A Practical Introduction to Data Structures and Algorithm Analysis

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Sec. 7.4 Mergesort 24736 20 17 13 28 14 23 1520 36 13 17 14 28 15 23131720361415232813 14 15 17 20 23 28 36Figure 7.7 An illustration of Mergesort. The first row shows eight numbers thatare to be sorted. Mergesort will recursively subdivide the list into sublists of oneelement each, then recombine the sublists. The second row shows the four sublistsof size 2 created by the first merging pass. The third row shows the two sublistsof size 4 created by the next merging pass on the sublists of row 2. The last rowshows the final sorted list created by merging the two sublists of row 3.Implementing Mergesort presents a number of technical difficulties. The firstdecision is how to represent the lists. Mergesort lends itself well to sorting a singlylinked list because merging does not require random access to the list elements.Thus, Mergesort is the method of choice when the input is in the form of a linkedlist. Implementing merge for linked lists is straightforward, because we need onlyremove items from the front of the input lists and append items to the output list.Breaking the input list into two equal halves presents some difficulty. Ideally wewould just break the lists into front and back halves. However, even if we know thelength of the list in advance, it would still be necessary to traverse halfway downthe linked list to reach the beginning of the second half. A simpler method, whichdoes not rely on knowing the length of the list in advance, assigns elements of theinput list alternating between the two sublists. The first element is assigned to thefirst sublist, the second element to the second sublist, the third to first sublist, thefourth to the second sublist, and so on. This requires one complete pass throughthe input list to build the sublists.When the input to Mergesort is an array, splitting input into two subarrays iseasy if we know the array bounds. Merging is also easy if we merge the subarraysinto a second array. Note that this approach requires twice the amount of spaceas any of the sorting methods presented so far, which is a serious disadvantage forMergesort. It is possible to merge the subarrays without using a second array, butthis is extremely difficult to do efficiently and is not really practical. Merging thetwo subarrays into a second array, while simple to implement, presents another difficulty.The merge process ends with the sorted list in the auxiliary array. Considerhow the recursive nature of Mergesort breaks the original array into subarrays, asshown in Figure 7.7. Mergesort is recursively called until subarrays of size 1 havebeen created, requiring log n levels of recursion. These subarrays are merged into
248 Chap. 7 Internal Sortingsubarrays of size 2, which are in turn merged into subarrays of size 4, and so on.We need to avoid having each merge operation require a new array. With somedifficulty, an algorithm can be devised that alternates between two arrays. A muchsimpler approach is to copy the sorted sublists to the auxiliary array first, and thenmerge them back to the original array. Here is a complete implementation formergesort following this approach:static
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xviPrefacemake the examples as clea
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xviiiPrefaceOne of the most importa
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xxPrefaceOthers at Prentice Hall wh
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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 13Data Typ
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Sec. 1.3 Design Patterns 151.3.3 Co
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Sec. 1.4 Problems, Algorithms, and
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Sec. 1.5 Further Reading 194. It mu
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Sec. 1.6 Exercises 21If you want to
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Sec. 1.6 Exercises 231.12 Imagine t
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2Mathematical PreliminariesThis cha
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Sec. 2.1 Sets and Relations 27examp
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Sec. 2.2 Miscellaneous Notation 29x
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Sec. 2.3 Logarithms 31positive inte
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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 37factorial of n
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Sec. 2.6 Mathematical Proof Techniq
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Sec. 2.7 Estimating 47Example 2.17
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Sec. 2.8 Further Reading 49typical
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Sec. 2.9 Exercises 51(f) {〈2, 1
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Sec. 2.9 Exercises 532.17 Prove by
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Sec. 2.9 Exercises 552.38 When buyi
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3Algorithm AnalysisHow long will it
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Sec. 3.1 Introduction 59compiled wi
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Sec. 3.1 Introduction 61Example 3.3
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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 67comp
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Sec. 3.4 Asymptotic Analysis 69fast
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Sec. 3.4 Asymptotic Analysis 71Exam
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Sec. 3.4 Asymptotic Analysis 73Rule
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Sec. 3.5 Calculating the Running Ti
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Sec. 3.5 Calculating the Running Ti
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Sec. 3.6 Analyzing Problems 79binar
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Sec. 3.7 Common Misunderstandings 8
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Sec. 3.9 Space Bounds 83pixel. Assu
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Sec. 3.9 Space Bounds 85A classic e
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Sec. 3.10 Speeding Up Your Programs
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Sec. 3.11 Empirical Analysis 893.11
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Sec. 3.13 Exercises 913.3 Arrange t
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Sec. 3.13 Exercises 93(f) sum = 0;f
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Sec. 3.14 Projects 95a summation fo
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4Lists, Stacks, and QueuesIf your p
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Sec. 4.1 Lists 101The next step is
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Sec. 4.1 Lists 103mentations, asser
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Sec. 4.1 Lists 105/** Array-based l
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Sec. 4.1 Lists 107Insert 23:1312 20
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Sec. 4.1 Lists 109currtailheadFigur
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Sec. 4.1 Lists 111// Linked list im
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Sec. 4.1 Lists 113curr... 23 12 ...
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Sec. 4.1 Lists 115// Singly linked
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Sec. 4.1 Lists 117determined size.
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Sec. 4.1 Lists 119pointer to each e
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Sec. 4.1 Lists 121// Insert "it" at
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Sec. 4.1 Lists 123curr... 20curr23
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Sec. 4.2 Stacks 125/** Stack ADT */
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Sec. 4.2 Stacks 127// Linked stack
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Sec. 4.2 Stacks 129Currptr β 1Curr
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Sec. 4.2 Stacks 131Example 4.3 The
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Sec. 4.3 Queues 133/** Queue ADT */
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Sec. 4.3 Queues 135frontfront20 512
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Sec. 4.3 Queues 137// Array-based q
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Sec. 4.4 Dictionaries 139enqueue pl
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Sec. 4.4 Dictionaries 141Method cle
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Sec. 4.4 Dictionaries 143// Contain
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Sec. 4.4 Dictionaries 145/** Dictio
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Sec. 4.5 Further Reading 1474.5 Fur
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Sec. 4.6 Exercises 149(a) The data
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Sec. 4.7 Projects 1514.3 Use singly
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5Binary TreesThe list representatio
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Sec. 5.1 Definitions and Properties
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Sec. 5.1 Definitions and Properties
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Sec. 5.2 Binary Tree Traversals 159
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Sec. 5.2 Binary Tree Traversals 161
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Sec. 5.3 Binary Tree Node Implement
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Sec. 5.4 Binary Search Trees 171Thi
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Sec. 5.4 Binary Search Trees 173120
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Sec. 5.4 Binary Search Trees 175/**
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Sec. 5.4 Binary Search Trees 177min
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Sec. 5.4 Binary Search Trees 17937
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Sec. 5.5 Heaps and Priority Queues
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Sec. 5.6 Huffman Coding Trees 189Le
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Sec. 5.6 Huffman Coding Trees 191St
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Sec. 5.6 Huffman Coding Trees 193/*
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Sec. 5.6 Huffman Coding Trees 195//
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Page 268 and 269: Sec. 7.5 Quicksort 249static
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298 Chap. 8 File Processing and Ext
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318 Chap. 9 SearchingThis and the f
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320 Chap. 9 Searchingelement in L,
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322 Chap. 9 SearchingThis is equal
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324 Chap. 9 Searching9.2 Self-Organ
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326 Chap. 9 SearchingWhen a frequen
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328 Chap. 9 Searchingand the total
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330 Chap. 9 SearchingThe set differ
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332 Chap. 9 Searchingprobability th
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334 Chap. 9 SearchingExample 9.6 A
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336 Chap. 9 Searchingand the next f
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338 Chap. 9 Searching01HashTable100
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340 Chap. 9 Searching/** Insert rec
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344 Chap. 9 SearchingExample 9.9 Co
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346 Chap. 9 Searchingproject at the
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348 Chap. 9 SearchingIf N and M are
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350 Chap. 9 SearchingDepending on t
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352 Chap. 9 SearchingBentley et al.
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354 Chap. 9 Searching(c) How many s
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356 Chap. 9 Searching9.5 Implement
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358 Chap. 10 Indexingfile is create
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360 Chap. 10 Indexing1 2003 5894 10
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362 Chap. 10 IndexingSecondaryKeyJo
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364 Chap. 10 Indexingkey value for
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366 Chap. 10 IndexingFigure 10.7 Br
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368 Chap. 10 Indexing/** 2-3 tree n
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370 Chap. 10 Indexing18331223 30 48
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372 Chap. 10 Indexingprivate TTNode
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382 Chap. 10 IndexingAs mentioned e
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PART IVAdvanced Data Structures387
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390 Chap. 11 Graphs04123147123(a)(b
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392 Chap. 11 Graphs0 1 2 3 40201 11
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394 Chap. 11 GraphsThe adjacency ma
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396 Chap. 11 Graphsedge list. Funct
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398 Chap. 11 Graphspublic int weigh
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400 Chap. 11 Graphs}// Store edge w
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402 Chap. 11 GraphsABABCCDDFFEE(a)(
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404 Chap. 11 Graphs// Breadth first
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406 Chap. 11 GraphsAInitial call to
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408 Chap. 11 Graphsstatic void tops
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410 Chap. 11 Graphs// Compute short
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412 Chap. 11 Graphs// Dijkstra’s
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414 Chap. 11 Graphs// Compute a min
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416 Chap. 11 GraphsMarkedVertices v
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418 Chap. 11 GraphsInitialA B C D E
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420 Chap. 11 Graphs2 511032041032 6
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422 Chap. 11 Graphstriangular matri
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424 Chap. 12 Lists and Arrays Revis
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PART VTheory of Algorithms479
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482 Chap. 14 Analysis Techniquesthi
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484 Chap. 14 Analysis TechniquesExa
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486 Chap. 14 Analysis Techniquesthe
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488 Chap. 14 Analysis Techniquesof
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490 Chap. 14 Analysis TechniquesHow
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492 Chap. 14 Analysis Techniques= 2
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494 Chap. 14 Analysis TechniquesNot
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496 Chap. 14 Analysis Techniquesfro
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498 Chap. 14 Analysis Techniquesbit
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500 Chap. 14 Analysis Techniquesfro
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502 Chap. 14 Analysis Techniques14.
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504 Chap. 14 Analysis Techniques}G.
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15Lower BoundsHow do I know if I ha
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Sec. 15.1 Introduction to Lower Bou
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Sec. 15.2 Lower Bounds on Searching
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Sec. 15.3 Finding the Maximum Value
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Sec. 15.4 Adversarial Lower Bounds
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Sec. 15.4 Adversarial Lower Bounds
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Sec. 15.5 State Space Lower Bounds
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Sec. 15.5 State Space Lower Bounds
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Sec. 15.6 Finding the ith Best Elem
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Sec. 15.7 Optimal Sorting 525search
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Sec. 15.8 Further Reading 527Merge
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Sec. 15.9 Exercises 52915.10 Show t
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16Patterns of AlgorithmsThis chapte
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Sec. 16.1 Dynamic Programming 535ar
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Sec. 16.2 Randomized Algorithms 537
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Sec. 16.3 Numerical Algorithms 545e
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Sec. 16.3 Numerical Algorithms 547d
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Sec. 16.3 Numerical Algorithms 549W
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Sec. 16.3 Numerical Algorithms 5511
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Sec. 16.3 Numerical Algorithms 555b
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Sec. 16.6 Projects 55716.8 What is
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560 Chap. 17 Limits to Computations
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562 Chap. 17 Limits to Computationf
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564 Chap. 17 Limits to ComputationP
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592 Chap. 17 Limits to Computationt
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594 BIBLIOGRAPHY[BG00] Sara Baase a
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596 BIBLIOGRAPHY[LLKS85] E.L. Lawle
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598 BIBLIOGRAPHY[WMB99][Zei07]I.H.
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600 INDEXbest fit, see memory manag
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602 INDEXrelation, 27, 50estimating
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604 INDEXinteger representation, 5,
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606 INDEXpath compression, 215-216,
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608 INDEXterminology, 236-237spatia
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A Practical Introduction to Data Structures and Algorithm Analysis

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