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Sec. 6.2 The Parent Pointer Implementation 203AB C DEAB C D E F G HIJ0 1 2 3 4 5 6 7 8 9FGHIJ(a)0 3 5 25A C FJDAB C D E F G HIJ0 1 2 3 4 5 6 7 8 9B H GIE(b)0 0 5 3 5 25AFJAB C D E F G HIJ0 1 2 3 4 5 6 7 8 9BCGID(c)HE50 0 5 3 5 25FJA B C D E F G H IJ0 1 2 3 4 5 6 7 8 9AGIDBCEH(d)Figure 6.7 An example of equivalence processing. (a) Initial configuration forthe ten nodes of the graph in Figure 6.6. The nodes are placed into ten independentequivalence classes. (b) The result of processing five edges: (A, B), (C, H), (G, F),(D, E), and (I, F). (c) The result of processing two more edges: (H, A) and (E, G).(d) The result of processing edge (H, E).
204 Chap. 6 Non-Binary Treesditional processing to be worth the effort, so we must settle for getting as close aspossible.A low-cost approach to reducing the height is to be smart about how two treesare joined together. One simple technique, called the weighted union rule, joinsthe tree with fewer nodes to the tree with more nodes by making the smaller tree’sroot point to the root of the bigger tree. This will limit the total depth of the tree toO(log n), because the depth of nodes only in the smaller tree will now increase byone, and the depth of the deepest node in the combined tree can only be at most onedeeper than the deepest node before the trees were combined. The total numberof nodes in the combined tree is therefore at least twice the number in the smallersubtree. Thus, the depth of any node can be increased at most log n times when nequivalences are processed.Example 6.3 When processing equivalence pair (I, F) in Figure 6.7(b),F is the root of a tree with two nodes while I is the root of a tree with onlyone node. Thus, I is set to point to F rather than the other way around.Figure 6.7(c) shows the result of processing two more equivalence pairs:(H, A) and (E, G). For the first pair, the root for H is C while the rootfor A is itself. Both trees contain two nodes, so it is an arbitrary decisionas to which node is set to be the root for the combined tree. In the caseof equivalence pair (E, G), the root of E is D while the root of G is F.Because F is the root of the larger tree, node D is set to point to F.Not all equivalences will combine two trees. If equivalence (F, G) is processedwhen the representation is in the state shown in Figure 6.7(c), no change will bemade because F is already the root for G.The weighted union rule helps to minimize the depth of the tree, but we can dobetter than this. Path compression is a method that tends to create extremely shallowtrees. Path compression takes place while finding the root for a given node X.Call this root R. Path compression resets the parent of every node on the path fromX to R to point directly to R. This can be implemented by first finding R. A secondpass is then made along the path from X to R, assigning the parent field of eachnode encountered to R. Alternatively, a recursive algorithm can be implemented asfollows. This version of FIND not only returns the root of the current node, butalso makes all ancestors of the current node point to the root.public Integer FIND(Integer curr) {if (array[curr] == null) return curr; // At rootarray[curr] = FIND(array[curr]);return array[curr];}
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Data Structures and AlgorithmAnalys
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ivContents2.6.1 Direct Proof 372.6.
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viContents6.1 General Tree Definiti
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viiiContents10.5.2 B-Tree Analysis
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xContents15.7 Optimal Sorting 50115
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PrefaceWe study data structures so
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PrefacexvA sophomore-level class wh
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Prefacexviithe form ofcalls to meth
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PrefacexixFor the second edition, I
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1Data Structures and AlgorithmsHow
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Sec. 1.1 A Philosophy of Data Struc
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Sec. 1.1 A Philosophy of Data Struc
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Sec. 1.2 Abstract Data Types and Da
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Sec. 1.2 Abstract Data Types and Da
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Sec. 1.3 Design Patterns 13that tra
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Sec. 1.3 Design Patterns 15will cal
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Sec. 1.4 Problems, Algorithms, and
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Sec. 1.5 Further Reading 19vides po
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Sec. 1.6 Exercises 21than 10,000 of
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2Mathematical PreliminariesThis cha
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Sec. 2.1 Sets and Relations 25A seq
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Sec. 2.2 Miscellaneous Notation 27E
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Sec. 2.3 Logarithms 29Unfortunately
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Sec. 2.4 Summations and Recurrences
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Sec. 2.4 Summations and Recurrences
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Sec. 2.5 Recursion 35(a)(b)Figure 2
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Sec. 2.6 Mathematical Proof Techniq
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Sec. 2.8 Further Reading 45one inch
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Sec. 2.9 Exercises 47(c) For nonzer
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Sec. 2.9 Exercises 492.16 Write a r
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Sec. 2.9 Exercises 512.38 How many
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3Algorithm AnalysisHow long will it
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Sec. 3.1 Introduction 55efficient.
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Sec. 3.1 Introduction 57n! 2 n2n 25
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Sec. 3.2 Best, Worst, and Average C
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Sec. 3.3 A Faster Computer, or a Fa
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Sec. 3.4 Asymptotic Analysis 633.4
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Sec. 3.4 Asymptotic Analysis 65If s
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Sec. 3.5 Calculating the Running Ti
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Sec. 3.10 Speeding Up Your Programs
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Sec. 3.11 Empirical Analysis 833.11
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Sec. 3.13 Exercises 853.13 Exercise
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Sec. 3.13 Exercises 87(f) sum = 0;f
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Sec. 3.14 Projects 893.24 Prove tha
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4Lists, Stacks, and QueuesIf your p
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Sec. 4.1 Lists 95list ADT. Interfac
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Sec. 4.1 Lists 97for (L.moveToStart
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Sec. 4.1 Lists 99public void moveTo
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Sec. 4.1 Lists 101/** Singly linked
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Sec. 4.1 Lists 103headcurrtailhead2
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Sec. 4.1 Lists 105/** Set curr at l
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Sec. 4.1 Lists 107curr...23 10 12(a
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Sec. 4.1 Lists 109/** Singly linked
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Sec. 4.1 Lists 111Array-based lists
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Sec. 4.1 Lists 113headcurrtail20 23
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Sec. 4.1 Lists 115/** Insert "it" a
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Sec. 4.2 Stacks 117The principle be
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Sec. 4.2 Stacks 119/** Array-based
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Sec. 4.2 Stacks 121top1top2Figure 4
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Sec. 4.2 Stacks 123you might want t
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Sec. 4.3 Queues 125/** Queue ADT */
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Sec. 4.3 Queues 127frontfront20 512
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Sec. 4.3 Queues 129/** Array-based
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Sec. 4.4 Dictionaries 1314.3.3 Comp
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Sec. 4.4 Dictionaries 133not get at
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Sec. 4.4 Dictionaries 135/** Contai
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Sec. 4.4 Dictionaries 137}/** @retu
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Sec. 4.6 Exercises 139(b) L2.append
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Sec. 4.7 Projects 1414.16 Re-implem
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Sec. 4.7 Projects 143‘a’ ‘b
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146 Chap. 5 Binary TreesABCDEFGHIFi
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148 Chap. 5 Binary TreesAny number
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150 Chap. 5 Binary Trees/** ADT for
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Sec. 7.9 Lower Bounds for Sorting 2
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Sec. 7.9 Lower Bounds for Sorting 2
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Sec. 7.10 Further Reading 257• Eq
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Sec. 7.11 Exercises 259algorithm to
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Sec. 7.12 Projects 2617.18 Which of
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Sec. 7.12 Projects 2637.6 It has be
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266 Chap. 8 File Processing and Ext
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302 Chap. 9 Searchingintroduced in
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304 Chap. 9 Searchingvalue in L gre
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306 Chap. 9 SearchingWe now show th
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308 Chap. 9 Searchingrecords by exp
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310 Chap. 9 SearchingThis is potent
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312 Chap. 9 Searchingand the total
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314 Chap. 9 SearchingThis method of
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316 Chap. 9 Searchingbirthday is si
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318 Chap. 9 Searching45674567319692
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320 Chap. 9 Searching01000953012330
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322 Chap. 9 Searching01HashTable100
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324 Chap. 9 Searching/** Insert rec
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326 Chap. 9 Searching09050090501100
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328 Chap. 9 SearchingExample 9.9 Co
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330 Chap. 9 Searching/** Dictionary
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332 Chap. 9 SearchingThe cost for a
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334 Chap. 9 Searchingthat is not in
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336 Chap. 9 SearchingA good introdu
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338 Chap. 9 Searching9.16 Assume th
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10IndexingMany large-scale computin
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Sec. 10.1 Linear Indexing 343Linear
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Sec. 10.2 ISAM 347In−memoryTable
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Sec. 10.3 Tree-based Indexing 349Fi
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Sec. 10.4 2-3 Trees 351/** 2-3 tree
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Sec. 10.5 B-Trees 355private TTNode
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Sec. 10.5 B-Trees 361331012 233348
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Sec. 10.6 Further Reading 365at mos
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Sec. 10.8 Projects 36710.7 Prove th
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PART IVAdvanced Data Structures369
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372 Chap. 11 Graphs04123147123(a)(b
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374 Chap. 11 Graphs0 1 2 3 40201 11
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376 Chap. 11 Graphstime. This is a
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378 Chap. 11 GraphsIt is reasonably
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380 Chap. 11 Graphs}/** @return an
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382 Chap. 11 Graphs}/** Set the wei
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384 Chap. 11 GraphsABABCCDDFFEE(a)(
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386 Chap. 11 Graphs/** Breadth firs
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388 Chap. 11 Graphs/** Recursive to
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390 Chap. 11 GraphsA103B2205D11C15E
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392 Chap. 11 Graphs/** Dijkstra’s
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394 Chap. 11 GraphsA7 5CB91 26ED 21
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396 Chap. 11 GraphsPrim’s algorit
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398 Chap. 11 GraphsInitialA B C D E
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400 Chap. 11 Graphs(a) IF an undire
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402 Chap. 11 Graphs11.8 Projects11.
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12Lists and Arrays RevisitedSimple
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Sec. 12.1 Multilists 407L1L2L3bcdL4
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Sec. 12.2 Matrix Representations 40
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Sec. 12.2 Matrix Representations 41
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Sec. 12.3 Memory Management 413/**
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Sec. 12.3 Memory Management 415Smal
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Sec. 12.3 Memory Management 417Tag
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Sec. 12.3 Memory Management 419tend
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Sec. 12.3 Memory Management 421is e
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Sec. 12.3 Memory Management 423AaBc
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Sec. 12.4 Further Reading 4251a3b42
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Sec. 12.6 Projects 42712.7 Write a
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13Advanced Tree StructuresThis chap
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Sec. 13.1 Tries 4310101010120001240
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Sec. 13.2 Balanced Trees 4353737244
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Sec. 13.2 Balanced Trees 437that is
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Sec. 13.2 Balanced Trees 439GPDSASP
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Sec. 13.3 Spatial Data Structures 4
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Sec. 13.4 Further Reading 45313.3.4
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Sec. 13.6 Projects 455(c) Show the
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Sec. 13.6 Projects 45713.11 Select
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14Analysis TechniquesOften it is ea
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Sec. 14.1 Summation Techniques 463w
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Sec. 14.2 Recurrence Relations 4671
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Sec. 14.2 Recurrence Relations 469n
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Sec. 14.2 Recurrence Relations 475T
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Sec. 14.3 Amortized Analysis 477(ch
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Sec. 14.4 Further Reading 479compar
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Sec. 14.5 Exercises 48114.14 For th
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Sec. 14.6 Projects 48314.24 Use mat
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486 Chap. 15 Lower Boundsthe concep
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488 Chap. 15 Lower Boundsat least o
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490 Chap. 15 Lower BoundsABCGDEFFig
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492 Chap. 15 Lower BoundsProof 1: T
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494 Chap. 15 Lower BoundsFigure 15.
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496 Chap. 15 Lower BoundsTo minimiz
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498 Chap. 15 Lower BoundsThis recur
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500 Chap. 15 Lower BoundsFigure 15.
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502 Chap. 15 Lower Boundsnot only t
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504 Chap. 15 Lower Bounds1234Figure
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506 Chap. 15 Lower Bounds(a) Assume
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16Patterns of AlgorithmsThis chapte
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Sec. 16.1 Dynamic Programming 513on
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Sec. 16.2 Randomized Algorithms 515
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Sec. 16.3 Numerical Algorithms 523
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Sec. 16.3 Numerical Algorithms 525t
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Sec. 16.3 Numerical Algorithms 527B
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Sec. 16.3 Numerical Algorithms 529o
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Sec. 16.3 Numerical Algorithms 531o
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Sec. 16.6 Projects 533value depth5
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536 Chap. 17 Limits to ComputationT
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540 Chap. 17 Limits to ComputationS
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542 Chap. 17 Limits to Computationa
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544 Chap. 17 Limits to Computation3
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548 Chap. 17 Limits to Computationp
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550 Chap. 17 Limits to ComputationE
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552 Chap. 17 Limits to Computation1
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554 Chap. 17 Limits to ComputationM
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556 Chap. 17 Limits to Computationw
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558 Chap. 17 Limits to Computationx
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560 Chap. 17 Limits to Computationt
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562 Chap. 17 Limits to Computationm
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564 Chap. 17 Limits to Computation(
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Bibliography[AG06] Ken Arnold and J
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BIBLIOGRAPHY 569[GI91] Zvi Galil an
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BIBLIOGRAPHY 571[SJH93] Clifford A.
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Index80/20 rule, 309, 333abstract d
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INDEX 575image space, 430key space,
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INDEX 577building, 172-177for memor
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INDEX 579octree, 451Ω notation, 65
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INDEX 581comparing algorithms, 224-
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