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Sec. 13.4 Further Reading 45313.3.4 Other Spatial Data StructuresThis section has barely scratched the surface of the field of spatial data structures.Dozens of distinct spatial data structures have been invented, many with variationsand alternate implementations. Spatial data structures exist for storing many formsof spatial data other than points. The most important distinctions between are thetree structure (binary or not, regular decompositions or not) and the decompositionrule used to decide when the data contained within a region is so complex that theregion must be subdivided.One such spatial data structure is the Region Quadtree for storing images wherethe pixel values tend to be blocky, such as a map of the countries of the world.The region quadtree uses a four-way regular decomposition scheme similar to thePR quadtree. The decomposition rule is simply to divide any node containing pixelsof more than one color or value.Spatial 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 popular
454 Chap. 13 Advanced Tree Structuresdata 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 13.1.13.5 (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).
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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. 1.3 Design Patterns 13that tra
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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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2Mathematical PreliminariesThis cha
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Sec. 2.1 Sets and Relations 25A seq
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Sec. 2.3 Logarithms 29Unfortunately
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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. 3.3 A Faster Computer, or a Fa
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Sec. 3.4 Asymptotic Analysis 633.4
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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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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 105/** Set curr at l
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Sec. 4.1 Lists 115/** Insert "it" a
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Sec. 4.2 Stacks 121top1top2Figure 4
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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 135/** Contai
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Sec. 4.4 Dictionaries 137}/** @retu
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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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152 Chap. 5 Binary Treestends to be
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154 Chap. 5 Binary Trees5.3 Binary
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156 Chap. 5 Binary TreesABCDEFGHIFi
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158 Chap. 5 Binary Treesto its call
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160 Chap. 5 Binary Trees5.3.2 Space
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162 Chap. 5 Binary Trees012345 678
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164 Chap. 5 Binary Trees12037422442
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166 Chap. 5 Binary Trees37244273240
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168 Chap. 5 Binary Treessubroot105
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170 Chap. 5 Binary Treesin the case
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172 Chap. 5 Binary Treessynonymous
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174 Chap. 5 Binary Trees/** Heapify
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176 Chap. 5 Binary TreesRH1H2Figure
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178 Chap. 5 Binary Trees5.6 Huffman
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180 Chap. 5 Binary TreesLetter C D
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182 Chap. 5 Binary Trees/** Huffman
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184 Chap. 5 Binary Trees/** Build a
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186 Chap. 5 Binary Treesa reverse p
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188 Chap. 5 Binary Trees18 bits, mo
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190 Chap. 5 Binary Trees5.12 Write
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192 Chap. 5 Binary Trees(c) What is
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194 Chap. 5 Binary Trees5.7 Complet
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196 Chap. 6 Non-Binary TreesRootRAn
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198 Chap. 6 Non-Binary TreesRABCD E
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200 Chap. 6 Non-Binary Trees/** Gen
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202 Chap. 6 Non-Binary Treesspond t
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204 Chap. 6 Non-Binary Treesditiona
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206 Chap. 6 Non-Binary Treeslence o
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208 Chap. 6 Non-Binary TreesR’RA
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210 Chap. 6 Non-Binary TreesRRABABC
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212 Chap. 6 Non-Binary Trees(a)(b)F
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216 Chap. 6 Non-Binary Treeschild v
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218 Chap. 6 Non-Binary TreesCAFBEHG
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7Internal SortingWe sort many thing
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Sec. 7.3 Shellsort 231Insertion Bub
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Sec. 7.4 Mergesort 233static
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Sec. 7.5 Quicksort 237Quicksort is
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Sec. 7.5 Quicksort 239static
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Sec. 7.5 Quicksort 241is still unli
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Sec. 7.6 Heapsort 243subarray, only
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Sec. 7.7 Binsort and Radix Sort 245
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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. 7.10 Further Reading 257• Eq
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Sec. 7.11 Exercises 259algorithm to
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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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322 Chap. 9 Searching01HashTable100
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324 Chap. 9 Searching/** Insert rec
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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. 10.4 2-3 Trees 351/** 2-3 tree
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Sec. 10.5 B-Trees 355private TTNode
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Sec. 10.6 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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506 Chap. 15 Lower Bounds(a) Assume
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16Patterns of AlgorithmsThis chapte
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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. 16.6 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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