6.006IntroductiontoAlgorithms

Lecture24:GeometryProf.ErikDemaine

Today• Computationalgeometry• Line‐segmentintersection

– Sweep‐linetechnique• Closestpairofpoints

– Divide&conquerstrikesback!

Motivation:CollisionDetectionphotobyfotios lindiakos

http://www.flickr.com/photos/fotios_lindiakos/342596118/

Motivation:CollisionDetectionphotobyfotios lindiakos

http://www.flickr.com/photos/fotios_lindiakos/342596118/

“GTA4Carmageddon!”bydot12321http://www.youtube.com/watch?v=4-620xx7yTo

Line‐SegmentIntersection• Input: linesegmentsin2D• Goal: Findthe intersections

ObviousAlgorithmforeverypair oflinesegments:checkforintersection

Sweep‐LineTechnique• Idea: Sweepaverticallinefromlefttoright• Maintainintersectionoflinewithinput

Sweep‐LineIntersections

Sweep‐LineEvents• Discretizecontinuousmotionofsweepline• Eventswhenintersectionchanges

– Segmentendpoints – Intersections

Sweep‐LineAlgorithmSketch

• Maintainsweep‐lineintersection• Maintainpriorityqueue of(possible)eventtimes ( coordinatesofsweepline)

• Untilqueueisempty:– Deleteminimumeventtime frompriorityqueue– Updatesweep‐lineintersectionfrom to– Updatepossibleeventtimesinpriorityqueue

“Discrete‐eventsimulation”

IntersectionDataStructure• Balancedbinarysearchtree(e.g.,AVLtree)tostoresorted ( ) orderofintersections

EndpointEvents• Foreachlinesegment :

– At :insert (binarysearchforneighborsthen)– At :delete

IntersectionEvents?• Howtoknowwhenhaveintersectionevents?• Thisisthewholeproblem!

IntersectionEvents• Computewhenneighboringsegmentswouldintersect,ifnothingchangesmeanwhile

• Ifsuchaneventisnext,thenitreallyhappens

while isnotempty:

if is :insert into (binarysearchingwith )

elif is :delete from

elif is :reportintersectionbetween andswapcontentsofnodesfor and in

foreachnode changedorneighboringchangedin :deleteall eventsinvolving fromadd & to

Sweep‐LineAlgorithm

Sweep‐LineAnalysis• Runningtime=• Numberofendpointevents=• Numberofmeetevents=number ofintersections=sizeofoutput

• Runningtime=• Outputsensitivealgorithm:runningtimedependsonsizeofoutput

• Bestalgorithmrunsin time

ClosestPairofPoints• Input: pointsin2D• Goal: findtwoclosestpoints

ObviousAlgorithmmin(distance

foreverypair ofpoints)

Divide‐and‐ConquerIdea• Initiallysortpointsby coordinate• Splitpointsintolefthalfandrighthalf• Recurseoneachhalf:findclosestpair• returnmin{closestpairineachhalf,

closestpairbetweentwohalves}

ClosestPairBetweenTwoHalves?

• Let =min{closestpairineachhalf}• Onlyinterestedinpairswithdistance• Restricttowindowofwidth aroundmiddle

ClosestPairBetweenTwoHalves

• Foreachleftpoint,interestedinpointsonrightwithindistance

• Pointsonrightsidecan’tbewithin ofeachother

• Soatmostthreerightpointstoconsiderforeachleftpoint– Dittoforeachrightpoint

• Cancomputein timebymergingtwosortedarrays

Divide‐and‐Conquerpresortpointsbydef :

sort bysort bymergeandfindclosestpairbetweentwolistsreturnmin{ ,closestdistancefrommerge}

FasterDivide‐and‐Conquer[Shamos &Hoey 1975]

presortpointsby and ,andcrosslinkpointsdef :

maptoypoints &findclosestpairbetweenlistsreturnmin{ ,closestdistancefrommerge}

OtherComputationalGeometryProblems

• Convexhull• Voronoi diagram• Triangulation• Pointlocation• Rangesearching• Motionplanning• …

6.850:GeometricComputing