New Lecture 7 - Stanford Universityweb.stanford.edu/.../cs/cs161/cs161.1176/Slides/Lecture7.pdf ·...

Lecture7BinarySearchTreesandRed-BlackTrees

Announcements

• HW3released!(DueFriday)

Today:binarysearchtrees• Briefforayintodatastructures!• SeeCS166formore!

• Whatarebinarysearchtrees?• YoumayrememberthesefromCS106B• Whyaretheygood?• Whyaretheybad?

thiswillleadusto…

• Self-BalancingBinarySearchTrees• Red-Black trees.

Whyarewestudyingself-balancingBSTs?

1. Thepunchlineisimportant:• AdatastructurewithO(log(n))INSERT/DELETE/SEARCH

2. TheideabehindRed-Black Treesisclever• It’sgoodtobeexposedtocleverideas.• Alsoit’sjustaestheticallypleasing.

Motivationforbinarysearchtrees• We’vebeenassumingthatwehaveaccesstosomebasicdatastructures:• (Sorted)linkedlists

• (Sorted)arrays

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Sortedlinkedlists

• O(1)insert/delete(assumingwehaveapointertothelocationoftheinsert/delete):

• O(n)search/select:

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42 871 3 5HEAD

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SortedArrays

• O(n)insert/delete:

• O(log(n))search,O(1)select:

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421 3

42 871 3 5Search:Binarysearchtoseeif3isinA.

8754.5

Select:ThirdsmallestisA[3].

Thebestofbothworlds

Sorted Arrays LinkedLists BinarySearchTrees*

Search O(log(n)) O(n) O(log(n))

Insert/Delete O(n) O(1) O(log(n))

Treeterminology

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1

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5Thisnodeistheroot

Thisisanode.Ithasakey (7).

Thesenodesareleaves.

Theleft childof is3 2

Therightchildof is3 4

Bothchildrenof areNIL1

Todayallkeysaredistinct.

BinarySearchTrees

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• It’sabinarytreesothat:• EveryLEFTdescendentofanodehaskeylessthanthatnode.• EveryRIGHTdescendentofanodehaskeylargerthanthatnode.

• Exampleofbuildingabinarysearchtree:

BinarySearchTrees

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BinarySearchTrees

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BinarySearchTrees

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Aside:thisshouldlookfamiliar

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kinda likeQuickSort

BinarySearchTrees

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• It’sabinarytreesothat:• EveryLEFTdescendent ofanodehaskeylessthanthatnode.• EveryRIGHTdescendentofanodehaskeylargerthanthatnode.

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BinarySearchTree NOT aBinarySearchTree

SEARCHinaBinarySearchTreedefinitionbyexample

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5EXAMPLE:Searchfor4.

EXAMPLE:Searchfor4.5• Itturnsoutitwillbeconvenient

toreturn4inthiscase• (thatis,return thelastnode

beforewewentoffthetree)

Ollietheover-achievingostrich

Writepseudocode(oractualcode)toimplementthis!

INSERTinaBinarySearchTree

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5EXAMPLE:Insert4.5

4.5

• INSERT(key):• x=SEARCH(key)• ifkey>x.key:

• Makeanewnodewiththecorrectkey,andputitastherightchildofx.

• ifkey<x.key:• Makeanewnodewiththe

correctkey,andputitastheleftchildofx.

• if x.key ==key:• return

x= 4

DELETEinaBinarySearchTree

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5EXAMPLE:Delete2

• DELETE(key):• x=SEARCH(key)• if x.key ==key:• ….deletex….

Thisisabitmorecomplicated…x= 2

DELETEinaBinarySearchTreeseveralcases(byexample)saywewanttodelete3

Thistriangleisacartoonforasubtree

3Case1:if3isaleaf,justdeleteit. 2

3

Case2:if3hasjustonechild,movethatup.

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Wewon’twritedownpseudocodeforthis– trytodoityourself!

DELETEinaBinarySearchTreectd.

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Case3:if3hastwochildren,replace3withit’simmediatesuccessor.(aka,nextbiggestthingafter3) • Howdowefindthe

immediatesuccessor?• SEARCH(3.right,3)

• Howdoweremoveitwhenwefindit?• RunDELETEforoneoftheprevioustwocases.

• Wait,whatifit’sTHIScase?(Case3).• It’snot.

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Howlongdotheseoperationstake?• SEARCH isthebigone.• EverythingelsejustcallsSEARCH andthendoessomesmallO(1)-timeoperation.

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Time=O(depthoftree)

TreeshavedepthO(log(n)).Done!

Luckythelackadaisicallemur.

Wait...

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• Thisisavalidbinarysearchtree.

• Theversionwithnnodeshasdepthn,not O(log(n)).

Couldsuchatreeshowup?InwhatorderwouldIhaveto

insertthenodes?

Insertingintheorder2,3,4,5,6,7,8woulddoit.

Sothiscould happen.

Whattodo?

• Keeptrackofhowdeepthetreeisgetting.

• Ifitgetstootall,re-doeverythingfromscratch.• AtleastΩ(n)everysooften….

•Otherideas?


Howoftenis“everysooften”intheworstcase?It’sactuallypretty

often!

Idea1:Rotations• MaintainBinarySearchTree(BST)property,whilemovingstuffaround.

BA

CY

XYOINK!

CLAIM:thisstillhasBSTproperty.

NomatterwhatlivesunderneathA,B,C,thistakestimeO(1).(Why?)

BA

C

Y

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B

A

C

Y

X

Thisseemshelpful

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YOINK!

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Doesthiswork?

• Wheneversomethingseemsunbalanced,dorotationsuntilit’sokayagain.

LuckytheLackadaisicalLemur

Evenformethisisprettyvague.Whatdowemeanby“seemsunbalanced”?What’s

“okay”?

Idea2:havesomeproxyforbalance

• Maintainingperfectbalanceistoohard.• Instead,comeupwithsomeproxyforbalance:• Ifthetreesatisfies[SOMEPROPERTY],thenit’sprettybalanced.• Wecanmaintain[SOMEPROPERTY] usingrotations.

Thereareactuallyseveralwaystodothis,buttodaywe’llsee…

• A Binary Search Tree that balances itself!• No more time-consuming by-hand balancing!• Be the envy of your friends and neighbors

with the time-saving…

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6Maintain balance by stipulating that

black nodes are balanced, and that there aren’t too many red nodes.

It’s just good sense!

Red-BlackTrees

Red-BlackTreestheserulesaretheproxyforbalance

• Everynodeiscoloredred orblack.• Therootnodeisablacknode.• NIL childrencountasblacknodes.• Childrenofarednodeareblacknodes.• Forallnodesx:• allpathsfromxtoNONE’shavethesamenumberofblacknodesonthem.

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Examples(?) • Everynodeiscoloredred orblack.• Therootnodeisablacknode.• NONEchildrencountasblacknodes.• Childrenofarednodeareblacknodes.• Forallnodesx:

• allpathsfromxtoNONE’shavethesamenumberofblacknodesonthem.

Yes!No! No! No!

Why???????

• Thisisprettybalanced.• Theblacknodes arebalanced• Therednodesare“spreadout”sotheydon’tmessthingsuptoomuch.

• Wecanmaintainthispropertyasweinsert/deletenodes,byusingrotations.

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Thisis“prettybalanced”• Toseewhy,intuitively,let’strytobuildaRed-BlackTreethat’sunbalanced.

Luckythelackadaisicallemur

Let’sbuildsomeintuition!

Onepathcouldbetwiceaslongasalltheothersifwepaditwithrednodes.

Conjecture:theheightofared-blacktreeisatmost2log(n)

Thatturnsoutthebebasicallyright.[proofsketch]

• Saythereareb(x)blacknodesinanypathfromxtoNONE.• (includingx).

• Claim:• Thenthereareatleast2b(x) – 1nodesinthesubtreeunderneathx.

• [Proofbyinduction– onboardiftime]Then:𝑛 ≥ 2$ %&&' − 1≥ 2

+,-.+/01 − 1

Rearranging:

𝑛 + 1 ≥ 234563'

71 ⇒ ℎ𝑒𝑖𝑔ℎ𝑡 ≤ 2log(𝑛 + 1)

x

y

usingtheClaim

b(root)>=height/2becauseofRBTree rules.

Okay,soit’sbalanced……butcanwemaintainit?

•Yes!• Fortherestoflecture:• sketchofhowwe’ddothis.

• SeeCLRSformoredetails.

InsertingintoaRed-BlackTree

• Makeanewrednode.• Insertitasyouwouldnormally.

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6Example:insert0

0

Whatifitlookslikethis?

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• Makeanewrednode.• Insertitasyouwouldnormally.• Fixthingsupifneeded.

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No!


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Can’twejustinsert0asablacknode?

0No!


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0

• Insteadrecolorlikethis.• Needtoargue:

• RB-Treepropertiesstillhold.• Whatabouttheredroot?

• if6isactuallytheroot,coloritblack.• Else,recursivelyre-colorupthetree.


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ish

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-1Nowtheproblemlookslikethis,whereI’minserting 6



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• Actually,thiscan’thappen?

• Itmighthappenthatwejustturned0redfromthepreviousstep.

• OritcouldhappenifisactuallyNIL.

0


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RecallRotations• MaintainBinarySearchTree(BST)property,whilemovingstuffaround.

BA

CY

XYOINK!

CLAIM:thisstillhasBSTproperty.

BA

C

Y

X

B

A

C

Y

X

InsertingintoaRed-BlackTree• Makeanewrednode.• Insertitasyouwouldnormally.• Fixthingsupifneeded.

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YOINK!3

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NeedtoarguethatifRB-Treepropertyheldbefore,itstill

does.

Thepunchline:

• AfewthingsstilllefttocheckforINSERT!• Anythingelsethatmighthappenlooksbasicallylikewhatwejustdid.

• Formallydealingwiththerecursion.• Youcheckthese!(orseeCLRS)

• DELETE issimilar.

That’sbasicallyit

• Red-BlackTreesalwayshaveheightatmost2log(n+1).• AswithgeneralBinarySearchTrees,alloperationsareO(height)• SoalloperationsareO(log(n)).

Pluckythepedanticpenguin

Conclusion:Thebestofbothworlds

Sorted Arrays LinkedListsBalanced

BinarySearchTrees

Search O(log(n)) O(n) O(log(n))

Insert/Delete O(n) O(1) O(log(n))

Recap

•Hashing!Nexttime

• Balancedbinarytreesarethebestofbothworlds!• Butweneedtokeepthembalanced.• Red-BlackTrees dothatforus.• WegetO(log(n))-timeINSERT/DELETE/SEARCH• Cleveridea:haveaproxyforbalance

New Lecture 7 - Stanford Universityweb.stanford.edu/.../cs/cs161/cs161.1176/Slides/Lecture7.pdf ·...

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