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6.851AdvancedDataStructures(Spring12)Prof.ErikDemaine

Problem10 SamplesolutionCompactbalancedparentheses.

1. ConstructagraphGwithavertexforeachblock,andanedgebetweenthetheblockcontain-ingeachparenthesisandtheblockcontaining itsmatch. ObservethatG isasimplegraph,since if twopioneer parenthesis lie inthe same block, their matchingparens cannot also liein the same block. We construct the following planar embedding for this graph: place theverticesinorderalongahorizontalline,anddrawtheedgesbelowthislinesuchthattheedgesdonotcross. Thisispossiblebecauseofthenestingpropertythattheparenthesispairsmustobey. Sinceaplanarembeddingexists,G isaplanargraphandthusthenumberofedges isatmostthreetimesthenumberofvertices. Thereare|S|/Bvertices,andthereisabijectionbetween

the

edges

and

pioneer

parenthesis,

thus

there

are

at

most

3|S|/B

=

O(|S|/B)

pioneer

parenthesis,asdesired. Forthetightboundof2|S|/B3,observethatthisgraphisalsoanouterplanargraph(asshownbythesameembedding).

2. DivideS intoblocksofsizeB= 1/2lg|S|.Makeatablegivingtheblockcontainingthematchingparenthesisofeachpioneerparen-

thesis.Makea rankdatastructureoverthebitvectorof pioneerparenthesis(1 forpioneer, 0

otherwise).MakearankdatastructureovertheopenparensinS,andanotheroverthecloseparens

inS.aMakealookuptableT1 whichgivenablockandanindexintotheblock( |S|1/2lg|S|

possibilities)returns itsmatchingparenif it is intheblock,orsay itisntthere.Make a lookup table T2 which given a block and an integer k between B and Ba

( |S| lg|S| possibilities) return the first closing paren in the block such that thenumberofopenparensminusthenumberofclosingparensintheblockuptothatparenequalsk.

Todoqueryonaparenp,firstcheckT1 foritsblock,ifitsclosingparenisthere,thenreturnit. Otherwise,userankoverpioneersto look intothepioneertabletofindwhichblockb psmatchingparenisin. ThenuseT2 forbwithkequaltothedifferencebetweenthenumberofopenparensandthenumberofcloseparensuptopminusdifferencebetweenthenumberofopenparensandthenumberofcloseparensuptob(computedusingrankoveropenparenandrankovercloseparenstructures).

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6.851 Advanced Data Structures

Spring 2012

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