CSE326:DataStructures Lecture#10 AmazinglyVexingLetters Bart - - PDF document

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CSE326:DataStructures Lecture#10 AmazinglyVexingLetters

Bart Niswonger SummerQuarter2001

Today’sOutline

• AVLTrees

– Deletion – buildTree – ThinkingaboutAVLtrees

• SplayTrees

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Deletion(ReallyEasyCase)

20 9 2 15 5 10 30 17 3 12

1 1 2 2 3

Delete(17)

Deletion(PrettyEasyCase)

20 9 2 15 5 10 30 17 3 12

1 1 2 2 3

Delete(15)

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Deletion(PrettyEasyCasecont.)

20 9 2 17 5 10 30 3 12

1 1 2 2 3

Delete(15)

Deletion(HardCase#1)

20 9 2 17 5 10 30 3 12

1 1 2 2 3

Delete(12)

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SingleRotationonDeletion

20 9 2 17 5 10 30 3

1 1 2 2 3

30 9 2 20 5 10 17 3

1 2 1 3

Whatisdifferent about deletionthaninsertion?

Deletion(HardCase)

Delete(9) 20 9 2 17 5 10 30 3 12

1 2 2 2 3 4

33 15 13

1

20 30 12 33 15 13

1

11 18

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DoubleRotationonDeletion

2 3 20 2 17 5 10 30 12

1 2 2 2 3 4

33 15 13

1 1

11 18 20 5 2 17 3 10 30 12

2 2 1 3 4

33 15 13

1 1

11 18

DeletionwithPropagation

20 5 2 17 3 10 30 12

2 2 1 3 4

33 15 13

1 1

11 18 Whatrotationdoweapply?

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PropagatedSingleRotation

30 20 17 33 12 15 13

1

5 2 3 10

4 3 2 1 2 1

11 20 5 2 17 3 10 30 12

2 2 1 3 4

33 15 13

1 1

11 18 18

PropagatedDoubleRotation

17 12 11 5 2 3 10

4 2 3 1

20 5 2 17 3 10 30 12

2 2 1 3 4

33 15 13

1 1

11 18 15

1

20 30 33

1

18 13

2

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AVLDeletionAlgorithm

Recursive

1.Ifatnode,delete it 2.Otherwiserecurse tofindit 3.Correctheights a.Ifimbalance#1, singlerotate b.Ifimbalance#2 (ordon’tcare), doublerotate

Iterative

1.Searchdownwardfor node,stacking parentnodes 2.Deletenode 3.Unwindstack, correctingheights a.Ifimbalance#1, singlerotate b.Ifimbalance#2 (ordon’tcare) doublerotate

FunwithAVLTrees

ToInsertasequenceofnkeys(unordered) 1934187 intoinitiallyemptyAVLtreetakes Ifwethenprintusinginordertraversaltaking O(n) whatdoweget?

1 1

( lo l g )

• g

log

n n i i

O n n i n

• ✁

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Whatcanweimprove?

PrintingeverynodeisO(n),nothingtodo Whataboutbuildingatree?

– CanwedoitinlessthanO(nlogn)

• Whatiftheinputissorted?

3471819

Ifitissorted,whybother? We’llseeinamoment!

AVLbuildTree

8 10 15 20 30 35 40 5 17 17 8 1015 5 20303540

Divide&Conquer

– Dividetheproblemintoparts – Solveeachpartrecursively – Mergethepartsintoa generalsolution Howlongdoes divide&conquertake?

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BuildTreeExample

35 17 15 5 8 10

3 2 2 1

30

1

40 20 8 10 15 20 30 35 40 5 17 8 10 15 5 8 5 30 35 40 20 30 20

BuildTreeAnalysis(Approximate)

T(1)=1 T(n)=2T(n/2)+1

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• ✁

2 1 n

• ✝
• ✂✞✂✆☎

2 1 n

2 1 n

2 1 n

BuildTreeAnalysis(Exact)

PreciseAnalysis:T(0)=b

T(n)=T()+T()+c Byinductiononn: T(n)=(b+c)n+b Basecase: T(0)=b=(b+c)0+b Inductionstep: T(n)=(b+c)+b+ (b+c)+b+c =(b+c)n+b QED:T(n)=(b+c)n+b=

1 2 1 2 1

n n n

Application:BatchDeletion

• Supposeweareusinglazy deletion
• Whentherearelots ofdeletednodes

(n/2),needtoflush themallout

• Batchdeletion:

– Printnon-deletednodesintoanarray

How?

– Divide&conquerAVLTreebuild – Totaltime:

Whywecared!

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ThinkingAboutAVL

• Observations

+WorstcaseheightofanAVLtreeisabout1.44logn +Insert,Find,DeleteinworstcaseO(logn) +Onlyone(singleordouble)rotationneededon insertion

• O(logn)rotationsneededondeletion

+Compatiblewithlazydeletion

• Heightfieldsmustbemaintained(or2-bitbalance)

AlternativestoAVLTrees

• Changethebalancecriteria:

– Weightbalancedtrees

• keepaboutthesamenumberofnodesineachsubtree
• notnearlyasnice
• Changethemaintenanceprocedure:

– Splaytrees

• “blind”adjustingversionofAVLtrees

– noheightinformationmaintained!

• insert/findalwaysrotatesnodetotheroot!
• worstcasetimeisO(n)
• amortizedtimeforalloperationsisO(logn)
• mysterious,butoftenfasterthanAVLtreesinpractice

(betterlow-orderterms)

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SplayTrees

• “blind”rebalancing

– noheightorbalanceinformationstored

• amortizedtimeforalloperationsisO(logn)
• worstcasetimeisO(n)
• insert/findalwaysrotatesnodetotheroot!

– Goodlocality

• mostcommonkeysmovehighintree

Idea

17 10

9 2 5 3

You’reforcedtomake areallydeepaccess: Sinceyou’redownthereanyway, fixupalotofdeepnodes!

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SplayOperations:Find

• Find(x)
• 1. doanormalBSTsearchtofindnsuch

that n->key=x

• 2. moventorootbyseriesofzig-zag and

zig-zig rotations,followedbyafinalzig if necessary

Zig-Zag*

g X p Y

n

Z W

*Thisisjustadoublerotation

n

Y g W p Z X Helped Unchanged Hurt

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Zig-Zig

n

Z Y p X g W g W X p Y

n

Z

Zig

p X

n

Y Z

n

Z p Y X root root

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WhySplayingHelps

• Nodenanditschildrenarealwayshelped(raised)
• Exceptforfinalzig,nodesthatarehurtbyazig-

zagorzig-zigarelaterhelpedbyarotationhigher upthetree!

• Result:

– shallow(zig)nodesmayincreasedepthbyoneortwo – helpednodesmaydecreasedepthbyalargeamount

• Ifanoden ontheaccesspathisatdepthd before

thesplay,it’sataboutdepthd/2 afterthesplay

– Exceptionsaretheroot,thechildoftheroot,andthe nodesplayed

Locality

• Assumem
• n accessinatreeofsizen

– TotalamortizedtimeO(m logn) – O(logn)peraccessonaverage

• Getsbetterwhenyouonlyaccessk

distinctitemsinthemaccesses.

– Exercise.

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SplayingExample

2 1 3 4 5 6 Find(6) 2 1 3 6 5 4 zig-zig

StillSplaying6

zig-zig 2 1 3 6 5 4 1 6 3 2 5 4

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AlmostThere,StayonTarget

zig 1 6 3 2 5 4 6 1 3 2 5 4

SplayAgain

Find(4) zig-zag 6 1 3 2 5 4 6 1 4 3 5 2

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ExampleSplayedOut

zig-zag 6 1 4 3 5 2 6 1 4 3 5 2

SplayTreeSummary

• AlloperationsareinamortizedO(logn)time
• Splayingcanbedonetop-down;better

because:

– onlyonepass – norecursionorparentpointersnecessary

• InventedbySleatorandTarjan(1985),now

widelyusedinplaceofAVLtrees

• Splaytreesarevery effectivesearchtrees

– relativelysimple – noextrafieldsrequired – excellentlocality properties:frequentlyaccessed keysarecheaptofind

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ToDo

• Studyformidterm!
• Readthroughsection4.7inthebook
• Comments&Feedback
• HomeworkIV(studying)
• ProjectII– partB

ComingUp

• MidtermnextWednesday
• AHuge SearchTreeDataStructure

(notonthemidterm)