Compact Data Structures: A Practical Approach

567 Pages·2016·4.869 MB·English

by  Gonzalo Navarro

Compact Data Structures
A Practical Approach
Gonzalo Navarro
DepartmentofComputerScience,
UniversityofChile
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
CambridgeUniversityPressispartoftheUniversityofCambridge.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107152380
©GonzaloNavarro2016
Firstpublished2016
PrintedintheUnitedStatesofAmericabySheridanBooks,Inc.
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Navarro,Gonzalo,1969–author.
Title:Compactdatastructures:apracticalapproach/GonzaloNavarro,
UniversidaddeChile.
Description:NewYork,NY:UniversityofCambridge,[2016]|Includes
bibliographicalreferencesandindex.
Identifiers:LCCN2016023641|ISBN9781107152380(hardback:alk.paper)
Subjects:LCSH:Datastructures(Computerscience)|Computeralgorithms.
Classification:LCCQA76.9.D35N382016|DDC005.7/3–dc23
LCrecordavailableathttps://lccn.loc.gov/2016023641
ISBN978-1-107-15238-0Hardback
Contents
ListofAlgorithms pagexiii
Foreword xvii
Acknowledgments xix
1 Introduction 1
1.1 WhyCompactDataStructures? 1
1.2 WhyThisBook? 3
1.3 Organization 4
1.4 SoftwareResources 6
1.5 MathematicsandNotation 7
1.6 BibliographicNotes 10
2 EntropyandCoding 14
2.1 Worst-CaseEntropy 14
2.2 ShannonEntropy 16
2.3 EmpiricalEntropy 17
2.3.1 BitSequences 18
2.3.2 SequencesofSymbols 20
2.4 High-OrderEntropy 21
2.5 Coding 22
2.6 HuffmanCodes 25
2.6.1 Construction 25
2.6.2 EncodingandDecoding 26
2.6.3 CanonicalHuffmanCodes 27
2.6.4 BetterthanHuffman 30
2.7 Variable-LengthCodesforIntegers 30
2.8 Jensen’sInequality 33
2.9 Application:PositionalInvertedIndexes 35
2.10 Summary 36
2.11 BibliographicNotes 36
3 Arrays 39
3.1 ElementsofFixedSize 40
3.2 ElementsofVariableSize 45
3.2.1 SampledPointers 46
3.2.2 DensePointers 47
3.3 PartialSums 48
3.4 Applications 49
3.4.1 Constant-TimeArrayInitialization 49
3.4.2 DirectAccessCodes 53
3.4.3 Elias-FanoCodes 57
3.4.4 DifferentialEncodingsandInvertedIndexes 59
3.4.5 CompressedTextCollections 59
3.5 Summary 61
3.6 BibliographicNotes 61
4 Bitvectors 64
4.1 Access 65
4.1.1 Zero-OrderCompression 65
4.1.2 High-OrderCompression 71
4.2 Rank 73
4.2.1 SparseSampling 73
4.2.2 ConstantTime 74
4.2.3 RankonCompressedBitvectors 76
4.3 Select 78
4.3.1 ASimpleHeuristic 78
4.3.2 AnO(loglogn)TimeSolution 80
4.3.3 ConstantTime 81
4.4 VerySparseBitvectors 82
4.4.1 Constant-TimeSelect 83
4.4.2 SolvingRank 83
4.4.3 BitvectorswithRuns 86
4.5 Applications 87
4.5.1 PartialSumsRevisited 87
4.5.2 PredecessorsandSuccessors 89
4.5.3 Dictionaries,Sets,andHashing 91
4.6 Summary 98
4.7 BibliographicNotes 98
5 Permutations 103
5.1 InversePermutations 103
5.2 PowersofPermutations 106
5.3 CompressiblePermutations 108
5.4 Applications 115
5.4.1 Two-DimensionalPoints 115
5.4.2 InvertedIndexesRevisited 116
5.5 Summary 117
5.6 BibliographicNotes 117
6 Sequences 120
6.1 UsingPermutations 121
6.1.1 Chunk-LevelGranularity 121
6.1.2 OperationswithinaChunk 123
6.1.3 Construction 126
6.1.4 SpaceandTime 127
6.2 WaveletTrees 128
6.2.1 Structure 128
6.2.2 SolvingRankandSelect 132
6.2.3 Construction 134
6.2.4 CompressedWaveletTrees 136
6.2.5 WaveletMatrices 139
6.3 AlphabetPartitioning 150
6.4 Applications 155
6.4.1 CompressiblePermutationsAgain 155
6.4.2 CompressedTextCollectionsRevisited 157
6.4.3 Non-positionalInvertedIndexes 157
6.4.4 RangeQuantileQueries 159
6.4.5 RevisitingArraysofVariable-LengthCells 160
6.5 Summary 161
6.6 BibliographicNotes 162
7 Parentheses 167
7.1 ASimpleImplementation 170
7.1.1 RangeMin-MaxTrees 170
7.1.2 ForwardandBackwardSearching 175
7.1.3 RangeMinimaandMaxima 180
7.1.4 RankandSelectOperations 188
7.2 ImprovingtheComplexity 188
7.2.1 QueriesinsideBuckets 190
7.2.2 ForwardandBackwardSearching 191
7.2.3 RangeMinimaandMaxima 196
7.2.4 RankandSelectOperations 200
7.3 Multi-ParenthesisSequences 200
7.3.1 NearestMarkedAncestors 201
7.4 Applications 202
7.4.1 SuccinctRangeMinimumQueries 202
7.4.2 XMLDocuments 204
7.5 Summary 207
7.6 BibliographicNotes 207
8 Trees 211
8.1 LOUDS:ASimpleRepresentation 212
8.1.1 BinaryandCardinalTrees 219
8.2 BalancedParentheses 222
8.2.1 BinaryTreesRevisited 228
8.3 DFUDSRepresentation 233
8.3.1 CardinalTreesRevisited 240
8.4 LabeledTrees 241
8.5 Applications 245
8.5.1 RoutinginMinimumSpanningTrees 246
8.5.2 GrammarCompression 248
8.5.3 Tries 252
8.5.4 LZ78Compression 259
8.5.5 XMLandXPath 262
8.5.6 Treaps 264
8.5.7 IntegerFunctions 266
8.6 Summary 272
8.7 BibliographicNotes 272
9 Graphs 279
9.1 GeneralGraphs 281
9.1.1 UsingBitvectors 281
9.1.2 UsingSequences 281
9.1.3 UndirectedGraphs 284
9.1.4 LabeledGraphs 285
9.1.5 Construction 289
9.2 ClusteredGraphs 291
9.2.1 K2-TreeStructure 291
9.2.2 Queries 292
9.2.3 ReducingSpace 294
9.2.4 Construction 296
9.3 K-PageGraphs 296
9.3.1 One-PageGraphs 297
9.3.2 K-PageGraphs 299
9.3.3 Construction 307
9.4 PlanarGraphs 307
9.4.1 OrderlySpanningTrees 308
9.4.2 Triangulations 315
9.4.3 Construction 317
9.5 Applications 327
9.5.1 BinaryRelations 327
9.5.2 RDFDatasets 328
9.5.3 PlanarRouting 330
9.5.4 PlanarDrawings 336
9.6 Summary 338
9.7 BibliographicNotes 338
10 Grids 347
10.1 WaveletTrees 348
10.1.1 Counting 350
10.1.2 Reporting 353
10.1.3 SortedReporting 355
10.2 K2-Trees 357
10.2.1 Reporting 359
10.3 WeightedPoints 362
10.3.1 WaveletTrees 362
10.3.2 K2-Trees 365
10.4 HigherDimensions 371
10.5 Applications 372
10.5.1 DominatingPoints 372
10.5.2 GeographicInformationSystems 373
10.5.3 ObjectVisibility 377
10.5.4 Position-RestrictedSearchesonSuffixArrays 379
10.5.5 SearchingforFuzzyPatterns 380
10.5.6 IndexedSearchinginGrammar-CompressedText 382
10.6 Summary 388
10.7 BibliographicNotes 388
11 Texts 395
11.1 CompressedSuffixArrays 397
11.1.1 ReplacingAwith(cid:2) 398
11.1.2 Compressing(cid:2) 399
11.1.3 BackwardSearch 401
11.1.4 LocatingandDisplaying 403
11.2 TheFM-Index 406
11.3 High-OrderCompression 409
11.3.1 TheBurrows-WheelerTransform 409
11.3.2 High-OrderEntropy 410
11.3.3 PartitioningLintoUniformChunks 413
11.3.4 High-OrderCompressionof(cid:2) 414
11.4 Construction 415
11.4.1 SuffixArrayConstruction 415
11.4.2 BuildingtheBWT 416
11.4.3 Building(cid:2) 418
11.5 SuffixTrees 419
11.5.1 LongestCommonPrefixes 419
11.5.2 SuffixTreeOperations 420
11.5.3 ACompactRepresentation 424
11.5.4 Construction 426
11.6 Applications 429
11.6.1 FindingMaximalSubstringsofaPattern 429
11.6.2 LabeledTreesRevisited 432
11.6.3 DocumentRetrieval 438
11.6.4 XMLRetrievalRevisited 441
11.7 Summary 442
11.8 BibliographicNotes 442
12 DynamicStructures 450
12.1 Bitvectors 450
12.1.1 SolvingQueries 452
12.1.2 HandlingUpdates 452
12.1.3 CompressedBitvectors 461
12.2 ArraysandPartialSums 463
12.3 Sequences 465
12.4 Trees 467
12.4.1 LOUDSRepresentation 469
12.4.2 BPRepresentation 472
12.4.3 DFUDSRepresentation 474
12.4.4 DynamicRangeMin-MaxTrees 476
12.4.5 LabeledTrees 479
12.5 GraphsandGrids 480
12.5.1 DynamicWaveletMatrices 480
12.5.2 Dynamick2-Trees 482
12.6 Texts 485
12.6.1 Insertions 485
12.6.2 DocumentIdentifiers 486
12.6.3 Samplings 486
12.6.4 Deletions 490
12.7 MemoryAllocation 492
12.8 Summary 494
12.9 BibliographicNotes 494
13 RecentTrends 501
13.1 EncodingDataStructures 502
13.1.1 EffectiveEntropy 502
13.1.2 TheEntropyofRMQs 503
13.1.3 ExpectedEffectiveEntropy 504
13.1.4 OtherEncodingProblems 504
13.2 RepetitiveTextCollections 508
13.2.1 Lempel-ZivCompression 509
13.2.2 Lempel-ZivIndexing 513
13.2.3 FasterandLargerIndexes 516
13.2.4 CompressedSuffixArraysandTrees 519
13.3 SecondaryMemory 523
13.3.1 Bitvectors 524
13.3.2 Sequences 527
13.3.3 Trees 528
13.3.4 GridsandGraphs 530
13.3.5 Texts 534
Index 549
Algorithms
2.1 Buildingaprefixcodegiventhedesiredlengths page24
2.2 BuildingaHuffmantree 27
2.3 BuildingaCanonicalHuffmancoderepresentation 29
2.4 ReadingasymbolwithaCanonicalHuffmancode 29
2.5 Variousintegerencodings 34
3.1 Readingandwritingonbitarrays 41
3.2 Readingandwritingonfixed-lengthcellarrays 44
3.3 Manipulatinginitializablearrays 52
3.4 Readingfromadirectaccesscoderepresentation 55
3.5 Creatingdirectaccesscodesfromanarray 56
3.6 Findingoptimalpiecelengthsfordirectaccesscodes 58
3.7 Intersectionofinvertedlists 60
4.1 Encodinganddecodingbitblocksaspairs(c,o) 67
4.2 Answeringaccessoncompressedbitvectors 69
4.3 Answeringrankwithsparsesampling 74
4.4 Answeringrankwithdensesampling 75
4.5 Answeringrankoncompressedbitvectors 77
4.6 Answeringselectwithsparsesampling 80
4.7 Buildingtheselectstructures 82
4.8 Answeringselectandrankonverysparsebitvectors 85
4.9 Buildingthestructuresforverysparsebitvectors 86
4.10 Buildingaperfecthashfunction 94
5.1 Answeringπ−1withshortcuts 105
5.2 Buildingtheshortcutstructure 107
5.3 Answeringπk withthecycledecomposition 108
5.4 Answeringπ andπ−1oncompressiblepermutations 112
5.5 Buildingthecompressedpermutationrepresentation,part1 113
5.6 Buildingthecompressedpermutationrepresentation,part2 114
6.1 Answeringquerieswiththepermutation-basedstructure 125
6.2 Buildingthepermutation-basedrepresentationofasequence 126
6.3 Answeringaccessandrankwithwavelettrees 131
6.4 Answeringselectwithwavelettrees 134
6.5 Buildingawavelettree 135
6.6 Answeringaccessandrankwithwaveletmatrices 143
6.7 Answeringselectwithwaveletmatrices 144
6.8 Buildingawaveletmatrix 145
6.9 BuildingasuitableHuffmancodeforwaveletmatrices 149
6.10 BuildingawaveletmatrixfromHuffmancodes 150
6.11 Answeringquerieswithalphabetpartitioning 153
6.12 Buildingthealphabetpartitioningrepresentation 155
6.13 Answeringπ andπ−1usingsequences 156
6.14 Invertedlistintersectionusingasequencerepresentation 158
6.15 Non-positionalinvertedlistintersection 159
6.16 Solvingrangequantilequeriesonwavelettrees 161
7.1 ConvertingbetweenleafnumbersandpositionsofrmM-trees 171
7.2 BuildingtheCtableforthermM-trees 174
7.3 BuildingthermM-tree 175
7.4 Scanningablockforfwdsearch(i,d) 177
7.5 Computingfwdsearch(i,d) 178
7.6 Computingbwdsearch(i,d) 181
7.7 Scanningablockformin(i, j) 182
7.8 ComputingtheminimumexcessinB[i, j] 183
7.9 Computingmincount(i, j) 186
7.10 Computingminselect(i, j,t) 187
7.11 Computingrank (i)onB 189
10
7.12 Computingselect (j)onB 189
10
7.13 Findingthesmallestsegmentofatypecontainingaposition 202
7.14 Solvingrmq with2nparentheses 204
A
7.15 BuildingthestructureforsuccinctRMQs 205
8.1 ComputingtheordinaltreeoperationsusingLOUDS 216
8.2 Computinglca(u,v)ontheLOUDSrepresentation 217
8.3 BuildingtheLOUDSrepresentation 218
8.4 ComputingthecardinaltreeoperationsusingLOUDS 220
8.5 ComputingbasicbinarytreeoperationsusingLOUDS 221
8.6 BuildingtheBPrepresentationofanordinaltree 223
8.7 ComputingthesimpleBPoperationsonordinaltrees 225
8.8 ComputingthecomplexBPoperationsonordinaltrees 227
8.9 BuildingtheBPrepresentationofabinarytree 230
8.10 ComputingbasicbinarytreeoperationsusingBP 231
8.11 ComputingadvancedbinarytreeoperationsusingBP 234
8.12 BuildingtheDFUDSrepresentation 235
8.13 ComputingthesimpleDFUDSoperationsonordinaltrees 239
8.14 ComputingthecomplexDFUDSoperationsonordinaltrees 240
8.15 ComputingtheadditionalcardinaltreeoperationsonDFUDS 241
8.16 ComputingthelabeledtreeoperationsonLOUDSorDFUDS 244
8.17 EnumeratingthepathfromutovwithLOUDS 247