RotationDistancesbetweenBinaryTrees

二元樹之間的旋轉距離研究生：陳嬿如指導教授：王有禮教授國立臺灣科技大學資訊管理系碩士學位論文1.Introduction1.1RotationdistanceTherotationdistance

d(T,T’)betweentwobinarytreesTandT’withthesamenumberofleavesistheminimumnumberofrotationsneededtotransformTintoT’.RotationLeftrotationRightrotation34LeftRotation5Rightrotation61.2SurveyItremainsanopenproblemwhetherrotationdistancebetweenbinarytreescanbecomputedinpolynomialtime[29].Pallo[22],

Rogers[27]gaveapproximationalgorithmsCulikandWood[7]upperboundto2n–2[17,30]2n–6forn≥11Pallo[20,21,23]rotationdistanceinthelattice7“Restricted”rotationdistanceBonninandPallo[4]onlyatnodeswithaleaf

Sundar[31]onlyasingledirectionofrotationcalledrightrotationCleary[5]therootortherightchildoftheroot

Pallo[24]atnodesalongtherightarmofthetree8RootorRightchildoftheroot9Rightarm102.Preliminaries2.1LW-sequenceTheweightwT

(i)iscalledtheleftweightofviinTandtheintegersequencewT=(wT

(1),wT

(2),...,wT

(n))iscalledtheleftweightsequence(LW-sequenceforshort)ofT[19]

.

vi

123456wT(i)12121612GiventheLW-sequenceofabinarytreeT,weshowthatthepreorderofTcanbedeterminedbyusingtheintervalrepresentation.Thestartpoint

ST

(i)=i

wT

(i).TheendingpointDT

(i)=i.viwT

(i)ST(i)DT(i)v1101v2202v3123v4224v5145v66060123456v1v2v3v4v5v6132.2AVL-rotationsLL(Rightrotation)RR(Leftrotation)LRRL14LRRL152.3Weightdifferencesδ(i)=wT’

(i)

wT

(i),1≤i≤n,astheweightdifferenceforeachnodeviinthetrees.vi123456wT’(i)123451wT(i)121216δ(i)00+2+2+45163.AlgorithmAlgorithm1Tree-TransformationInput:TheLW-sequenceswTandwT’ofthesourcetreeTandthedestinationtreeT’,respectively.Output:

dist(T,T’),anestimationoftherotationdistancebetweenTandT’.Step1.Letdist(T,T’)=0,andcompute,,and.Step2.while

δ(i)≠0forsomei

{1,...,n}do

Substep2.1ifthereexistvi,vp,vqinthecurrenttreesuchthatviistherightchildofvpandvpistheleftchildofvqforsomei

{1,...,n}andδ(i)>0>δ(q)

then

doLR-rotation(i);18

Substep2.2

elseifthereexistvi,vp,vq

inthecurrenttreesuchthatviistheleftchildofvpandvpistherightchildofvqforsomei

{1,...,n}andδ(i)>0>δ(p)

thendoRL-rotation(i);

Substep2.3

elseifthereexistvi,vp

inthecurrenttreesuchthatviistherightchildofvpforsomei

{1,...,n}andδ(i)>0thendoRR-rotation(i);

Substep2.4

elseifthereexistvi,vl

inthecurrenttreesuchthatvlistheleftchildofviforsomei

{1,...,n}andδ(i)<0then

doLL-rotation(i);endif

Substep2.5

dist(T,T’)=dist(T,T’)+1,and

recompute,and.endwhileStep3.Outputdist(T,T’).193.1Anexample20213.2AnalysisTheorem8GiventheLW-sequencesoftwobinarytreesTandT’withthesamenumberofinternalnodes,sayn,thealgorithmTree-TransformationproducesasequenceofrotationstoconvertTintoT’inO(Δn)time,where.O(n2)224.ConclusionUsinganamortizedanalysisComputingtheaveragerotationdistanceIn[20],Pallopresentedtherotationdistanceinthelatticeofbinarytrees.(3)=1.5(3)=1.424References[1]G.M.Adelson-VelskyandE,M.Landis,“Analgorithmfororganizationofinformation,”SovietMathematicsDoklady3(1962)1259–1263.[2]A.Andersson,“Generalbalancedtrees,”JournalofAlgorithms30(1999)1-18.[3]R.Bayer,“SymmetricbinaryB-trees:datastructureandmaintenancealgorithms,”Acta

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