A Practical Introduction to Data Structures and Algorithm Analysis

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Sec. 13.4 Further Reading 473Spatial data structures can also be used to store line object, rectangle object,or objects of arbitrary shape (such as polygons in two dimensions or polyhedra inthree dimensions). A simple, yet effective, data structure for storing rectangles orarbitrary polygonal shapes can be derived from the PR quadtree. Pick a thresholdvalue c, and subdivide any region into four quadrants if it contains more than cobjects. A special case must be dealt with when more than c object intersect.Some of the most interesting developments in spatial data structures have todo with adapting them for disk-based applications. However, all such disk-basedimplementations boil down to storing the spatial data structure within some varianton either B-trees or hashing.13.4 Further ReadingPATRICIA tries and other trie implementations are discussed in Information Retrieval:Data Structures & Algorithms, Frakes and Baeza-Yates, eds. [FBY92].See Knuth [Knu97] for a discussion of the AVL tree. For further reading onsplay trees, see “Self-adjusting Binary Search” by Sleator and Tarjan [ST85].The world of spatial data structures is rich and rapidly evolving. For a goodintroduction, see Foundations of Multidimensional and Metric Data Structures byHanan Samet [Sam06]. This is also the best reference for more information onthe PR quadtree. The k-d tree was invented by John Louis Bentley. For furtherinformation on the k-d tree, in addition to [Sam06], see [Ben75]. For informationon using a quadtree to store arbitrary polygonal objects, see [SH92].For a discussion on the relative space requirements for two-way versus multiwaybranching, see “A Generalized Comparison of Quadtree and Bintree StorageRequirements” by Shaffer, Juvvadi, and Heath [SJH93].Closely related to spatial data structures are data structures for storing multidimensionaldata (which might not necessarily be spatial in nature). A populardata structure for storing such data is the R-tree, which was originally proposed byGuttman [Gut84].13.5 Exercises13.1 Show the binary trie (as illustrated by Figure 13.1) for the following collectionof values: 42, 12, 100, 10, 50, 31, 7, 11, 99.13.2 Show the PAT trie (as illustrated by Figure 13.3) for the following collectionof values: 42, 12, 100, 10, 50, 31, 7, 11, 99.13.3 Write the insertion routine for a binary trie as shown in Figure 13.1.13.4 Write the deletion routine for a binary trie as shown in Figure 13.1.
474 Chap. 13 Advanced Tree Structures13.5 (a) Show the result (including appropriate rotations) of inserting the value39 into the AVL tree on the left in Figure 13.4.(b) Show the result (including appropriate rotations) of inserting the value300 into the AVL tree on the left in Figure 13.4.(c) Show the result (including appropriate rotations) of inserting the value50 into the AVL tree on the left in Figure 13.4.(d) Show the result (including appropriate rotations) of inserting the value1 into the AVL tree on the left in Figure 13.4.13.6 Show the splay tree that results from searching for value 75 in the splay treeof Figure 13.10(d).13.7 Show the splay tree that results from searching for value 18 in the splay treeof Figure 13.10(d).13.8 Some applications do not permit storing two records with duplicate key values.In such a case, an attempt to insert a duplicate-keyed record into a treestructure such as a splay tree should result in a failure on insert. What isthe appropriate action to take in a splay tree implementation when the insertroutine is called with a duplicate-keyed record?13.9 Show the result of deleting point A from the k-d tree of Figure 13.11.13.10 (a) Show the result of building a k-d tree from the following points (insertedin the order given). A (20, 20), B (10, 30), C (25, 50), D (35,25), E (30, 45), F (30, 35), G (55, 40), H (45, 35), I (50, 30).(b) Show the result of deleting point A from the tree you built in part (a).13.11 (a) Show the result of deleting F from the PR quadtree of Figure 13.16.(b) Show the result of deleting records E and F from the PR quadtree ofFigure 13.16.13.12 (a) Show the result of building a PR quadtree from the following points(inserted in the order given). Assume the tree is representing a space of64 by 64 units. A (20, 20), B (10, 30), C (25, 50), D (35, 25), E (30,45), F (30, 35), G (45, 25), H (45, 30), I (50, 30).(b) Show the result of deleting point C from the tree you built in part (a).(c) Show the result of deleting point F from the resulting tree in part (b).13.13 On average, how many leaf nodes of a PR quadtree will typically be empty?Explain why.13.14 When performing a region search on a PR quadtree, we need only searchthose subtrees of an internal node whose corresponding square falls withinthe query circle. This is most easily computed by comparing the x and yranges of the query circle against the x and y ranges of the square correspondingto the subtree. However, as illustrated by Figure 13.13, the x andy ranges might overlap without the circle actually intersecting the square.Write a function that accurately determines if a circle and a square intersect.
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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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Sec. 5.6 Huffman Coding Trees 197A
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Sec. 5.8 Exercises 199Many techniqu
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Sec. 5.8 Exercises 2015.19 Write a
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Sec. 5.9 Projects 203(d) The record
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6Non-Binary TreesMany organizations
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Sec. 6.1 General Tree Definitions a
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Sec. 6.2 The Parent Pointer Impleme
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Sec. 6.3 General Tree Implementatio
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Sec. 6.3 General Tree Implementatio
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Sec. 6.4 K-ary Trees 221RRABABCDEFC
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Sec. 6.5 Sequential Tree Implementa
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Sec. 6.5 Sequential Tree Implementa
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Sec. 6.7 Exercises 2276.2 Write an
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Sec. 6.7 Exercises 229CAFBEHGDIFigu
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Sec. 6.8 Projects 231proved perform
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7Internal SortingWe sort many thing
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Sec. 7.2 Three Θ(n 2 ) Sorting Alg
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Sec. 7.3 Shellsort 24559 20 17 13 2
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Sec. 7.4 Mergesort 24736 20 17 13 2
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Sec. 7.5 Quicksort 249static
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Sec. 7.5 Quicksort 251static
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Sec. 7.5 Quicksort 253InitialPass 1
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Sec. 7.5 Quicksort 255The initial c
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Sec. 7.6 Heapsort 257and fast. The
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Sec. 7.7 Binsort and Radix Sort 259
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Sec. 7.8 An Empirical Comparison of
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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 271• A
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Sec. 7.11 Exercises 2737.7 Recall t
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Sec. 7.12 Projects 2757.16 (a) Devi
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Sec. 7.12 Projects 277classmates? W
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280 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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374 Chap. 10 Indexing24152033 45 48
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382 Chap. 10 IndexingAs mentioned e
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384 Chap. 10 Indexing(b) Show the i
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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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Page 526 and 527: 15Lower BoundsHow do I know if I ha
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Page 532 and 533: Sec. 15.3 Finding the Maximum Value
Page 534 and 535: Sec. 15.4 Adversarial Lower Bounds
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Page 538 and 539: 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 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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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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