Red Black Trees Top Down Deletion

RecalltherulesforBSTdeletion RedBlackTrees Top-DownDeletion

Whatcangowrong? 1. Ifthedelete Ifthedeletenod nodeisre eisred? d? Notaproblem– noRBpropertiesviolated 2. Ifthedelete Ifthedeletedno dnodeisb deisblack? lack? Ifthenodeisnottheroot,deletingitwill changetheblack-heightalongsomepath

Bottom-Upvs.Top-Down • Bottom-Upisrecursive – BSTdeletiongoingdownthetree(windingup therecursion) – FixingtheRBpropertiescomingbackupthe tree(unwindingtherecursion)

1. Ifvertextobedeletedisaleaf,justdelete it. 2. Ifvertextobedeletedhasjustonechild, replaceitwiththatchild 3. Ifvertextobedeletedhastwochildren, replacethevalue ofbyit’sin-order predecessor’svaluethendeletetheinorderpredecessor(arecursivestep)

ThegoalofT-DDeletion • Todeletear Todeletearedle edleaf af • Howdoweensurethat’swhathappens? – Aswetraversethetreelookingfortheleafto delete,wechangeeverynodeweencounterto red. – IfthiscausesaviolationoftheRBproperties, wefixit

Terminology • MatchingWeisstextsection12.2 – Xisthenodebeingexamine Xisthenodebeingexamined d – TisX’ssibling TisX’ssibling – PisX’s(andT’s)parent PisX’s(andT’s)parent – RisT’srightchild RisT’srightchild – LisT’sleftchild LisT’sleftchild

• Top-Downisiterative – Restructurethetreeonthewaydownsowe don’thavetogobackup

• ThisdiscussionassumesXistheleftchildofP. Asusual,thereareleft-rightsymmetriccases.

1

BasicStrategy

Step1– Examinetheroot

• Aswetraversethetree,wechangeevery nodewevisit,X,toRed.

1. Ifbothoftheroot’schildrenareBlack a. MaketherootRed

• WhenwechangeXtoRed,weknow

b. MoveXtotheappropriatechildoftheroot c. Proceedtostep2

– PisalsoRed(wejustcamefromthere)

2. OtherwisedesignatetherootasXand proceedtostep2B.

– Tisblack(sincePisRed,it’schildrenare Black)

Step2– themaincase

Case2A XhastwoBlackChildren

Aswetraversedownthetree,wecontinually encounterthissituationuntilwereachthe nodetobedeleted XisBlack,PisRed,TisBlack

2A1.Thas2BlackChildren

P X

2A2.T’sleftchildisRed

T

WearegoingtocolorXRed,thenrecolor othernodesandpossiblydorotation(s) basedonthecolorofX’sandT’schildren 2A.Xhas2Blackchildren 2B.XhasatleastoneRedchild

2A3.T’srightchildisRed **ifbothofT’schildrenareRed, wecandoeither2A2or2A3

Case2A1 XandThave2BlackChildren

Case2A2 Xhas2BlackChildrenandT’sLeftChildisRed

P

RotateLaroundT,thenLaroundP RecolorXandPthencontinuedownthetree

P

L X

T

X

P

T X L

T

P

T X

L1

L2

JustrecolorX,PandTandmovedownthetree L1

L2

2

Case2B XhasatleastoneRedchild

Case2A3 Xhas2BlackChildrenandT’sRightChildisRed RotateTaroundP RecolorX,P,TandRthencontinuedownthetree

Continuedownthetreetothenextlevel IfthenewXisRed,continuedownagain

T

P X

T

IfthenewXisBlack(TisRed,PisBlack) R

P

RotateTaroundP RecolorPandT

R R1

X L

R2

Backtomaincase– step2

L R1

R2

Case2BDiagram

Step3

P X

Eventually,findthenodetobedeleted– aleaforanodewith onenon-nullchildthatisaleaf.

T

DeletetheappropriatenodeasaRedleaf Movedownthetree. P

Step4 ColortheRootBlack

P

X

T

T

X

IfmovetoBlackchild(2B2) IfmovetotheRedchild(2B1) RotateTaroundP;RecolorPandT Movedownagain Backtostep2,themaincase

Example1 Delete10fromthisRBTree

Example1(cont’d)

15

15

6

X

17 12

3 10

20

16 13

18

7

Step1– Roothas2Blackchildren.ColorRootRed Descendthetree,movingXto6

6

17 12

3 10

23

20

16 13

18

23

7

OneofX’schildrenisRed(case2B).Descenddownthe tree,arrivingat12.SincethenewX(12)isalsoRed(2B1), continuedownthetree,arrivingat10.

3



15

15 6

6

17 12

3 10

13

X

20

16 18

17 12

3 23

13

7

20

16 18

23

7

Step3-Since10isthenodetobedeleted,replaceit’svalue withthevalueofit’sonlychild(7)anddelete7’srednode

Thefinaltreeafter7hasreplaced10and7’srednode deletedand(step4)theroothasbeencoloredBlack.



15

X

6

17 12

3 2

4

10

6 20

16

15

12

3 2

13

17

4

10

20

16 13

XhasatleastoneRedchild(case2B).Proceeddown thetree,arrivingat6.Since6isalsoRed(case2B1), continuedownthetree,arrivingat12.

Step1– therootdoesnothave2Blackchildren. Colortherootred,SetX=rootandproceedtostep2


Example2(cont’d) 15

15 6

3

6

P

T

X

4

10

12

16 13

Xhas2Blackchildren.X’ssibling(3)alsohas2blackchildren. Case2A1– recolorX,P,andTandcontinuedownthetree,arriving at10.

P

3

20 2

2

17

17

4

X

10

12

T

16

20

13

Xisnowtheleaftobedeleted,butit’sBlack,sobacktostep 2. Xhas2BlackchildrenandThas2Blackchildren– case2A1 RecolorX,PandT. Step3-- Nowdelete10asaredleaf. Step4-- Recolortherootblack

4


Example2Solution 15

15 6

17 10 12

3

16

20 12

5 2

4

13 3 2

7 4

11 6

Validand unaffected Rightsubtree

13

9

Step1– roothas2Blackchildren.ColorRootred. SetXtoappropriatechildofroot(10)



15 10

10 12

5 3 2

7 4

15

X 5

11 6

3

13

9

2

XhasoneRedchild(case2B) Traversedownthetree,arrivingat12.

P

T

X

7 4

12

11 6

13

9

Sincewearrivedatablacknode(case2B2)assuringTisred andPisblack),rotateTaroundP,recolorTandP Backtostep2


Example3(cont’d) 15

15 5 5 10 3

7

10

P

3

X

T

4

6

9

12

12 2

2

P

7

11

13

NowXisBlackwithRedparentandBlacksibling. XandTbothhave2Blackchildren(case2A1) JustrecolorX,PandTandcontinuetraversal

4

6

9

11

X

13

T

Havingtraverseddownthetree,wearriveat11,theleafto bedeleted,butit’sBlack,sobacktostep2. XandTbothhavetwoBlackchildren.RecolorX,PandT. Step3-- delete11asaredleaf. Step4-- Recolortherootblack

5

Example3Solution 15

5 10 3

2

7

4

6

12

9

13

6

Red Black Trees Top Down Deletion

Recommend Documents