
 1
 i ( i  1)
 L


 2
2
iM
with equality holding if and only σi = σj, for all i,j in
M, where σi is the size of ith component of M.
 Objective function is best when components are of
average size.
Theoretical Results
 Lemma 2: Let M1 and M2 be a two sets of
partitions of G = (V,E) obtained by deleting a set
D1 and D2 sets of nodes respectively, where |D1|
= |D2| = k. Let L1 and L2 be the number of
components in M1 and M2 respectively, and L1 ≥
L2. If σi = σj, for all i,j in M1, then we obtain a better
objective function value by deleting D1.
Proof of NP-Completeness
 NP-complete: Reduction from Independent Set
Problem by a simple transformation and the result
follow from the above Lemmas.
Formulation
 Let ui,j = 1, if i and j are in the same component of
G(V \ A), and 0 otherwise.
 Let vi = 1, if node i is deleted in the optimal solution,
and 0 otherwise.
 We can formulate the CNP as the following integer
linear program
Formulation
Formulation
If i and j in different
components and
there is an edge
between them, at
least one must be
deleted
Formulation
Number of nodes
deleted is at most
k.
Formulation
For all triplets
(i,j,k), if (i,j) in
same comp and
(j,k) in same comp,
then (i,k) in same
comp.
Heuristics
 We implement a heuristic based on Maximal
Independent Sets
 Why? Because induced subgraph is empty
 Maximum Independent Set provides upper bound on #
of components in optimal solution.
 Greedy type procedure
 Enhanced with local search procedure
 Results are excellent
 Heuristic obtains optimal solutions in fraction of time
required by CPLEX
 Runs in O(k2 + |V|k) time.
Results
•This is the case you just saw!!
•Optimal solutions computed for all values of k for this graph
•The solutions are computed very quickly
Conclusions and Future Work
 Identified nodes of sparse
 Breakdown communication
 Integer Programming and Heuristics
 Approximation algorithms
 Weighted version of the problems
Reference
 A. Arulselvan, C.W. Commander, P.M. Pardalos, and
O. Shylo. Managing network risk via critical node
identification. Risk Management in
Telecommunication Networks, N. Gulpinar and B.
Rustem (editors), Springer, 2009
Conclusions
 Applications
 Health care
 Military
 Security and law enforcement
 Satellite surveillance
 … essentially Everywhere!
 Research Directions
 Computational complexity
 Stochasticity
 Robustness
 …
Thank You!
Questions?