Visibility Graphs of Staircase Polygons Yulia Alexandr Mentor: Prof. James Abello NSF grant CCF-1559855
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.