Look at the following picture. This is a...
Transcript of Look at the following picture. This is a...
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Lookatthefollowingpicture.Thisisarepresentationofastreetnetwork.Thebluedotsrepresentparkingmeters.Makeagraphicalrepresentationofthisstreetnetwork.Notethatanysidewalkswithoutparkingmetersarenotrepresentedbyedgesbecausethosestreetsarenotnecessaryinournetwork.Verticesshouldrepresentstreetintersections.
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DoesthisnetworkhaveanEulerCircuit?DoesthisnetworkhaveanEulerPath?WhywouldanEulerPathnotbeoptimalforthisspecificproblemaboutparkingmeters?
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Sowhatisthebestrouteforcollectingthemoneyfromthemeters?Canyoumakeacircuitforthisgraph?(rememberacircuitjuststartsandendsatthesameplace,itcanpassthesameedgemorethanonce)
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Isthereawaytoaddonlyafew(oronly1)edgetomakethisintoanEulercircuit?
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Byaddingjustoneedge,wecanmakeanEulercircuit,butnote,theedgeisnotreallynew,itisjustreused
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Whatwejustdidiscalledeulerizingagraph.• Wetookagraphandaddededgesbyduplicatingexistingedgestomakeaconnectedevenvalentgraph
• WefoundanEulercircuitontheeulerizedgraph
EulerizingaGraph—addingedgesthatduplicateexistingedgestoaconnectedgraphtomakeallvalenceseven.
Goal → make m Euler Circuit
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EulerizingaGraphExample:1) locatetheverticeswithoddvalence
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2) addoneendofanedgeateachoddvertex,matchupthenewedgewithanexistingedgeintheoriginalgraph
duplicated
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→
•
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3) createanEulercircuit
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4) “Squeeze”theEulercircuitontotheoriginalgraph.Notethattherewillbetworeusesofedgesandeachreusecorrespondstoanaddededge
GeneralRule:Thenumberofreusesofedgesequalsthenumberofaddededgesduringeulerization
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Howdowefindthebesteulerization?Didwefindthemostefficientpathforthelastgraph?
Isthereawaywherewecanhaveonlyonereuseinsteadoftwo?
.
Optimal colorization
$-4
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Obviouslyonereusededgeisbetterthantwo.Sothisisabettereulerization.Thereisanalgorithmtofindthebesteulerization,butitisacomplicatedalgorithmthatwearenotgoingtoexamine.