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Sec. 13.3 Spatial Data Structures 447CADBEFFigure 13.14 Searching in the k-d treeof Figure 13.11. (a) The k-d tree decompositionfor a 128 × 128-unit region containing seven data points. (b) The k-d treefor the region of (a).private void rshelp(KDNode rt, int[] point,int radius, int lev) {if (rt == null) return;int[] rtkey = rt.key();if (InCircle(point, radius, rtkey))System.out.println(rt.element());if (rtkey[lev] > (point[lev] - radius))rshelp(rt.left(), point, radius, (lev+1)%D);if (rtkey[lev] < (point[lev] + radius))rshelp(rt.right(), point, radius, (lev+1)%D);}Figure 13.15 The k-d tree region search method.Figure 13.15 shows an implementation for the region search method. Whena node is visited, function InCircle is used to check the Euclidean distancebetween the node’s record and the query point. It is not enough to simply checkthat the differences between the x- and y-coordinates are each less than the querydistances because the the record could still be outside the search circle, as illustratedby Figure 13.13.13.3.2 The PR quadtreeIn the Point-Region Quadtree (hereafter referred to as the PR quadtree) each nodeeither has exactly four children or is a leaf. That is, the PR quadtree is a full fourwaybranching (4-ary) tree in shape. The PR quadtree represents a collection ofdata points in two dimensions by decomposing the region containing the data pointsinto four equal quadrants, subquadrants, and so on, until no leaf node contains morethan a single point. In other words, if a region contains zero or one data points, then
448 Chap. 13 Advanced Tree Structures00127BACDnwne swseAB(40,45)(15,70)C D(70, 10) (69,50)E F (b)127(a)EF(55,80)(80, 90)Figure 13.16 Example of a PR quadtree. (a) A map of data points. We definethe region to be square with origin at the upper-left-hand corner and sides oflength 128. (b) The PR quadtree for the points in (a). (a) also shows the blockdecomposition imposed by the PR quadtree for this region.it is represented by a PR quadtree consisting of a single leaf node. If the region containsmore than a single data point, then the region is split into four equal quadrants.The corresponding PR quadtree then contains an internal node and four subtrees,each subtree representing a single quadrant of the region, which might in turn besplit into subquadrants. Each internal node of a PR quadtree represents a singlesplit of the two-dimensional region. The four quadrants of the region (or equivalently,the corresponding subtrees) are designated (in order) NW, NE, SW, and SE.Each quadrant containing more than a single point would in turn be recursively dividedinto subquadrants until each leaf of the corresponding PR quadtree containsat most one point.For example, consider the region of Figure 13.16(a) and the correspondingPR quadtree in Figure 13.16(b). The decomposition process demands a fixed keyrange. In this example, the region is assumed to be of size 128 × 128. Note that theinternal nodes of the PR quadtree are used solely to indicate decomposition of theregion; internal nodes do not store data records. Because the decomposition linesare predetermined (i.e, key-space decomposition is used), the PR quadtree is a trie.Search for a record matching point Q in the PR quadtree is straightforward.Beginning at the root, we continuously branch to the quadrant that contains Q untilour search reaches a leaf node. If the root is a leaf, then just check to see if thenode’s data record matches point Q. If the root is an internal node, proceed tothe child that contains the search coordinate. For example, the NW quadrant ofFigure 13.16 contains points whose x and y values each fall in the range 0 to 63.The NE quadrant contains points whose x value falls in the range 64 to 127, and
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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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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. 2.3 Logarithms 29Unfortunately
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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. 2.8 Further Reading 45one inch
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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. 3.10 Speeding Up Your Programs
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Sec. 3.13 Exercises 853.13 Exercise
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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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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. 7.4 Mergesort 235static
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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. 7.10 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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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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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 511Dy
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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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