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Zip TreesRobertE.Tarjan

PrincetonUniversity&IntertrustTechnologies

Pitea,Sweden,9January2018

ToDon,whoinspiredmetodesignandanalyzealgorithms

ObservationsOverthelast60years,computerscientistshavedevelopedmanybeautifulandtheoreticallyefficientalgorithms.

Butcomputerscienceisstill ayoungfield.

Wehaveoftensettledforthefirst(goodenough)solution.

Itmaynotbethebest– thedesignspaceisrich.

Goal:simplicityIdentifythesimplestpossibleefficientmethodstosolvebasicproblems:algorithmsfrom“thebook”

ala“proofsfromthebook”(Erdős)Algorithmsassimpleaspossible,withprovable resourcebounds

forimportantinputclasses,andefficientinpractice

“Makeeverythingassimpleaspossible,butnotsimpler”- Einstein

Sometimes,simplicityiscritical,notjustdesirable,notablyinconcurrent algorithms.

Makingaconcurrentalgorithmcorrectisnotoriously hard.Thesimplertheunderlyingidea,thegreaterthechanceofsuccess.

Exampleofsimplicity:Zip Trees

(imagefromStreetArtonPinterest)

Dictionary:containsasetS ofitems,eachwithassociatedinformation.

Operations:Access(x):Determineifx isinS.Ifso,returnx’s information.

Insert(x):(x notinS)Insertx anditsinformation.

Delete(x):(x inS)Deletex anditsinformation.

BinarysearchUniverseofitems(orofaccesskeysorindexvalues)istotallyordered,allowingbinarycomparison

Binarysearch:MaintainS insortedorder.Tofindx inS:IfS empty,stop(failure).IfS non-empty,comparex tosomeitemy inS.Ifx =y,stop(success).Ifx <y,searchin{z inS|z <y}.Ifx >y,searchin{z inS|z >y}.

Implementation:binarysearchtree

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Binarytree:Eachnodex hasaleftchildx.leftandarightchildx.right,eitherorbothofwhichcanbenull.Anodeisbinary,unary,oraleaf ifithas0,1,or2nullchildren,respectively.

n =#(non-null)nodesdepthofx:lengthofpathfromroottox,roothasdepth0

heightofx:maxlengthofapathfromx toaleaf,leaveshavedepth0

Binarysearchtree

Items(keyplusdata)innodes,onepernode,insymmetricorder(in-order):itemsinleftsubtreeareless,itemsinrightsubtree aregreaterinkeyorder.

TofindanitemtakesO(d +1)time,whered =depthofitem’snode,orofnullnodereachedbysearchifitemisnotintree.

Binarysearchtree

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BestcaseAllleaveshavedepthswithin1:depth⌊lgn⌋.

(lg:base-twologarithm)

Canachieveiftreeisstatic(orinsertionorderdeterminedoff-line)

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(Leaf)InsertionSearch.Replacenullbynodewithitem.InsertR F

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LeafInsertionSearch.Replacenullbynodewithitem.InsertR F

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WorstcaseNaturalbutbadinsertionorder:sorted.InsertA,B,C,D,E,F,G,…

Depthoftreeisn – 1.Worst-caseaccesscostisn.

=list!

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Waystoimproveefficiency

• Balance: AVLtrees(Adelson-Velskii &Landis1962),red-blacktrees(Bayer1972;Guibas &Sedgewick1978),weakAVLtrees(Sen&Tarjan 2009)…

• Self-adjustment: Splaytrees(Sleator &Tarjan1983)

• Randomization:Zip trees

AnAVLtree(imagefromWikipedia)

Ared-blacktree(imagefromWikipedia)

ASplayTree

(imagebyJorgeStolfi)

Rotation(restructuringprimitive)

rotateatxrotateatyy

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Zip Trees

Idea:Oninsertion,chooseaheightforanitemandinsertitatthegivenheight,orclosetoit.Chooseheightslikethoseinabest-caseBST:½thenodesatheight0,¼atheight1,1/8atheight2…

Choosetheheightsrandomly.

Wecannotchooseheightsexactly.

Instead,foreachnodetobeinsertedwechoosearank,asfollows:flipafaircoinandcountthenumberofheadsbeforethefirsttail.Therankofanodedoesnotchangewhileitisinthetree.

Therankofanodehasageometric distribution:anodehasrankk withprobability1/2k +1.

Wewanttheheightofanodetobewithinaconstantfactorofitsrank.

Zip*TreeAbinarysearchtreeinwhicheachnodehasarankchosenrandomlyoninsertion,withnodessymmetricallyorderedbykeyandheaporderedbyrank,breakingranktiesinfavorofsmallerkey:

x.left.key <x.key <x.right.keyx.left.rank <x.rankx.right.rank ≤x.rank

*Zip:“tomoveveryfast”

AZip Tree

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Zip treeinsertion?

Rootinsertion(Stephenson1980)

Letx betheitemtobeinserted.Followthesearchpathforx,unzipping itbysplittingitintoapathPofnodeswithkeyslessthanthatofxandapathQofnodeswithkeysgreaterthanthatofx.MakethetopnodeofPtheleftchildofx andthetopnodeofQ therightchildofx.

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InsertN

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InsertN N

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InsertNUnzipsearchpath

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Ziptreeinsertion:hybridofleaf&rootinsertion

(Sprugnoli 1980;Martinez&Roura 1998)

Toinsertx: Chooseitsrank.Searchforxuntilreachingthenodey thatx shouldreplace. Unziptherestofthesearchpath(fromydown)bysplittingitintoapathPofnodeswithkeyslessthanx’sandapathQofnodeswithkeysgreaterthanx’s.MakethetopofPtheleftchildofxandthetopofQ therightchildofx.Makex achildoftheoldparentofy.

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InsertJ

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InsertJrank=2

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InsertJrank=2ReplaceH

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InsertJrank=2ReplaceHUnzippathfromH

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InsertJrank=2ReplaceHUnzippathfromH J2

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Aziptree(imagefromStreetArtonPinterest)

Ziptreedeletion:Inverseofinsertion

Searchforthenodex tobedeleted.Zip thepathfromx toitspredecessor(inkeyorder)withthepathfromx toitssuccessor(inkeyorder),bymergingthemindecreasingrankorder,breakingtiesinfavorofsmallerkey,toformasinglepathP.ReplacexasachildofitsparentbythetopnodeofP.

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DeleteP

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DeletePZipthepathsfromPtoMandPtoQ

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Staticpropertiesofziptrees

Expectedrootrank:lgn +O(1)RootrankisO(logn)withhighprobabilityE(nodedepth)=1.5lgn+O(1)(Prodinger 1996)TreedepthisO(logn)withhighprobabilityTreestructureisuniquelydeterminedbyranks:

History-independentdatastructure

Dynamicpropertiesofzip trees

Expectedrestructuringtimetoinsertordeleteanodeofrankk =O(k).ExpectedrestructuringtimeisO(1)ProbabilityofrestructuringtakingO(k)timeisexponentiallysmallinkInsertion/deletioncanbedonepurelytop-downNorotationcases,justunzippingandzippingNoswappingondeletion

Arezip treesnew?

Azip treeisatreap (Aragon&Seidel1996)withadifferentwayofchoosingranksanddifferentinsertion/deletionalgorithms.Zip treesallowrankties;treaps don’t.Azip treeisabinary-treerepresentationofaskiplist(Pugh1990)

Atreap(imagefromWikipedia)

Askiplist(imagebyIgorOstrovsky)

Zip Trees:Summary

Insertionsanddeletionsaresimpleandefficient,takeO(1)expectedrestructuring,andarepurelytop-down

History-independentNorestructuringonaccessStandardbinarytreerepresentation:lessspacethanskiplists

O(lglgn)balancebitspernodeCanbreakranktiesbyusingfractionalranks,

improvesE(depth)by8%Canmodifytosupportfrequency-biasedaccess

Currentwork

Develop,analyze,andimplementefficientnon-blockingconcurrent zip trees

Thanks!