VisibilityGraphsofStaircasePolygons

Yulia AlexandrMentor:Prof.JamesAbelloNSF grant CCF-1559855

Letmeremindyou…• Weconsiderasimplenon-degeneratecollectionofpointsintheplanethatproducesapolygon• Inparticular,welookatstaircasepolygonpaths• Twoverticesofapolygonarecalledinternallyvisibleiftheclosedlinesegmentbetweenthemiseitheranedgeofthepolygonorliesentirelyintheinteriorofthepolygon(Abello etal)• Thevisibilitygraphofapolygonisagraphwhosevertexsetisthesameasthevertexsetofthepolygonandwhoseedgesarethestraight-linesegmentsbetweeninternallyvisiblevertices

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BalancedTableau

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• Hookofacell isthecollectionofcellsthatincludesthechosencellwithallthecellsaboveitandallthecellstotheright

• Mate cellswithrespecttothechosencell• Atableauisbalanced ifthevalueofeverycellliesinbetweeneverypairofmatecellsinitshook

• (!)Tableaurepresentssloperanks inastaircasepathonnvertices

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LocalMax(Min)Rule

• Applytheruletoobtaintheadjacencymatrix

ProblemStatement

• TheproblemisknowntobePSPACE• WealsowanttoknowwhetheritisNPorP

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ProblemStatement:Input: AbalancedtableauTnOutput: Buildastaircasepolygons.t. itsvisibilitygraphisisomorphictolocalmax (Tn)

WhatItried:

• RandomStuff L

• ConvexHullApproach L

• InductiveApproach L /J• VisibilityRegionsApproach 😍

VisibilityRegionsApproach

• Startsbuildingfromthemiddle• Takesadvantageofunboundedness• Formsavisibilityregiontoplaceeachnewvertex

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Example

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Example

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Example

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Example1

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Example1

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• Canvisibilityregionsbeempty?

Toogoodtobetrue…

• Canvisibilityregionsbeempty?Yep.

Toogoodtobetrue…

• Canvisibilityregionsbeempty?Yep.• Why?

Toogoodtobetrue…

• Canvisibilityregionsbeempty?Yep.• Why? Researchishard.

Toogoodtobetrue…

• Canvisibilityregionsbeempty?Yep.• Why? Researchishard.• Whatmakesthemempty?

Toogoodtobetrue…

• Canvisibilityregionsbeempty?Yep.• Why? Researchishard.• Whatmakesthemempty? Notpreservingsloperanksoffarthestseenvertices!!

Toogoodtobetrue…

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• Regionsareneveremptyaslongaswepreservesloperanksoffarthestseenverticesateachstageofconstruction

• Concave-concave(convex-convex)• Concave-convex(convex-concave)• Generalcase

WhatIproved:

• Regionsareneveremptyaslongaswepreservesloperanksoffarthestseenverticesateachstageofconstruction

• Concave-concave(convex-convex)• Concave-convex(convex-concave)• Generalcase

• Itisalwayspossibletopreservesloperanksoffarthestseenvertices

WhatIproved:

• Determinecomplexity• Doublecheckandpolishproofs• Finalizeresultsforpublication

What’sleft:

Acknowledgements:

• Prof.JamesAbello• DIMACSandProf.Gallos• NSF grant CCF-1559855

Thanks!J

•References:• [1]Abello etal, VisibilityGraphsofStaircasePolygonsandtheWeakBruhatOrder,I:fromVisibilityGraphstoMaximalChains*.Discrete&ComputationalGeometry.1995.331-358.