localization goldenberg defense template

8/17/2019 Localization Goldenberg Defense Template

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Fine-Grained

Localization in Sensorand Ad-Hoc Networks

David GoldenbergDissertation Advisor: Y. Richard Yang

Committee Members: Jim Aspnes, A. Stephen Morse,Avi Silberschatz, itin !aid"a #$%$C&

Ph.D. Dissertation Defense


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Overview

(his dissertation provides a theoretical basis)or the localization problem, demonstratingconditions )or its solvability and de*ning its

computational complexity . +e appl" or )ndamental reslts on

localization to identi)" conditions nder-hich the problem is efciently solvable andto develop localization algorithms )or abroader class o) net-ors than previosapproaches cold localize.


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Collaborators (2003-

2006) 0rian D.1. Anderson #Astralia ational $niversit" and %C(A& James Aspnes 2.. 0elhmer #Colmbia $niversit"& 2ascal 0ihler Ming Cao (olga 3ren Jia 4ang Arvind 5rishnamrth" Jie #Archer& 6in +esle" Maness A. Stephen Morse 0rad Rosen Andreas Savvides +alter +hitele" #Yor $niversit"& Y. Richard Yang Anthon" Yong


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Outline

%ntrodction to 6ocalization

Conditions )or $ni8e 6ocalization

Comptational Comple9it" o)6ocalization

6ocalization in Sparse et-ors


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Why are Locations

Important +ireless ad;hoc net-ors are an important emergingtechnolog" Small, lo-;cost, lo-;po-er, mlti;)nctional sensors -ill soon be a realit". Accrate locations o) individal sensors are se)l )or man" applications

“Sensing data -ithot no-ing the sensor location is meaningless.”


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!"ample# $reat %uc& Islan'

ensor etwor& Monitoring breeding o) 6eachBs

Storm 2etrels -ithot hmanpresence.

minte hman visit leads to '=oEspring mortalit".

Sensors need to be small toavoid disrpting bird behavior.


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$reat %uc& Islan' %eployment

$oals 1ccpanc" pattern o) nests 3nvironmental changes arond

the nests over time 3nvironmental variation across

nests Correlation -ith breeding

sccess

6ight, temperatre, in)rared, andhmidit" sensors installed.

%n)rared sensors detect presenceo) birds in nests.

10m

Single hop weatherSingle hop burrowMulti hop weatherMulti hop burrow

Sensor locations critical to interpreting data. 6ocations determined b" manal

con*gration, bt this -ill not be possible inthe general case.


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H!"ample# *ebraet ensoretwor&

0iologists -ant to trac animals to std":

%nteractions bet-een individals. %nteractions bet-een species. %mpact o) hman development.

Crrent tracing technolog": !I4 collar transmitters +ishlist:

'7@F position, data, and interaction logs. +ireless connectivit" )or mobilit". Data storage to tolerate an intermittent base station.

ebraet: Mobile sensor net -ith intermittent base station. Records position sing G2S ever" / mintes. Records Sn@shade in)o. Detailed movement in)ormation #speed, movement

signatre& / mintes each hor. 4tre: head p@head do-n, bod" temperatre, heart

rate, camera. Goal, )ll ecos"stem monitoring #zebras, h"enas,

lionsL&.


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+ilitary ,pplications %ntelligence gathering #troop movements, events o)

interest&. Detection and localization o) chemical, biological,

radiological, nclear, and e9plosive materials. Sniper localization. Signal Namming over a speci*c area.

!isions )or sensor net-or deplo"ment: Dropped in large nmbers )rom $A!. Mortar;6anched.

!


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ine-$raine' Localiation(avvi'es1 200) 2h"sicall":

et-or o) n nodes, m o) -hich have known location, e9isting inspace at locations: O9CL9m,9mPC,L,9nQ.

Set o) some pair;-ise inter;node distance measrements. $sall" bet-een pro9imal nodes #iE d r in unit disk networks&.

Abstraction

Given: Graph G, O9C,...,9mQ, and , the edge -eight )nction. 4ind: Realization o) the graph.

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3 1

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{x1,x2,x3}

{x4, x5}

Beacons: nodes with known position

Regular nodes: nodes with unknown position

{d14, d24, d25, d35, d45}


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an4in4 ystems (DoA T (ime DiEerence o) Arrival

$ses ltrasond and radio signalsto determine distance.

Range o) meters, cm accrac".

2ossible to increase sensingrange b" increasing transmissionpo-er.

M%( cricet mote

$C6A medsa mote #'==&

$C6A medsa mote ' #'=='&

Yale 3A6A0 UY Motes


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Our Contributions Graph;theoretic conditions or the unique solvability o) the

localization problem in the plane.

2roo) that the problem is !"complete even )or the idealizedcase o) nit;dis net-ors.

#onstructive characterization o) classes o) ni8el" localizableand easil" localizable net-ors )or the plane and /D.

A localization algorithm that localizes a -ider class o)

net-ors than -as possible -ith e9isting approaches.

%n;depth std" o) the localizability properties o randomnetworks: e- adaptive localizabilit";optimizing deplo"ment strategies. %mpact o) non;ni8el" localizable nodes on net-or

per)ormance.


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7

Outline



Comptational Comple9it" o)6ocalization



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5niue Localiability

et-or is uniquely localizable i) there is e9actl"one set o) points O9mPC,L,9nQ consistent -ith G,O9C,L,9mQ and :3 to R.

Can -e determine localizabilit" b" graphproperties alone #as opposed to the properties o)).

%n the plane, "es #more or less&. 2roperties o) thegraph determine solvabilit" in the generic case. 2robabilit" )or randoml" generated node locations.


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%e4enerate Cases ool ,bstraction

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2

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{x1, x2, x3}

{d14, d24, d34}

probability 1 case:

probability 0 case:

first case: {x4}

second case: ???

2

1

3

?

?

In general, this netor! is"ni#"ely locali$able%

In degenerate case, it is not:

&he constraints are red"ndant%

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Continuous on-5niueness

Continos non;ni8eness: Can move points )rom one con*gration to another -hile

respecting constraints. 39cess degrees o) )reedom present in con*gration. A )ormation is RIGID i) it cannot be continosl" de)ormed.


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Con'ition 7or i4i'ity

2rel" combinatorial characterizationo) generic rigidit" in the plane.

'n;/ edges necessar" )or rigidit", and:

'a(an)s condition:'a(an)s condition: * graph + ith 2n3 edges is rigid in to di(ensions * graph + ith 2n3 edges is rigid in to di(ensions

if and only if no s"bgraph +) has (ore than 2n)3 edges-%if and only if no s"bgraph +) has (ore than 2n)3 edges-%- here n) is the n"(ber of .ertices in +)- here n) is the n"(ber of .ertices in +)

'a(an)s condition is a state(ent that any rigid graph ith n .ertices ("st ha.e a set

of 2n3 well-distributed edges%

/ot eno"gh edges no"gh edges b"t not ell distrib"ted "st right

n o t r i g i d ! n o t

r i g i d !

Rigid!


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5niue $raphealiation

a

e

b

f

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d

a

c

b

e

d

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Solution:

+ ("st be 3connected%

+ ("st be redundantly rigid:It ("st re(ain rigid "pon

re(o.al of any single edge%

+ ("st be rigid.

A graph has a unique realization in the plane if it is

redundantly rigid and 3-connected (globally rigid ).Iendricson, V7


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Is .he etwor& 5niuelyLocaliable

2roblem: 0" looing onl" at the ph"sical connectivit" strctre, -e-old neglect or a priori no-ledge o) beacon positions. Soltion: (he distances bet-een beacons are implicitl" no-nW

y adding all edges !et"een !eacons to G#$ "e get theGrounded Graph o% the net"or& , "hose propertiesdetermine net"or& localiza!ility.

Theorem' A net"or& is generically uniquely localizable ifits grounded graph is glo!ally rigid and it contains at least

three !eacons.

0" agmenting graph strctre in this -a", -e )ll" e9press allavailable constraint in)ormation in a graph.

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Is this locali$able?1

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!"amples o7 $ 4raphs -Constructions 3ver" globall" rigid graph has a spanning sbgraph that is minimall" globall" rigid. 3ver" minimall" globall" rigid graph can be constrcted indctivel" starting )rom 5 7

b" a series o) extensions #0erg;Jordan ‘=&: e- node w and edges uw and " replace edge uv $ 3dge wx added )or some node x distinct )rom u, v $

Minimal globall" rigid graphs have %n"% edges.

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6ight edges are those sbdivided b" the e9tension operation.


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!"amples o7 $lobal i4i'it

Random net-or T avg node degree . Reglarized random net-or T avg node degree 7.

+lobally rigid co(ponents in green%


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Construction 5sin4.rilateration

A position is ni8el" determined b" three distances to three non;collinear

re)erences. Minimal trilateration graphs )ormed b" trilateration e9tension:

e- node w and edges uw, vw, xw added, )or u, v , x distinct. Minimal trilateration graphs are globall" rigid. Minimal trilateration graphs have &n"' edges.

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6ight edges are those removed in e9tension )or minimall" GR graph bt not in trilate


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.rilateration $raphs

A trilateration graph G is one -ith an trilateratie ordering:an ordering o) the vertices ,..$,n sch that the complete graphon the initial & vertices is in ( and )rom ever" verte9 ) * &, thereare at least & edges to vertices earlier in the se8ence.

(rilateration graphs are globall" rigid.

Iand;made trilateration T avg degree . (rilateration graph )rom mobile net-or T avg degree .


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8.riple'9 Connecte' $raphsare .rilateration $raphs

Theorem:6et G X #!,3& be a connected graph.

6et G/ X #!,3∪ 3'∪ 3/& be the graph )ormed )rom Gb" adding an edge bet-een an" t-o vertices

connected b" paths o) ' or / edges in 3. (hen G/ is a trilateration graph.

xa(ple here

+ is a path%


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8%ouble'9 2-connecte' $raphsare$lobally i4i' in 2% Theorem:6et G be a ';connected graph.

(hen G' is globall" rigid.

xa(ple here

+ is a cycle%

Minimall" GR graphb" e9tension:

Dobled c"cle:

Dobled c"cles al-a"shave t-o edges morethan a minimall" GRgraph, so the" aregloball" rigid.

+ne gets (% by doubling sensing radiusor measuring angles between ad)acentedges$

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8.riple'9 :iconnecte'

$raphs are $lobally i4i'in 3% (here is no no-n generic

characterization o) global rigidit" in/D, bt or reslt on dobled graphse9tends to /D.

Theorem:

6et G be a ';connected graph.

(hen G/ is globall" rigid in /D.

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ummary o7 ConstructiveCharacteriation o7 $lobally

i4i' $raphs 'D /;connectivit" necessar" )or GR. G' GR i) G ';connected. G/ GR i) G connected.

/D G/ GR in /D i) G ';connected. G7 GR in /D i) G connected.

$ni8e localizabilit" b" increasing sensing range,given initial connectivit".

Conditions nder -hich additional in)ormation canhelp.

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Outline



Comptational Comple9it" o)

6ocalization


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Localiation

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Decision problem Search problem

Rigidit"theor"

Does this have a

ni8e realization

Yes@o

G r o , n

d e d

g r a p h

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/

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O9,9',9/Q

Od7, d'7, d', d/, d7Q

(his graph has ani8e realization.

+hat is it

O97,9Q

(his problem isin general 2;hard.

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Computational

Comple"ity

Sppose all edgedistances no-n

)or smalltriangles.

6ocalization goes-oring ot )roman" beacon.

(rianglereectionpossibilities gro-e9ponentiall"L.

Land reection

possibilities are onl"sorted ot -hen onegets to anotherbeacon.

%ntitivel", reection possibilities are lined-ith comptational comple9it"

//


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Comple"ity o7 $ $raphealiation

%) a net-or is localizable, ho- does one go abotlocalizing it %t is 2;hard to localize a net-or in R' even -hen it is

no-n to be ni8el" localizable.

+e -ill se t-o tools in or argment: (he 2;hard set;partition problem. (he globall" rigid -heel graph +n.

&he set partition problem:

Inp"t: * set of n"(bers S%6"tp"t: 7an S be partitioned into to s"bsets A

and S-A s"ch that the s"( of n"(bers

in each s"bset is e#"al?

/7


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/-har'ness o7 ealiationTheorem:

Realization of globally rigid weighted graphs that are realizable is NP-hard

8roof s!etch:

*ss"(e e ha.e algorith( X that ta!es as inp"t a reali$able globally rigid

eighted graph and o"tp"ts its "ni#"e reali$ation%

e ill find the setpartition of the partitionable set S scaled %l%o%g so that thes"( of ele(ents in S is less than π 92 by "sing calls to X.

"ppose e ha.e S;{s1,s2,s3,s4} ith a setpartition% 7onstr"ct a graph + along

ith its edge eights for X :

0

1

23

4 s1s3

s4s2

s1 s 1 9 2 ?

s i n > s 1 9 2 ?

/


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Localiation Comple"ity 7or parseetwor&s

2roblem -ith previos reslt is that edges e9ist arbitraril". Graphs sed in previos proo) nliel" to arise in practice.

%n realistic net-ors, edges are more liel" to e9ist bet-een closenodes, and do not e9ist bet-een distant nodes. $nit Dis Graphs: edge present i) distance bet-een nodes less than

parameter r . (here)ore: i) edge absent, distance bet-een nodes is greater than r .

Does this in)ormation help s solve the localization problem

0

1

23

4

Red edge -old e9ist in nit dis graph, sonit dis graph localization -old not solveSet 2artition.

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Comple"ity o7 Localiin4 5nit%is& $raphs

heorem: 6ocalization )or sparse sensor net-ors is 2;hard. Method:

Redction )rom #ircuit -atis.ability to /nit 0isk (raph Reconstruction.

Redction is b" constrction o) a )amil" o) graphs that represent 0ooleancircits.

Rigid bodies in the graph represent -ires. Relative position o) rigid bodies in the graph represent signals on -ires. 1( and AD gates bilt ot o) constraints bet-een these bodies e9pressed in

the graph strctre. (here is a pol"nomial;time redction )rom #ircuit -atis.ability to /nit 0isk (raph

Reconstruction, in -hich there is a one;to;one correspondence bet-een satis)"ingassignments to the circit and soltions to the reslting localization problem.

/nit 0isk (raph Reconstruction 1decision problem2%npt: Graph G along -ith a parameter r , and thes8are o) each edge length 1luv 2

% #to avoid irrational

edge lengths&.

1tpt: Y3S iE there e9ists a set o) points in R% sch

that distance )rom u to v is luv i) uv is an edge in ( and greater than r other-ise.

#ircuit -atis.ability 1 2;hard 2:%npt: A boolean combinatorialcircit.Composed o) AD, 1R, and 1(gates

1tpt: Y3S iE the circit issatis*able.

/F


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Localiation in.rilateration $raphs

As one adds more edges, localization becomes easier: (here are classeso) globall" rigid graph -hich are eas" to localize.

(rilateration graphs are localizable in pol"nomial time. Remember: 1ne gets a trilateration graph )rom a connected net-or b"

tripling the sensing radis.

Algorithm: %) initial / vertices no-n,

localize vertices one at a timentil all vertices localized.

3lse starting -ith each triangle in

the graph, proceed as aboventil all localized.

1#Z!Z'& or 1#Z!Z&.

/H


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(he )ollo-ing garantees Gn#r& is;connected -ith high probabilit")or some constant c largeenogh and constant k :

2enrose, Vr

Connectivity in an'ometwor&s

(he random geometric graph (n1r2 is therandom graph associated -ith )ormations-ith n vertices -ith all lins o) length lessthan r, -here the vertices are points in 'generated b" a t-o dimensional 2oissonpoint process o) intensit" n$

cn

nr

n =

∞→ loglim

2

ote: eed nr % 31log n2 * c, )or some c,to garantee even connectivit".

heorem: %) nr % 31log n2 * 4, -ith highprobabilit", Gn#r& is a trilateration graph.

(his identi*es conditions nder -hicha simple iterated trilaterationalgorithm -ill scceed in localization.

/.rilateration in an'om


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/.rilateration in an'ometwor&s

Loalized mode:@roadcast position%

!nloalized mode:

'isten for broadcast%

if broadcast fro( >x,y heard,

Aeter(ine distance to >x,y% if three broadcasts heard

Aeter(ine position

itch to locali$ed (ode

Iterati.e &rilaterationIterati.e &rilateration

@"t ho

fast?

Sensors have ' modes.

Sensors determine distance )rom heardtransmitter. All sensors are pre;placed and plgged in

7=


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,symptotics o7 .rilateration in an'ometwor&s

0eacons Sensingradis

3

)log( n nO

)log( nO)log(n

nO

)log(n

nO

)log( nn

O

)log(

n

nO

)(nO )1(O

)1(O

B"nning ti(es to co(plete locali$ation "sing trilateration for different beacon densities%

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/-har'ness o7

Localiation 5ine"grained localization is !"hard de to 2;hardness o) realizing globall" rigid graphs.

(his means that localization o) net-ors incomplete generalit" is nliel" to be e?cientl"solvable.

Motivates search )or reasonable special cases and

heristics. 39plains hit;or;miss character o)previos approaches. Changing sensing radis can predictabl" convert

connectedness to global rigidit" and trilateration.

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Outline



Comptational Comple9it" o)

6ocalization


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+otivation

0eing able to precisel" localize onl"trilateration net-ors is nsatis)"ing. (rilateration net-ors contain signi*cantl"

more constraints than necessar" )or ni8elocalizabilit".

Can -e localize net-ors -ith closer to theminimal nmber o) constraints

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(rilateration graph Globall" rigid sbgraph

Red edgesnnecessar" )orni8e localizabilit".

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:ilateration $raphs A bilateration graph G is one -ith a !ilateration

ordering: an ordering o) the vertices ,..$,n sch that thecomplete graph on the initial & vertices is in ( and )romever" verte9 ) * &, there are at least 2 edges to verticesearlier in the se8ence.

Theorem: 0ilateration graphs are rigid #bt not globall"rigid&.

Theorem: 6et G X #!,3& be a connected graph. (hen G' is a bilateration graph.

0ilateration graphs are nitely localiza!le in 1#'Z!Z& time.

Algorithm: %) initial / vertices no-n, *nitel" localizevertices one at a time b" compting all

possible positions consistent -ith neighborpositions ntil all vertices *nitel" localized.

3lse starting -ith each triangle in the graph,proceed as above ntil all *nitel" localized.

0

1

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33)

4

4) 4))

4)))

55)

))

)

7

L li i i % bl '


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Localiation in %ouble'Cycles

0ased on *nite localization o) bilaterationgraphs, localization is uniquely comptable)or globall" rigid dobled c"cles.

Completes in 1#'Z!Z& time. Assmes nodes in general position$

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4) 4))

4)))

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)

CS-eepD Algorithm: 4i9 the position o) three vertices.

$ntil no progress made: 4initel" localize each verte9connected to t-o *nitel" localizedvertices.

Remove possibilities -ith noconsistent descendants.

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Localiation in %ouble' 2-connecte' $raphs ';connected graphs are a nion o) c"cles #the" have an 3ar Decomposition&.

(he ear decomposition gives a ordering in -hich c"cles ma" be localizedsing previos algorithm.

ote: (his means i) -e have angles, -e can localize ';connected net-ors.

0iconnected net-or -ith its ear decomposition. Dobled biconnected net-or.

7F

L li ti $ l


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Localiation on $eneral parseetwor&s +orst;case e9ponential time algorithm )or

localization in sparse net-ors:

4or -hich t"pes o) net-or does s-eep localization -or

Theorem: Shell s-eep *nitel" localizes bilateration net-ors. Theorem: Shell s-eep ni8el" localizes globall" rigid bilateration

net-ors$

%) G is connected, -hen rn on G', shell s-eep prodces all possiblepositions )or each node. %) G' globall" rigid, gives the ni8e positions.

ue!tion: Io- man" globall" rigid net-ors are also bilaterations

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03)

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("pical random graph. Starting nodes

randoml" chosen. Shell s-eep ni8el"

localizes localizableportion.

Also non;ni8el"localizes nodes rigidl"connected to localizedregion.

hell weepon an'om

etwor&

7


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== node graph -ithconsiderableanisotrop" and 7.

average degree. Shell s-eep

comptes in seconds[ -ith nointermediate positionset e9ceeding 'H.

/er7ormance on

Lar4eetwor&

[ As a JA!A applet on a zoonode -ith a dal '.HGIz C2$and 'G0 RAM

=


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ailin4Case

Globall" rigid net-or.

Connection bet-eenclsters nbridgeableb" bilateration.

!"tent o7 weep


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!"tent o7 weepLocaliation

S-eeps localizes more nodes thantrilateration, and almost all localizable nodesW

%n reglar net-ors, s-eeps localizessigni*cantl" more nodes than trilateration.

Most incremental localization algorithms aretrilateration based.

*ey point' 6any globally rigid randomgeometric graphs are bilateration graphs$

S-eeps in Random et-or S-eeps in Reglar et-or

'

7 L li ti % it


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ummary o7 Localiation %ensitypectrum 6ocalization is 2;hard in general, bt there are classes o) graphs

that are eas" to localize. Complete graphs. (rilateration graphs.

Graphs that -e no- ho- to localize in -orst;case e9ponentialtime: Dobled biconnected graphs.

0asic idea: more edges mae localization easier. Goal: to nderstand -hich net-ors can be localized and -hich are

problematic.

"onsider all possible networ#s on n sensors

Some networ#s an be

loalized in $%&'&():

&rilateration graphs ith "n!non

ordering

!nloalizableSome networ#s an be

loalized in e*ponential

time:

Ao"bled biconnected graphs

+lobally rigid bilateration graphs

Some networ#s an be

loalized in $%&'&+) time:

&rilateration graphs ith !non

ordering

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When %oes Localiation:ecome !asy

parse

Aense

Ensol.able

asy

+lobally Bigid /8hard

&rilateration +raph8olyno(ial ti(e

Number of edges "omple*ity of

realization

7o(plete +raph

3connected r 3

3r 1

2r 2

Sensing radius in ,n

%r)

0

1

xponential@ilateration +raph

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Conclusion an' uture

Wor& 4ormalized the localization problem and itssolvabilit". Sho-ed that the problem is )ndamentall"

comptationall" hard. Constrctivel" characterized easil" localizable

net-ors. 2rovided algorithm that localizes more nodes than

previos incremental algorithms.

e9t: 6ocalization sing maps. 6ocalization sing anglar order in)ormation. 6ocalization in net-ors o) mobile nodes. 6ocalization in /D or on /D sr)aces. 4ll s"stem )rom deplo"ment to localization.


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Our Wor& in the iel' \Rigidity$ +omputation$ and Randomization in #et"or& ,ocalization] ;

Comptational comple9it" reslts.

\A 0heory o% #et"or& ,ocalization] ; Algorithm )or localization in sparse ad;hoc net-ors.

\,ocalization in 1artially ,ocaliza!le #et"or&s] ; 6ocation;a-are controlled node;mobilit" algorithm )or sensor net-or

optimization.


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,c&nowle'4ements % -old lie to than all m" collaborators, -ithot -hom this -or -old

not have been possible.

0rian D.1. Anderson #Astralia ational $niversit" and %C(A& James Aspnes 2.. 0elhmer #Colmbia $niversit"&

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localization goldenberg defense template

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