Dept. of Industrial and

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7 Νοε 2013 (πριν από 4 χρόνια και 3 μέρες)

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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,

air
-
conditioning

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


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:


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



Tomorrow (April, 23)


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

Decision variable:

Penalty for not observing site
i

at time
t
:



-

fixed penalty;


-

variable penalty;


-

time of last observing site
i

before time moment
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





i
a
t
i
b
,
t
i
y
,
t
i
y
,
Single Sensor Scheduling


Problem Formulation:


Single Sensor Scheduling



Reference



This problem was first formulated for one sensor in




M. Yavuz and D.E. Je
ff
coat. 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)


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




where
σ
i

is the cardinality of component
i
, for all
i

in
M
?







M
i
i
i
K
2
)
1
(


Theoretical Results



Lemma 1
: Let
M

be a partition of
G = (V,E)

in to
L

components obtained by deleting a set
D
, where
|D| = k

Then the objective function




with equality holding if and only
σ
i

=

σ
j
, for all
i,j

in
M
, where
σ
i

is the size of
i
th

component of
M
.


Objective function is best when components are of
average size.
















M
i
i
i
L
k
V
k
V
2
1
|
|
)
|
(|
2
)
1
(


Theoretical Results



Lemma 2:

Let
M
1

and

M
2

be a two sets of
partitions of
G = (V,E)

obtained by deleting a set
D1 and D2
sets of nodes respectively, where
|D
1
|
= |D
2
| = k.

Let
L
1

and
L
2

be the number of
components in
M
1

and

M
2

respectively, and
L
1


L
2
. If
σ
i

=

σ
j
, for all
i,j

in
M
1
, then we obtain a better
objective function value by deleting
D
1
.


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
u
i,j

= 1
, if
i

and
j

are in the same component of
G(V
\

A)
, and
0

otherwise.


Let
v
i

= 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(k
2

+ |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?