503_lecture8

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UMass Lowell Computer Science 91.503

Graduate Algorithmsro!. "aren #aniels

Spring$ %005

Computational GeometryComputational Geometry

Overview from Cormen, et al.Overview from Cormen, et al.

Chapter 33Chapter 33

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&'er'iew

Computational Geometr( )ntroductionComputational Geometr( )ntroduction

Line Segment )ntersectionLine Segment )ntersection

Con'e* +ull AlgorithmsCon'e* +ull Algorithms

,earest ,eigh-orsClosest oints ,earest ,eigh-orsClosest oints

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)ntroduction

/hat is/hat is

Computational Geometr(Computational Geometr(

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Computational Geometry

TelecommunicationsTelecommunications

VisualizationVisualization

ManufacturingManufacturing

ComputerComputer

GraphicsGraphics

Design Analyze

Apply

CADCAD

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Sample Application Areas

Computer GraphicsComputer Graphics

GeographicGeographic

Information SystemsInformation Systems

RoboticsRobotics

BioinformaticsBioinformatics

Astrophysics Astrophysics

MeicalMeical

ImagingImaging

TelecommunicationsTelecommunications

Data Mining !Data Mining !

VisualizationVisualization

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(pical ro-lems

-in pac2ing -in pac2ing

oronoi diagramoronoi diagram

simpli!(ingsimpli!(ing pol(gons pol(gons

shape similarit(shape similarit(

con'e* hullcon'e* hull

maintaining linemaintaining line

arrangementsarrangements

pol(gon partitioning pol(gon partitioning

nearest neigh-or searchnearest neigh-or search

2d4trees2d4trees

S&UC6S&UC6: Steve Skiena’s Algorithm Design Manual: Steve Skiena’s Algorithm Design Manual

(for problem descriptions see graphics gallery at(for problem descriptions see graphics gallery at http7www.cs.sun(s-.edu8algorith ! !

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Common Computational

Geometr( Structures

Voronoi DiagramVoronoi DiagramCon"e# $ullCon"e# $ull

%e& 'oint%e& 'oint

source: O’Rourke, Computational Geometry in C source: O’Rourke, Computational Geometry in C

Delaunay TriangulationDelaunay Triangulation

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Sample ools o! the rade

Algorithm #esign atternsechniues7Algorithm #esign atternsechniues7 -inar( search -inar( search di'ide4and4conuer di'ide4and4conuer dualit(dualit(

randomi:ationrandomi:ation sweep4linesweep4line

derandomi:ationderandomi:ation parallelism parallelismAlgorithm Anal(sis echniues7Algorithm Anal(sis echniues7

as(mptotic anal(sis$ amorti:ed anal(sisas(mptotic anal(sis$ amorti:ed anal(sis

#ata Structures7#ata Structures7

winged4edge$ uad4edge$winged4edge$ uad4edge$ range tree$ 2d4treerange tree$ 2d4treeheoretical Computer Scienceheoretical Computer Science principles7 principles7

,4completeness$ hardness ,4completeness$ hardness

Gro&th of (unctionsGro&th of (unctions

SummationsSummations

RecurrencesRecurrences

SetsSets

'robability'robability

MATHMATH 'roofs'roofs GeometryGeometry

Graph TheoryGraph TheoryCombinatoricsCombinatorics

)inear Algebra)inear Algebra

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Computational Geometr(in Conte*t

TheoreticalTheoretical

Computer Computer

ScienceScience

Applie Computer Science Applie Computer Science

Applie Applie

MathMath

GeometryGeometry

ComputationalComputational

GeometryGeometry

EfficientEfficientGeometric AlgorithmsGeometric Algorithms

DesignAnalyze

Apply

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Sample o! Supporting Algorithmic ;

Math Areas ; echniues

ComputationalComputationalGeometr(Geometr( Con'e* hullsCon'e* hulls

isi-ilit( pol(gonsisi-ilit( pol(gons ArrangementsArrangements

MathematicalMathematicalrogrammingrogramming Linear programmingLinear programming

)nteger programming)nteger programming Lagrangian rela*ationLagrangian rela*ation Upper$ lower -oundingUpper$ lower -ounding

#(namic #ata#(namic #ataStructuresStructures

Algorithm #esign atternsAlgorithm #esign atterns search space su-di'isionsearch space su-di'ision -inar( search -inar( search di'ide4and4conuer di'ide4and4conuer sweep4linesweep4line discrete4e'ent simulationdiscrete4e'ent simulation

Algorithm Anal(sisAlgorithm Anal(sisechniuesechniues

Comple*it( heor(Comple*it( heor( ,4completeness$ hardness ,4completeness$ hardness

#iscrete Math#iscrete Math Min2ows2i sumMin2ows2i sum Monotone matricesMonotone matrices LatticesLattices Set operations7 union$Set operations7 union$

intersection di!!erence

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Line Segment )ntersections

<%#=

)ntersection o! % Line Segments)ntersection o! % Line Segments

)ntersection o! > %Line Segments)ntersection o! > %Line Segments

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Cross4roduct4?ased

Geometric rimiti'es

source: 91.!" te#t$ook Cormen et al.source: 91.!" te#t$ook Cormen et al.

pp**

pp++

pp,,

-,.-,.

pp,,

pp//

pp++

-+.-+.

pp++

pp,,

pp//

pp00

-/.-/.

Some funamental geometric 1uestions2Some funamental geometric 1uestions2

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Cross4roduct4?ased

Geometric rimiti'es7 <1=

source: 91.!" te#t$ook Cormen et al.source: 91.!" te#t$ook Cormen et al. A"antage A"antage2 less sensiti"e to accumulate roun3off error 2 less sensiti"e to accumulate roun3off error

1%%1

%1

%1

%1 det y " y " y y

" " p p −=

=×

pp**

pp++

pp,,

-,.-,.

//4,

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Cross4roduct4?ased

Geometric rimiti'es7 <%=


pp**

pp++

pp,,

-+.-+.

//4+

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)ntersection o! % Line Segments


pp++

pp,,

pp//

pp00

-/.-/.

Step 1Step 1::

Bounding BoxBounding Box

TestTest

Step 2Step 2: Does eac: Does eac

segment straddlesegment straddle

te line containingte line containing

te oter!te oter!

//4/

p/ an p0 on opposite sies of p,p+p/ an p0 on opposite sies of p,p+

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Segment#Segment $ntersection

@inding the@inding the actual intersection point actual intersection point Approach7 parametric 's. slopeinterceptApproach7 parametric 's. slopeintercept

parametric generali:es to more comple* intersections parametric generali:es to more comple* intersections e.g. segmenttrianglee.g. segmenttriangle

arameteri:e each segmentarameteri:e each segment

Intersection2 "alues ofIntersection2 "alues of ss55 tt such thatsuch that p-s.p-s. 661-t.1-t. 22 a7sAa7sA66c7tCc7tC

aa

bb

cc

))abab))cc

aa

bbcc

))abab))cc

A6b3a A6b3a

p-s.6a7sAp-s.6a7sA

1-t.6c7tC1-t.6c7tCC63cC63c

source: O’Rourke, Computational Geometry in C source: O’Rourke, Computational Geometry in C + e1uations in un8no&ns+ e1uations in un8no&ns ss55 tt 2 , for #5 , for y2 , for #5 , for y

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#emo

Segment%Segment $ntersectionSegment%Segment $ntersectionhttp7cs.smith.edu8orour2e-oo2sCompGeomCompGeom.htmlhttp7cs.smith.edu8orour2e-oo2sCompGeomCompGeom.html

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)ntersection o! >% Line Segment


S&eep3)ine Algorithmic 'araigm2S&eep3)ine Algorithmic 'araigm2

//40

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S&eep3)ine Algorithmic 'araigm2S&eep3)ine Algorithmic 'araigm2

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Time toTime to detect detect if any +if any +

segments intersect2segments intersect2"#n lg n$"#n lg n$

source: 91.!" te#t$ook Cormen et al.

//49

%ote that it e#its as soon as%ote that it e#its as soon as oneone intersection is etecte4intersection is etecte4

Balance BST stores segments in orerBalance BST stores segments in orer

of intersection &ith s&eep line4of intersection &ith s&eep line4

Associate operations ta8e :-lgn. time4 Associate operations ta8e :-lgn. time4

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$ntersection of Segments

GoalGoal7 &utput4si:e sensiti'eB line segment intersection algorithm7 &utput4si:e sensiti'eB line segment intersection algorithmthat actually computesthat actually computes all all intersection pointsintersection points

?entle(4&ttmann plane sweep7?entle(4&ttmann plane sweep7 &<<n2=log<n2==D &<<n2=logn=&<<n2=log<n2==D &<<n2=logn= timetime

2 D num-er o! intersection points in output2 D num-er o! intersection points in output

)ntuition7 sweep hori:ontal line downwards)ntuition7 sweep hori:ontal line downwards Eust -e!ore intersection$ % segments are adEacent in sweep4line intersection structure Eust -e!ore intersection$ % segments are adEacent in sweep4line intersection structure chec2 !or intersectionchec2 !or intersection onlyonly adEacent segmentsadEacent segments insert intersection e'ent into sweep4line structureinsert intersection e'ent into sweep4line structure e'ent t(pes7e'ent t(pes7

top endpoint o! a segmenttop endpoint o! a segment -ottom endpoint o! a segment -ottom endpoint o! a segment intersection bet&een ' segmentsintersection bet&een ' segments

s&ap order s&ap order

Impro"e to :-nlogn78. ;Chazelle<=elsbrunner>Impro"e to :-nlogn78. ;Chazelle<=elsbrunner>source: O’Rourke, Computational Geometry in C source: O’Rourke, Computational Geometry in C

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Con'e* +ull Algorithms

#e!initions#e!initionsGi!t /rappingGi!t /rapping

Graham ScanGraham ScanFuic2+ullFuic2+ull

)ncremental)ncremental#i'ide4and4Conuer #i'ide4and4Conuer

Lower ?ound inLower ?ound in ΩΩ<nlgn=<nlgn=

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Con'e*it( ; Con'e* +ulls

A con'e* com-ination o! pointsA con'e* com-ination o! points

**11$ ...$ *$ ...$ *2 2 is a sum o! the !ormis a sum o! the !orm

αα11**11...... αα2 2 **2 2 wherewhere

Con'e* hull o! a set o! points is theCon'e* hull o! a set o! points is the

set o! all con'e* com-inations o!set o! all con'e* com-inations o!

points in the set. points in the set.

noncon"e# polygonnoncon"e# polygon

con"e# hull of a point setcon"e# hull of a point set

10 1 =++∀≥ k i and i α α α


source: 91.!" te# t$ook Cormen et al.source: 91.!" te#t$ook Cormen et al.

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,ai'e Algorithms

!or 6*treme oints

AlgoritmAlgoritm2 I%T=RI:R ':I%TS2 I%T=RI:R ':I%TS

for each i ofor each i o

for each ? 6 i ofor each ? 6 i o

for each 8 6 ? 6 i ofor each 8 6 ? 6 i o for each ) 6 8 6 ? 6 i ofor each ) 6 8 6 ? 6 i o

if pif p)) in triangle-pin triangle-p

ii5 p5 p ? ?5 p5 p

88..

then pthen p)) is none#tremeis none#treme "#n"#n%%$$

AlgoritmAlgoritm2 =@TR=M= =DG=S2 =@TR=M= =DG=Sfor each i ofor each i o

for each ? 6 i ofor each ? 6 i o

for each 8 6 ? 6 i ofor each 8 6 ? 6 i o

if pif p88 is not left or on -pis not left or on -p

ii5 p5 p ? ?..

then -pthen -pii 5 p5 p ? ?. is not e#treme. is not e#treme "#n"#n&&

$$source: O’Rourke, Computational Geometry in C source: O’Rourke, Computational Geometry in C

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Algorithms: %# Gi!t /rapping

Use one e*treme edge as anUse one e*treme edge as an

anchor !or !inding the ne*tanchor !or !inding the ne*t

θθ

"#n"#n22$$

AlgoritmAlgoritm2 GI(T RA''I%G2 GI(T RA''I%G

ii** ine# of the lo&est point ine# of the lo&est point

i ii i**repeatrepeat for each ? 6 ifor each ? 6 i Compute countercloc8&ise angleCompute countercloc8&ise angle θθ from pre"ious hull egefrom pre"ious hull ege 8 ine# of point &ith smallest8 ine# of point &ith smallest θθ :utput -p:utput -p

ii 5 p5 p88. as a hull ege. as a hull ege

i 8i 866 source: O’Rourke, Computational Geometry in C source: O’Rourke, Computational Geometry in C

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Gi!t /rapping source: 91.!" te#t$ook Cormen et al.source: 91.!" te#t$ook Cormen et al.

//4

"utput Sensiti'ity"utput Sensiti'ity: "#n: "#n22$ run(time is actually "#n$$ run(time is actually "#n$

)ere is te num*er o+ 'ertices o+ te con'ex ull,)ere is te num*er o+ 'ertices o+ te con'ex ull,

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Algorithms: 3# Gi!t /rapping

C#%ull AnimationsC#%ull Animations2 http2<<&&&4cse4uns&4eu4au<lambert<?a"a</<hull4html2 http2<<&&&4cse4uns&4eu4au<lambert<?a"a</<hull4html

"#n"#n22$$ timetime

;output sensiti"e2 :-n(. for ( faces on hull>;output sensiti"e2 :-n(. for ( faces on hull>

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Algorithms: %# Fuic2+ull

Concentrate on points close toConcentrate on points close tohull -oundar(hull -oundar(

,amed !or similarit( to ,amed !or similarit( toFuic2sortFuic2sort

aa

bb

"#n"#n22

$$

AlgoritmAlgoritm2 EICF $E))2 EICF $E))

function uic8$ull-a5b5S.function uic8$ull-a5b5S. if S 6 * return-.if S 6 * return-. elseelse c ine# of point &ith ma# istance from abc ine# of point &ith ma# istance from ab A points strictly right of -a5c. A points strictly right of -a5c. B points strictly right of -c5b.B points strictly right of -c5b.

return uic8$ull-a5c5A. 7 -c. 7 uic8$ull-c5b5B.return uic8$ull-a5c5A. 7 -c. 7 uic8$ull-c5b5B.source: O’Rourke, Computational Geometry in C source: O’Rourke, Computational Geometry in C

fin&s one of upper or lo'er hull fin&s one of upper or lo'er hull

cc

A A

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Algorithms: 3# Fuic2+ull


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Algorithms: >D %#

hull2 http2<<&&&4geom4umn4eu<soft&are<1hull<hull2 http2<<&&&4geom4umn4eu<soft&are<1hull<

Con"e# $ullCon"e# $ull

bounary isbounary is

intersection ofintersection of

hyperplanes5 sohyperplanes5 so

&orst3case&orst3casecombinatorial sizecombinatorial size-not necessarily running-not necessarily running

time.time. comple#ity iscomple#ity is

in2in2 =< %d nΘ

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Grahams Algorithm

oints sorted angularl( pro'ideoints sorted angularl( pro'idestar4shapedB starting pointstar4shapedB starting point

re'ent dentsB as (ou go 'iare'ent dentsB as (ou go 'ia

con'e*it( testingcon'e*it( testing

"#nlgn$"#nlgn$

AlgoritmAlgoritm2 GRA$AM SCA%5 Version B2 GRA$AM SCA%5 Version B

(in rightmost lo&est point label it p(in rightmost lo&est point label it p**44

Sort all other points angularly about pSort all other points angularly about p**44 In case of tie5 elete point-s. closer to pIn case of tie5 elete point-s. closer to p

**44

Stac8 S -pStac8 S -p,,5 p5 p

**. 6 -p. 6 -ptt5 p5 p

t3,t3,. t ine#es top. t ine#es top

i +i +&hile i H n o&hile i H n o

if pif pii is strictly left of pis strictly left of p

t3,t3,pptt

then 'ush-pthen 'ush-pii5 S. an set i i 7,5 S. an set i i 7,

θθ


( ( multipop) multipop)

pp**

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Graham Scan


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Graham Scan


//4

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Graham Scan

source: 91.!" te#t$ook Cormen et al source: 91.!" te#t$ook Cormen et al.

//4

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Graham Scan


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Graham Scan


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Algorithms: %# )ncremental

Add points$ one at a timeAdd points$ one at a time update hull !or each new pointupdate hull !or each new point

"e( step -ecomes adding a single point to"e( step -ecomes adding a single point toan e*isting hull.an e*isting hull. @ind % tangents@ind % tangents

esults o! % consecuti'e L6@ tests di!!er esults o! % consecuti'e L6@ tests di!!er

)dea can -e e*tended to 3#.)dea can -e e*tended to 3#.

"#n"#n22$$

AlgoritmAlgoritm2 I%CR=M=%TA) A)G:RIT$M2 I%CR=M=%TA) A)G:RIT$M

)et $)et $++ Con"e#$ullJp Con"e#$ullJp

** 5 p5 p,, 5 p5 p

++ KK

for 8 / to n 3 , ofor 8 / to n 3 , o

$$88 Con"e#$ullJ $ Con"e#$ullJ $83,83, E pE p88 KK can $e impro*e& to O+nlgncan $e impro*e& to O+nlgn


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Algorithms: 3# )ncremental


"#n"#n22$$ timetime

Al h

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Algorithms:

%# #i'ide4and4Conuer #i'ide4and4Conuer in a geometric setting#i'ide4and4Conuer in a geometric setting

&<n= merge step is the challenge&<n= merge step is the challenge @ind upper and lower tangents@ind upper and lower tangents

Lower tangent7 !ind rightmost pt o! A ;Lower tangent7 !ind rightmost pt o! A ;

le!tmost pt o! ?H then wal2 it downwardsBle!tmost pt o! ?H then wal2 it downwardsB )dea can -e e*tended to 3#.)dea can -e e*tended to 3#.

AlgoritmAlgoritm2 DIVID=3an3C:%E=R2 DIVID=3an3C:%E=R

Sort points by # coorinateSort points by # coorinate

Di"ie points into + sets A an B2Di"ie points into + sets A an B2

A contains left n<+ points A contains left n<+ points

B contains right n<+ pointsB contains right n<+ points

Compute Con"e#$ull-A. an Con"e#$ull-B. recursi"elyCompute Con"e#$ull-A. an Con"e#$ull-B. recursi"ely

Merge Con"e#$ull-A. an Con"e#$ull-B.Merge Con"e#$ull-A. an Con"e#$ull-B. "#nlgn$"#nlgn$

A ABB


Al h

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Algorithms:

3# #i'ide and Conuer


"#n log n$"#n log n$ time Ltime L

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Lower ?ound o! &<nlgn=

/orst4case time to !ind con'e* hull o! n points/orst4case time to !ind con'e* hull o! n pointsin alge-raic decision tree model is inin alge-raic decision tree model is in ΩΩ<nlgn=<nlgn=

roo! uses sorting reduction7roo! uses sorting reduction7

Gi'en unsorted list o! n num-ers7 <*Gi'en unsorted list o! n num-ers7 <*11$*$*%% $I$ *$I$ *nn== @orm unsorted set o! points7 <*@orm unsorted set o! points7 <*ii$ *$ *ii

%%= !or each *= !or each *ii

Con'e* hull o! points producesCon'e* hull o! points produces sorted list sorted list JJ ara-ola7 e'er( point is on con'e* hullara-ola7 e'er( point is on con'e* hull

eduction is &<n= <which is in o<nlgn==eduction is &<n= <which is in o<nlgn==

@inding con'e* hull o! n points is there!ore at least@inding con'e* hull o! n points is there!ore at leastas hard as sorting n points$ so worst4case time is inas hard as sorting n points$ so worst4case time is inΩΩ<nlgn=<nlgn=

'arabola for sorting +5,5/'arabola for sorting +5,5/


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,earest ,eigh-orClosest air o! oints

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Closest air


GoalGoal2 Gi"en n -+D. points in a set 5 fin the closest pair2 Gi"en n -+D. points in a set 5 fin the closest pair

uner the =ucliean metric inuner the =ucliean metric in :-n lgn.:-n lgn. time4time4

Di'ide(and(Con-uer Strategy:

3. / points sorted *y increasing x3 0 / points sorted *y increasing y

3Di'ide: partition )it 'ertical line L into 3 4

3Con-uer: recursi'ely +ind closest pair in 3 4

3 δ3 δ4 are closest(pair distances

3 δ

min# δ

3δ

4 $3Com*ine: closest(pair is eiter δ or pair straddles partition line L

3 Cec5 +or pair straddling partition line L

3 *ot points must *e )itinδ

o+ L- create array 06 / 0 )it only points in 2δ strip3 +or eac point p in 06

3 +ind #7/ 8$ points in 06 )itinδ

o+ p

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Closest air


Correctness

//4,,

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Closest air

4unning Time:

Fey 'ointFey 'oint2 'resort points5 then at each step form2 'resort points5 then at each step form

sorte subset of sorte array in linear timesorte subset of sorte array in linear time

=lg<3i! =1<3i! =<=%<%=< nn

nnnn) n) =

≤>+=

-ike opposite of ERGE step in ERGE/OR0-ike opposite of ERGE step in ERGE/OR0

source: 91.!" te#t$ook Cormen et al.source: 91.!" te#t$ook Cormen et al.RR

))

Additi l C t ti l G t

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Additional Computational Geometr

esources

Computational Geometry in C Computational Geometry in C second edition second edition -( Koseph &our2e -( Koseph &our2eCam-ridge Uni'ersit( ressCam-ridge Uni'ersit( ress199199

*eb Site: http:%%cs+smith+edu%,orourke%books%compgeom+html

See also -.+/01 Course *eb Site:

http:%%&&&+cs+uml+edu%,kdaniels%courses%A2G3/01+html

Documents

503_lecture8