Algorithmic methods for lowest common ancestor - mediaTUM

122 Pages·2006·0.98 MB·English

by  Johannes Nowak

Technische Universit(cid:228)t M(cid:252)nchen
Fakult(cid:228)t f(cid:252)r Informatik
Lehrstuhl f(cid:252)r E(cid:30)ziente Algorithmen
Algorithmic Methods for Lowest Common Ancestor Problems in
Directed Acyclic Graphs
Johannes Nowak
Vollst(cid:228)ndiger Abdruck der von der Fakult(cid:228)t f(cid:252)r Informatik der Technischen Universit(cid:228)t M(cid:252)nchen
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. A. Kemper, Ph.D.
Pr(cid:252)fer der Dissertation:
1. Univ.-Prof. Dr. E. W. Mayr
2. Univ.-Prof. Dr. F. J. Esparza Estaun
Die Dissertation wurde am 04.02.2008 bei der Technischen Universit(cid:228)t M(cid:252)nchen
eingereicht und durch die Fakult(cid:228)t f(cid:252)r Informatik am 15.09.2009 angenommen.

Abstract
Lowest common ancestor (LCA) problems in directed acyclic graphs (dags) have attracted scientific
interestintherecentyears. Directedacyclicgraphsarepowerfultoolsformodelingcausalitysystems
or other kind of entity dependencies, and efficient solutions to the respective lowest common ances-
tor problems are indispensable computational tools with regard to proper analysis of these systems.
Similar problems in trees are widely understood, however, the generalizations to dags fall short of
achievingcomparableefficiencies.
In this work, we develop and analyze algorithmic techniques for solving LCA problems in dags.
Themainfocusisonall-pairsLCAproblems,i.e.,problemsthatrequiretheanswerstotherespective
LCA queries for all vertex pairs in the dag. In particular, the basic problems studied are finding
one arbitrary (representative) LCA for all vertex pairs and listing all LCAs for all vertex pairs. We
identify and describe in-depth three basic algorithmic approaches that lead to efficient solutions to
theseproblems,namelydynamicprogramming,matrixmultiplication,andapathcoverapproach.
Thedynamicprogrammingmethodincombinationwithusingtransitivereductionyieldsalgorithms
that are efficient in the average case and – as a result of an experimental study also described in this
thesis – in practice. However, the running times depend on the number of edges in the transitive
reduction of the input dags and are hence vulnerable to special constructs such as dense bipartite
graphs.
Matrix multiplication approaches lead to general upper time bounds for the two problems that
improve upon hitherto best solutions. More specifically, we prove that representative LCAs for all
vertexpairscanbecomputedintimeO(n2.575),andallLCAsforallvertexpairscanbecomputedin
time O(n3.334) for a dag with n vertices. We note in this place that any improvement of the matrix
multiplicationexponentautomaticallyimprovestheseboundsfortheLCAproblems.
Thethirdalgorithmicapproach,apathcovertechnique,yieldsefficientsolutionsforthetwoprob-
lems in dags G with small width w(G), namely O(n2w(G)logn) for computing representative and
O(n2w(G)2log2n) for computing all LCAs. However, perhaps the most important result connected
with the path cover technique is an improved algorithm for finding representative LCAs in general
that restricts the class of dags for which the upper bound O(n2.575) is actually attained significantly.
Furtherresearchinthisdirectionmightultimatelyimprovethisbound.
Additionally,wereviewalgorithmicapplicationsofthepresentedtechniques,namelyproblemvari-
antsindynamicsettings,inweighteddags,andunderspace-efficiencyconsiderations. Althoughsome
oftheuppertimeboundsthatwepresentinthisworkmightturnouttobetight,furtherstudyofthese
andalikeproblemsseemstobeapromisingdirectionforfutureresearch.
Finally,wepresenttheresultsofanexperimentalstudyofsomeofthealgorithmsdescribedinthis
work, in particular, the algorithms that are based on dynamic programming. As a consequence, we
concludethatbothrepresentativeandallLCAsforallvertexpairscanbecomputedreasonablyfastin
practice,i.e.,withruntimeclosetoO(n2).
Acknowledgments
Inthefirstplace,IthankmyadvisorErnstW.Mayrforhishelpful,encouragingguidanceandgener-
oussupportthroughoutmytimeofdoctoralstudies. Furthermore,Iamindebtedtoallthecurrentand
formermembersoftheEfficientAlgorithmsGroupforprovidingastimulatingandmotivatingwork-
ing environment. In particular, I am thankful to Stefan Eckhardt, Moritz Maaß, and Hanjo Täubig
who have contributed significantly to my academic progress. Special thanks goes to Arno Buchner
(and his wife) for providing food on the week-ends. Finally, I thank my parents exceptionally, not
onlyfortheircontributiontothisthesisbutformyeducationingeneral.
CONTENTS
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Preliminaries 7
2.1 ElementaryConcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 AnalysisofAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 CommonAncestorProblemsinDirectedAcyclicGraphs . . . . . . . . . . . . . . . 13
3 Dynamic Programming Algorithms 19
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 All-PairsRepresentativeLCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 All-PairsAllLCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Matrix-Multiplication-Based Algorithms 35
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 RepresentativeLCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 AllLCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4.1 Fixed-VertexLCAVariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.2 All-PairsAllLCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5 Path Cover Techniques 51
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
v
vi Contents
5.2 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 AFirstNon-TrivialAll-PairsAllLCASolution . . . . . . . . . . . . . . . . . . . . 53
5.4 GeneralizedPathCoverApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.5 CombiningSmallWidthandLowDepth . . . . . . . . . . . . . . . . . . . . . . . . 59
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6 Algorithmic Applications 67
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 (L)CAProblemsinWeightedDags . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2.1 CommonAncestorProblemVariants. . . . . . . . . . . . . . . . . . . . . . 70
6.2.2 LowestCommonAncestorProblemVariants . . . . . . . . . . . . . . . . . 76
6.3 DynamicAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3.1 FullyDynamicLowestCommonAncestors . . . . . . . . . . . . . . . . . . 79
6.3.2 IncrementalLCAAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.4 Space-EfficientLCAComputations . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7 Experimental Analysis 91
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 ExperimentalSetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2.2 TestData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3.1 All-PairsRepresentativeLCA . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3.2 All-PairsAllLCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Summary of Notation 105
Bibliography 107
LIST OF FIGURES
2.1 TransitiveClosureandReduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 (Lowest)CommonAncestors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 LowerBoundforallLCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 DynamicProgrammingforRepresentativeLCAs . . . . . . . . . . . . . . . . . . . 24
3.2 SimpleAlgorithmforallLCAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 DynamicProgrammingforallLCAs . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 NaiveMerging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 CommonAncestorExistencetoReachability . . . . . . . . . . . . . . . . . . . . . 36
4.2 MatrixSampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 MaximumWitnessesbyRectangularMatrixMultiplication . . . . . . . . . . . . . . 40
4.4 MatrixMultiplicationtoAll-PairsFixedVertexLCA . . . . . . . . . . . . . . . . . 43
4.5 MatrixMultiplicationtoFixed-Vertex-PairsAllLCA . . . . . . . . . . . . . . . . . 45
5.1 SpecialDFSTree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 ExampleoftheCombinedAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.1 ShortestDistanceCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 All-PairsShortestDistanceCAtoAll-PairsShortestDistances . . . . . . . . . . . . 72
6.3 MatrixSampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.4 ProblemRelationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.1 RealWorldDensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2 AverageVertexDegrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3 ComparisonofRepresentativeLCAAlgorithms . . . . . . . . . . . . . . . . . . . . 96
7.4 AsymptoticEvaluationofRMQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.5 AsymptoticEvaluationofDP-LCA . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.6 AsymptoticEvaluationofRMQandDP-LCA(RealWorld) . . . . . . . . . . . . . . 99
vii
viii ListofFigures
7.7 ComparisonofallLCAAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.8 AsymptoticEvaluationofDP-APA-lazy . . . . . . . . . . . . . . . . . . . . . . . . 100
7.9 AsymptoticEvaluationofDP-APA-it . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.10 AsymptoticEvaluationofTC-APAandPC-APA . . . . . . . . . . . . . . . . . . . 101
7.11 EvaluationofLCASetSizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
LIST OF ALGORITHMS
1 GreedyChainCoverConstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 DP-AlgorithmforRepresentativeLCA . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 SimpleAlgorithmforallLCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 DP-AlgorithmforallLCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 MergewithForbiddenSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 AllLCAUsingRepresentativeLCAs . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7 AllLCAUsingRepresentativeLCAs(Improved) . . . . . . . . . . . . . . . . . . . . 55
8 PreprocessingforCombinedAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 61
9 AlgorithmforRepresentativeLCA . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
10 DP-AlgorithmforShortestDistanceCA . . . . . . . . . . . . . . . . . . . . . . . . . 74
ix