lecture 7 - stanford universityweb.stanford.edu/class/archive/cs/cs161/cs161.1182/... · •idea 0:...

Lecture7BinarySearchTreesandRed-BlackTrees

Announcements

• HW3released!(DueFriday)

• Specialguestlecturer:SamKim!

Roadmap

Graphs!

Asymptotic

Analysis

Dynamic

ProgrammingGreedyAlgs

MIDTERM

The

Future!

Moredetailedscheduleonthewebsite!

Wearehere

Today

• Beginabriefforayintodatastructures!

• SeeCS166formore!

• Binarysearchtrees

• YoumayrememberthesefromCS106B

• Theyarebetterwhenthey’rebalanced.

thiswillleadusto…

• Self-BalancingBinarySearchTrees

• Red-Black trees.

Whyarewestudyingself-balancingBSTs?

1. Thepunchlineisimportant:

• AdatastructurewithO(log(n))INSERT/DELETE/SEARCH

2. TheideabehindRed-Black Treesisclever

• It’sgoodtobeexposedtocleverideas.

• Alsoit’sjustaestheticallypleasing.

Somedatastructuresforstoringobjectslike(aka,nodes withkeys)

• (Sorted)arrays:

• (Sorted)linkedlists:

• Somebasicoperations:

• INSERT,DELETE,SEARCH

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SortedArrays

• O(n)INSERT/DELETE:

• O(log(n))SEARCH:

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eg,Binarysearchtoseeif3isinA.

8754.5

Sortedlinkedlists

• O(1)INSERT/DELETE:• (assumingwehaveapointertothelocationoftheinsert/delete)

• O(n)SEARCH:

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MotivationforBinarySearchTrees

Sorted Arrays LinkedListsBinarySearch

Trees*

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

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

Binarytreeterminology

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1

3

5

Thisnodeis

theroot

Thisisanode.

Ithasakey (7).

Thesenodes

areleaves.

Theleft child

of is3 2

Therightchild

of is3 4

Bothchildren

of areNIL1

Fortodayallkeysaredistinct.

Eachnodehas

twochildren

isadescendant

of

2

5Eachnodehasa

pointertoitsleft

child,rightchild,

andparent.

BinarySearchTrees

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2

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1

35

• It’sabinarytreesothat:• EveryLEFTdescendantofanodehaskeylessthanthatnode.

• EveryRIGHTdescendantofanodehaskeylargerthanthatnode.

• Exampleofbuildingabinarysearchtree:

BinarySearchTrees

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BinarySearchTrees

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BinarySearchTrees

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Q:Isthistheonly

binarysearchtreeI

couldpossiblybuild

withthesevalues?

A:No. Imade

choicesabout

whichnodesto

choosewhen.Any

choiceswould

havebeenfine.

Aside:thisshouldlookfamiliar

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2

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35

kinda likeQuickSort

BinarySearchTrees

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• It’sabinarytreesothat:• EveryLEFTdescendant ofanodehaskeylessthanthatnode.


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BinarySearchTreeNOT aBinary

SearchTree

Whichof

theseisa

BST?

Rememberthegoal

FastSEARCH/INSERT/DELETECanwedothese?

SEARCHinaBinarySearchTreedefinitionbyexample

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5EXAMPLE:Searchfor4.

EXAMPLE:Searchfor4.5• Itturnsoutitwillbeconvenient

toreturn4inthiscase

• (thatis,return thelastnode

beforewewentoffthetree)

Ollietheover-achievingostrich

Writepseudocode

(oractualcode)to

implementthis!Howlongdoesthistake?

O(lengthoflongestpath)= O(height)

INSERTinaBinarySearchTree

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1

3

5EXAMPLE:Insert4.5

4.5

• INSERT(key):• x=SEARCH(key)

• Insertanewnodewith

desiredkeyatx…

x= 4

Youthoughtaboutthison

yourpre-lectureexercise!

(Seehiddenslidefor

pseudocode.)

INSERTinaBinarySearchTree

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5EXAMPLE:Insert4.5

4.5

• INSERT(key):• x=SEARCH(key)

• ifkey>x.key:

• Makeanewnodewiththe

correctkey,andputitasthe

rightchildofx.

• ifkey<x.key:

• Makeanewnodewiththe

correctkey,andputitasthe

leftchildofx.

• if x.key ==key:

• return

x= 4

Thisslide

skippedin

class– here

forreference

DELETEinaBinarySearchTree

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EXAMPLE:Delete2

• DELETE(key):• x=SEARCH(key)

• if x.key ==key:

• ….deletex….

Youthoughtaboutthisinyourpre-

lectureexercisetoo!

Thisisabitmorecomplicated…see

thehiddenslidesforsomepictures

ofthedifferentcases.

x= 2

DELETEinaBinarySearchTreeseveralcases(byexample)saywewanttodelete3

Thistriangle

isacartoon

forasubtree

3

Case1:if3isaleaf,

justdeleteit.2

3

Case2:if3hasjustonechild,

movethatup.

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2

55

Siggi theStudiousStork

Writepseudocodeforallof

these!(OrseeIPython

NotebookforLecture7)

Thisslideskipped

inclass– herefor

reference!

DELETEinaBinarySearchTreectd.

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3.1

2

5

Case3:if3hastwochildren,

replace3withit’simmediatesuccessor.(aka,nextbiggestthingafter3)

• DoesthismaintaintheBST

property?• Yes.

• Howdowefindthe

immediatesuccessor?• SEARCHfor3inthesubtree

under3.right

• Howdoweremoveitwhen

wefindit?• If[3.1]has0or1children,

dooneofthepreviouscases.

• Whatif[3.1]hastwo

children?• Itdoesn’t.

4

3.1

Thisslideskipped

inclass– herefor

reference!

Howlongdotheseoperationstake?

• SEARCH isthebigone.

• EverythingelsejustcallsSEARCH andthendoessomesmallO(1)-timeoperation.

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Time=

O(heightoftree)

Treeshavedepth

O(log(n)).Done!

Luckythelackadaisicallemur.

Howlongdoessearchtake?

Wait...

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2

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7

3

5

6

• Thisisavalidbinarysearchtree.

• Theversionwithnnodeshas

depthn,not O(log(n)).

Couldsuchatreeshowup?

InwhatorderwouldIhaveto

insertthenodes?

Insertingintheorder

2,3,4,5,6,7,8woulddoit.

Sothiscould happen.

Whattodo?

• Goal:FastSEARCH/INSERT/DELETE

• AllthesethingstaketimeO(height)

• Andtheheightmightbebig!!!L

• Idea0:

• Keeptrackofhowdeepthetreeisgetting.

• Ifitgetstootall,re-doeverythingfromscratch.

• AtleastΩ(n)everysooften….

• Turnsoutthat’snotagreatidea.Insteadweturnto…


Howoftenis“everyso

often”intheworstcase?

It’sactuallyprettyoften!

Self-BalancingBinarySearchTrees

Idea1:Rotations

• MaintainBinarySearchTree(BST)property,whilemovingstuffaround.

BA

CY

XYOINK!

CLAIM:

thisstillhasBSTproperty.

NomatterwhatlivesunderneathA,B,C,

thistakestimeO(1).(Why?)

BA

C

Y

X

B

A

C

Y

X

Bfell

down.

Note:A

,B,C,X,Yareva

riablenames,n

otth

e

contentso

fthenodes.

Thisseemshelpful

4

2

8

7

3

6

5

YOINK!

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Doesthiswork?

• Wheneversomethingseemsunbalanced,dorotationsuntilit’sokayagain.

LuckytheLackadaisicalLemur

Evenformethisis

prettyvague.Whatdo

wemeanby“seems

unbalanced”?What’s

“okay”?

Idea2:havesomeproxyforbalance

• Maintainingperfectbalanceistoohard.

• Instead,comeupwithsomeproxyforbalance:

• Ifthetreesatisfies[SOMEPROPERTY],thenit’sprettybalanced.

• Wecanmaintain[SOMEPROPERTY] usingrotations.

Thereareactuallyseveral

waystodothis,buttoday

we’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:

• allpathsfromxtoNIL’shavethesamenumberofblacknodesonthem.

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NIL NIL NIL NIL NIL NIL NIL NIL

I’mnotgoingtodrawtheNIL

childreninthefuture,butthey

aretreatedasblacknodes.

Examples(?)• Everynodeiscoloredred orblack.

• Therootnodeisablacknode.

• NIL childrencountasblacknodes.

• Childrenofarednodeareblacknodes.

• Forallnodesx:

• allpathsfromxtoNIL’shavethesamenumberofblacknodesonthem.

Yes!

No! No! No!

Why???????

• Thisisprettybalanced.

• Theblacknodes arebalanced

• Therednodesare“spreadout”sotheydon’tmessthingsuptoomuch.

• Wecanmaintainthispropertyasweinsert/deletenodes,byusingrotations.

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Thisisthereallycleveridea!

This Red-Blackstructureisaproxyforbalance.It’sjustasmidgeweakerthanperfectbalance,butwecanactuallymaintainit!

Thisis“prettybalanced”

• Toseewhy,intuitively,let’strytobuildaRed-BlackTreethat’sunbalanced.

Luckythe

lackadaisical

lemur

Let’sbuildsomeintuition!

Onepathcouldbetwiceas

longanotherifwepaditwith

rednodes.

Conjecture:theheightofa

red-blacktreeisatmost2log(n)

Thatturnsouttobebasicallyright.[proofsketch]

• Saythereareb(x)blacknodesinanypathfromxtoNIL.

• (excludingx,includingNIL).

• Claim:

• Thenthereareatleast2b(x) – 1non-NILnodesinthesubtreeunderneathx.(Includingx).

• [Proofbyinduction– onboardiftime]

Then:

𝑛 ≥ 2$ %&&' − 1

≥ 2+,-.+/

01 − 1Rearranging:

𝑛 + 1 ≥ 234563'

71 ⇒ ℎ𝑒𝑖𝑔ℎ𝑡 ≤ 2log(𝑛 + 1)

x

y

usingtheClaim

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

z

NIL

Okay,soit’sbalanced……butcanwemaintainit?

•Yes!

• Fortherestoflecture:• sketchofhowwe’ddothis.

• SeeCLRSformoredetails.

• (Youarenotresponsibleforthedetailsforthisclass– butyoushouldunderstandthemainideas).

Manycases

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• Supposewewanttoinserthere.• eg,wanttoinsert0.

• Andthenthereare9morecasesforallofthevarious

symmetriesofthese3cases…

InsertingintoaRed-BlackTree

• Makeanewrednode.

• Insertitasyouwouldnormally.

73

6Example:insert0

0

Whatifitlookslikethis?

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Manycases

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• Fixthingsupifneeded.

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6Example:insert0

0

No!


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6Example:insert0

Can’twejustinsert0as

ablacknode?

0No!


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Weneedabitmorecontext

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Example:insert0


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-1


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Example:insert0

0


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• Add0asarednode.

-1


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Example:insert0

0


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Flip

colors!

• Add0asarednode.

• Claim: RB-Treepropertiesstillhold.

-1

Butwhatifthat wasred?

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Example:insert0

0


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-1

Morecontext…

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Example:insert0

0


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-1

-3

Morecontext…

6

Example:insert0


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-1

-3

Nowwe’re

basicallyinserting

6intosome

smallertree.

Recurse!

Manycases

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Example:Insert0.

• Actually,thiscan’t

happen?• 6-3 pathhasoneblacknode

• 6-7-… hasatleasttwo

• Itmighthappenthatwe

justturned0redfrom

thepreviousstep.

• Oritcouldhappenif

isactuallyNIL.

0


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RecallRotations

• MaintainBinarySearchTree(BST)property,whilemovingstuffaround.

BA

CY

XYOINK!

CLAIM:

thisstillhasBSTproperty.

BA

C

Y

X

B

A

C

Y

X





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0


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6

YOINK!

3

60

7

Needtoarguethat

ifRB-Treeproperty

heldbefore,itstill

does.

Manycases

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DeletingfromaRed-Blacktree

Funexercise!


That’salotofcases

• Youarenotresponsible forthenitty-grittydetailsofRed-BlackTrees.(Forthisclass)

• Thoughimplementingthemisagreatexercise!

• Youshouldknow:

• WhatarethepropertiesofanRBtree?

• And(moreimportant)whydoesthatguaranteethattheyarebalanced?

Whatwasthepointagain?

• Red-BlackTreesalwayshaveheightatmost2log(n+1).

• AswithgeneralBinarySearchTrees,alloperationsareO(height)

• SoalloperationsareO(log(n)).

Conclusion:Thebestofbothworlds

Sorted Arrays LinkedLists

Balanced

BinarySearch

Trees

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

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

Today

• Beginabriefforayintodatastructures!

• SeeCS166formore!

• Binarysearchtrees

• YoumayrememberthesefromCS106B

• Theyarebetterwhenthey’rebalanced.

thiswillleadusto…

• Self-BalancingBinarySearchTrees

• Red-Black trees.

Recap

Recap

• Balancedbinarytreesarethebestofbothworlds!

• Butweneedtokeepthembalanced.

• Red-BlackTrees dothatforus.

• WegetO(log(n))-timeINSERT/DELETE/SEARCH

• Cleveridea:haveaproxyforbalance

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Before nexttime

• Pre-lectureexerciseforLecture8

• (More)funwithprobability!

Nexttime

• Hashing!

lecture 7 - stanford universityweb.stanford.edu/class/archive/cs/cs161/cs161.1182/... · •idea 0:...

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