Post on 11-Jul-2020
D7013E Computational Geometry - Håkan Jonsson 1
D7013ELecture2Linesegment
intersection
Booleanoperationsonpolygons
Subdivisions
Mapoverlay
Doubly-ConnectedEdgeLists
Planesweep
Closestpairs
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1. Computing intersections among line segments
6 line segments, 15 intersections 7 line segments, 3 intersections
Line segment intersection • Computeandreportallintersectionsamongasetofnline
segments– Intersectionmeans”haveatleastonepointincommon”
• Atrivialsolutionbasedonbrute-force:– Foreachpairoflinesegments:Computethepointofintersection
betweenthelinesthatcontainthelinesegments;ifthispointlieonbothlinesegments,wehavefoundanintersection
– ThereareO(n2)pairssothistakesO(n2)time
• So,dowereallyneedtospendO(n2)time?– Yes,thereareΩ(n2)intersectionsintheworstcase– Buttherecouldalsobejustafew…
• Interestingquestion:Isthereawaytorelatetherunningtimetothenumberofintersectionsactuallyreported?
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Output-sensitive algorithms • Analgorithmforwhichthetime(orspace)complexity
dependsnotonlyonthesizeoftheinput(ninourcase)butalsothesizeoftheoutput– AswithJarvismarchforcomputingconvexhulls
• Letkbethenumberofintersectionsfound– ”Numberofpairsofintersectinglinesegments.”
• Wewillnowlearnaboutanoutput-sensitivealgorithmthatcomputesallintersectionsin
O(nlogn+klogn)time
• (ThereexistsfasteralgorithmsthatruninO(nlogn+k)time.Theseareoptimal,butmorecomplicated,andwillnotbecovered.)
Lower bound: Ω(nlogn+k) • Thetermkisobvious;wehavetooutputtheresult• Thetermnlogncanbeshownbyareductionfrom:• ELEMENT-UNIQUENESS:
– Givenasetofnnumbers,{x1,x2,…,xn},aretwoofthemequal?
– RequiresΩ(nlogn)comparisons[Ben-Or1983]– (Easytosolvebysorting)
• Recallthatinourmodelallcommonbinaryoperatorslike<,>,=,AND,OR,NOT,+,-,*,/etccanbeperformedinunit(constant)time– So,ELEMENT-UNIQUENESSrequiresΩ(nlogn)time
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Lower bound reduction • LetAbeanyalgorithmthatcomputesintersections
amonglinesegments• WecanthensolveELEMENT-UNIQUENESSby
constructingshortlinesegmentsforthenumbersintheset,runA,andchecktheresult– Mapxito,forinstance,thelinesegment[(xi,0),(xi,1)]– Then,Awillreportintersectionsifftherearei≠j,xi=xj
• SinceeverythingelsethanAtakesO(n)time,itmustbeAthattakesΩ(nlogn)time
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{x1,x2,…,xn}=>
x1 x5 x4 x3 x2 xn … …
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Observation 1 about our intersections
• Onlylinesegmentsthatoverlapifprojectedontothey-axismightintersect!
Algorithmic technique: Plane sweep • Imagineahorizontallinesweepingdownwardsoverthe
plane,startingabovealllinesegments– Alsocalledlinesweep(orjustsweep)
• Asthissweeplineismoved,wekeeptrackofthesetoflinesegmentsthatitintersects-thestatus– Thestatuscontainstheintersectedlinesegmentsorderedin
thex-directionbyhowtheyintersectthesweepline• Ingeneral,thestatusisinvariant.Itonlychangeswhen
thesweeplineverticallyreachesaneventpoint• Aneventpointiseither
– anendpointofalinesegment,or– apointofintersectionbetweentwolinesegments
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Observation 2 about our intersections
• Iftwolinesegmentsintersect,theymustatsometimeduringthesweeplienexttoeachotherinthestatus!
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Algorithm • Useaneventqueue(apriorityqueue)tostore
eventpointsorderedbytheiry-coordinate• Insertall2nendpointsoflinesegmentsinthe
beginning• Startwithanemptystatus,andgo(”sweep”)from
oneeventpointtothenextwhilekeepinginvariantthat:
”Abovethesweeplineallpointsofinter-sectionhavebeencomputed.”
• Important:
– Thestatusdoesnotcontainthecoordinateswherethesweeplineintersects.Itholdstheorderinwhichlinesegmentsintersectthesweepline
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Algorithm (cont.) • Thishappensataneventpointwhen…
– Anupperendpointisreached:Insertthenewlinesegment(intothestatus)andtestitforintersectionswithitsneighborsbelowthesweepline;inserttheintersectionpointsaseventpoints,ifany
– Apointofintersectionisreached:Swapthelinesegmentsthatintersectandtestthemforintersectionsbelowthesweeplinewiththeirnewneighbors.Iftheyintersect,insertthepointsasneweventpoints
– Alowerendpointisreached:Deletethelinesegmentandtestwhetherthetwonewneighborsintersectbelow.Iftheyintersect,insertthepointasaneweventpoint
Report intersections!
Data structures used • Eventqueue
– Abalancedbinarysearchtree• AVL-trees,red-blacktrees,etcthatsupport
insertion,deletion,neighbor(successorandpredecessor),andlookupinO(logm)timewhenthetreecontainsmitems
– Defineanorderbetweentheeventpoints(highestfirst,leftmosttobreakties)
• Status– Abalancedbinarysearchtree
• (Onecanstoreanything-notjustnumbers-insuchatreeaslongastheitemsarecomparable)
• NB!Comparisonsbetweenlinesegmentsisdonebycomputingpointsofintersectionwiththesweeplinewhenneededonly
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A degenerate case
Special delicate care must be taken if line segments share end points
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Analysis
• Weinserted2nendpointsintotheeventqueue:– O(nlogn)time.
• Eachintersectionwasinsertedasaneventpoint:– O(klogn)time.
• Total:– O(nlogn+klogn)=O((n+k)logn)time.
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Reflection • O((n+k)logn)time…?Betteranalysispossible-seeLemma2.3!• Thisisworsethanthebrute-forcesolution(O(n2))ifkismorethan
O(n2/logn)butbetterwhenkisless– Itis,however,output-sensitive
• ComparewiththeincrementalalgorithmthatcomputesconvexhullsfromLecture1– Ouralgorithmisalsosort-ofincremental…
• …makesmonotoneprogressandkeepsaninvariant– butstilldifferent
• …considersnotonlyinputpoints(=endpointsoflinesegments)butalsointersectionsthatarefoundduringthecourseofthealgorithm
• Todetermineifthereareintersections,wejustneedtostopthelinesweepassoonasthefirstintersectionisfound,ifitexists– So,this(simpler)problemcanbesolvedinO(nlogn)time
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Doubly-Connected Edge Lists (DCEL)
• Adatastructuresuitabletokeeptrackofsubdivisionsoftheplaneintodifferentkindsofregions.
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Some basic definitions (Cont.)
• Apolygonalchainisafiniteconcatenationoflinesegmentswhereconsecutivesegmentsshareoneendpoint(”Oneendswherethenextstarts”)
• Asimplepolygonisa”closedpolygonalchainthatdoesn’tself-intersect”– Simplepolygonshaveverticesandedgesthatformstheirboundaries
– Aconvexhullisasimplepolygonvertex
edge
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DCEL:s • Geometricaldata:
– Coordinates• Topologicaldata:
– Pointerstoneighboringitems
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DCEL:s • Storeddata:
Lists/arrays
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The overlay of two subdivisions Euler’sformula
• Foranembeddingofaplanargraphwithvvertices,eedges,andffaces(regionsboundedbyedges,includingtheouter,infinitely-large,region)withoutcrossingedges,
v−e+f=2
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(v=7, e=8, f=3)
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Applying plane sweep
• Maintainadoubly-connectededgelistduringthesweepoverthesubdivisions.
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An application: Boolean operations
Application:ComputingClosestPairsusingPlaneSweep• Problem:
– Givenasetofpoints,findapairofpointswiththesmallestdistancebetweenthem
• RequiresΩ(nlogn)time– Simplereduction
• SolvedinO(nlogn)timebymanyalgorithms,forinstanceplanesweep
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LowerBoundonClosestPairComputations• ReductiontoElementUniqueness:• Givenasetofnumbers{xi}(and
facedwithhavingtodecideifallareunique),we– 1)constructpointspi=(xi,0),– 2)runanyClosestPointalgorithm,and– 3)checkifthedistancebetweenthe
closestpointsreturnedis0ornot• Ifnot0,allareuniqueandotherwise
thereareduplicates• WehavesolvedElementUniqueness
thatrequiresΩ(nlogn)time,andapartfromtheClosestPointalgorithm,everythingwedotakesO(n)time
• So,itmustbetheClosestPointalgorithmthattakesΩ(nlogn)time.
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