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CS106B,Lecture20AdvancedBinaryTrees

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Plan for Today • DiscusshowtomakeaBSTclass(hint:usefulforMiniBrowser)• AdvancedBSTs

–  Balancing–  Red-BlackTrees–  SplayTrees

• Non-BSTbinarytrees– Heaps–  CartesianTrees

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ImplementingTreeSetandTreeMap

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A BST set class //TreeSet.h//Asetofintegersrepresentedasabinarysearchtree.classTreeSet{members;...private:TreeNode*root;//NULLforanemptytree};

–  ThisisbasicallyhowStanfordlibrary'sSetclassisimplemented.

–  ClientcodetalkstotheTreeSet,nottothenodeobjectsinsideit.

– MembersofTreeSetcreateandmanipulatenodesandpointers.

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root

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Tree member template returnTypeTreeSet::functionName(parameters){helperName(root,parameters);}returnTypehelperName(TreeNode*node,parameters){...}•  Treemethodsareoftenimplementedrecursivelyin2steps:

–  apublicfunctionintendedtobecalledbytheclient–  a"helper"functionthatacceptsapointertothenodetoprocess

–  thepublicfunctiontypicallyjustcallsthehelperandpassesrootnode

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Tree maps • Convertingatreesetintoatreemap:

–  Eachtreenodewillstorebothakeyandavalue–  treeisBST-orderedbyitskeys–  keysmustbecomparable(havea<operator)forordering

structTreeMapNode{stringkey;intvalue;TreeMapNode*left;TreeMapNode*right;};

root

key ="Locke"val =51

key ="Jack"val =36

key ="Kate"val =28

key ="Sayid"val =36

key ="Sawyer"val =49

key ="Desmond"val =49

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Tree map details • Eachtreesetoperationcorrespondstooneinthetreemap:

–  add(value) → put(key,value)–  contains(value) → containsKey(key)–  remove(value) → remove(key)–  mustaddanoperation: get(key)

– WhataboutcontainsValue?• WoulditscodebesimilartothecodeforcontainsKey?

root

key ="Locke"val =51

key ="Jack"val =36

key ="Kate"val =28

key ="Sayid"val =36

key ="Sawyer"val =49

key ="Desmond"val =49

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Announcements • YoushouldbemostlydonewithCacheinMiniBrowser.LineManagerishard

• Homework3isgraded.Here'sthegradedistribution:

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BalancedTrees

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Trees and balance • balancedtree:OnewhereforeverynodeR,theheightofR'ssubtreesdifferbyatmost1,andR'ssubtreesarealsobalanced.–  Runtimeofadd/remove/containsarecloselyrelatedtoheight.–  Balancedtree'sheightisroughlylog2N.UnbalancedisclosertoN.

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height=4(balanced)

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root

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6height=7(unbalanced)

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BST balance question • AddingthefollowingnodestoanemptyBSTinthefollowingorderproducesthetreeatright:22,9,34,18,3.

• Q:Whatisanorderinwhichwecouldhaveaddedthenodestoproduceanunbalancedtree?A.18,9,34,3,22B.9,18,3,34,22C.9,22,3,18,34D.noneoftheabove

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AVL trees • AVLtree:Abinarysearchtreethatusesmodifiedaddandremoveoperationstostaybalancedasitselementschange.–  basicidea:Whennodesareadded/removed,repairtreeshapeuntilbalanceisrestored.• rebalancingisO(1);overalltreemaintainsanO(logN)height

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rotate

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Red-Black trees •  red-blacktree:Giveseachnodea"color"ofredorblack.(video)

–  Rootisblack.Root'sdirectchildrenarered.Allleavesareblack.–  Ifanodeisred,itschildrenmustallbeblack.–  Everypathdownwardfromanodetothebottommustcontainthesamenumberof"black"nodes.

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Splay trees • splaytree:Rotateseachelementyouaccesstothetop/root

–  veryefficientwhenthatelementisaccessedagain(happensalot)–  easytoimplementanddoesnotneedheightfieldineachnode

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Non-BSTBinaryTrees

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Heaps

• Whatifyouwanttofindthek-smallestelementsinanunsortedVector?–  Findthetop10studentsinaclass?

• Whatifyouwantedtoconstantlyinsertandremoveinsortedorder?– Modelahospitalemergencyroomwhereindividualsareseeninorderoftheirurgency

–  PriorityQueue• What'sagoodchoice?

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Heaps

•  Idea:ifweuseaVector,ittakesalongtimetoinsertorremoveinsortedorder(orsearchtheVectorforthesmallestelement)

•  Ifweuseabinarysearchtree,it'sfasttoinsertandremove(O(logN))butit'sslowtofindtheminimum/maximumelement(O(logN))

•  Idea:useatree,butstoretheminimum/maximumelementastheroot–  Treeshavelog(N)insertion/deletion–  LookingattherootisO(1)

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Heaps

• heap:Acompletebinarytreewithverticalordering:– min-heap:allchildrenmustbe≥parent'svalue– max-heap:allchildrenmustbe≤parent'svalue

–  completetree:alllevelsarefullofchildrenexceptperhapsthebottomlevel,inwhichallexistingnodesaremaximallytotheleft.• Nicecorollary:heapsarealwaysbalanced

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Heap enqueue • Whenaddingtoaheap,thevalueisfirstplacedatbottom-right.

–  Torestoreheapordering,thenewlyaddedelementisshifted("bubbled")upthetreeuntilitreachesitsproperplace(wereachtheroot,ortheelementissmallerthanitsparent[min-heap]).

–  Enqueue15atbottom-right;bubbleupuntilinorder.

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Heap dequeue • Removetheroot,andreplaceitwiththefurthest-rightancestor• Torestoreheaporder,theimproperrootisshifted("bubbled")downthetreebyswappingwithitssmallerchild.–  dequeueminof10;swapupbottom-rightleafof65;bubbledown.

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Cartesian Trees

• Howwouldyouquicklyfindtheminimum/maximumelementinanrange?– Maximumelevationonahike?–  Besttimetobuy/sellastockwithinacertainrangeoftimes?

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Cartesian Trees

• Therootstorestheminimum(ormaximum)elementintheentirearray

• Theleftsubtreeisthentheminimum(ormaximum)elementintherangetotheleftoftheroot;therightsubtreeistheminimum(ormaximum)elementintherangetotherightoftheroot–  Followsthemin-(ormax)-heapproperty:everyparentissmaller(orbigger)thanitschild

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Cartesian Trees

• WhatwouldtheCartesiantreelooklikeforthisarrayifwe'retryingtofindtheminimumvalueinarange?

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Cartesian Trees

• WhatwouldtheCartesiantreelooklikeforthisarrayifwe'retryingtofindtheminimumvalueinarange?

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Cartesian Trees

• Howwouldwewritethefollowingfunction:findMinElemInRange(CartesianNode*node,intstart,intend)structCartesianNode{intindex;CartesianNode*left;CartesianNode*right;};

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Exam • ThisexamwasalittleharderthanIintended.Youalldidreallywellandshowedalotofknowledgeofhardtopics.

• Commonmistakes:–  BigOquestion,partc:theinnerforloopisactuallyO(1)–  ADTtraceproblem:misreadingthefirstforloop–  Recursivetrace:integerdivisionwiththeparameters,wrongindiceswithsubstring

–  RecursiveBacktracking:modifiedparametersaftertherecursivecall,goingoutofboundsonthegrid,notdeclaringandreturningtheGrid,improperbasecases(returningtrue/falseorfailingtoprunethetree)

–  ADTwrite,parta:badscopingforthevaluesoftheMapandOBOBforstringparsing

–  ADTwrite,partb:incompletesubmissions(lowontime?)

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Exam