Data Structures and Algorithm Analysis - Computer Science at ...

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Sec. 5.3 Binary Tree Node Implementations 159/** Base class: Composite */public interface VarBinNode {public boolean isLeaf();public void traverse();}/** Leaf node: Composite */class VarLeafNode implements VarBinNode {private String operand;}// Operand valuepublic VarLeafNode(String val) { operand = val; }public boolean isLeaf() { return true; }public String value() { return operand; }public void traverse() {Visit.VisitLeafNode(operand);}/** Internal node: Composite */class VarIntlNode implements VarBinNode { // Internal nodeprivate VarBinNode left;// Left childprivate VarBinNode right;// Right childprivate Character operator;// Operator value}public VarIntlNode(Character op,VarBinNode l, VarBinNode r){ operator = op; left = l; right = r; }public boolean isLeaf() { return false; }public VarBinNode leftchild() { return left; }public VarBinNode rightchild() { return right; }public Character value() { return operator; }public void traverse() {Visit.VisitInternalNode(operator);if (left != null) left.traverse();if (right != null) right.traverse();}/** Preorder traversal */public static void traverse(VarBinNode rt) {if (rt != null) rt.traverse();}Figure 5.11 A second implementation for separate internal and leaf node representationsusing Java class inheritance and virtual functions using the compositedesign pattern. Here, the functionality of traverse is embedded into the nodesubclasses.
160 Chap. 5 Binary Trees5.3.2 Space RequirementsThis section presents techniques for calculating the amount of overhead required bya binary tree implementation. Recall that overhead is the amount of space necessaryto maintain the data structure. In other words, it is any space not used to storedata records. The amount of overhead depends on several factors including whichnodes store data values (all nodes, or just the leaves), whether the leaves store childpointers, and whether the tree is a full binary tree.In a simple pointer-based implementation for the binary tree such as that ofFigure 5.7, every node has two pointers to its children (even when the children arenull). This implementation requires total space amounting to n(2P + D) for atree of n nodes. Here, P stands for the amount of space required by a pointer, andD stands for the amount of space required by a data value. The total overhead spacewill be 2P n for the entire tree. Thus, the overhead fraction will be 2P/(2P + D).The actual value for this expression depends on the relative size of pointers versusdata fields. If we arbitrarily assume that P = D, then a full tree has about twothirds of its total space taken up in overhead. Worse yet, Theorem 5.2 tells us thatabout half of the pointers are “wasted” null values that serve only to indicate treestructure, but which do not provide access to new data.In Java, the most typical implementation is not to store any actual data in anode, but rather a reference to the data record. In this case, each node will typicallystore three pointers, all of which are overhead, resulting in an overhead fraction of3P/(3P + D).If only leaves store data values, then the fraction of total space devoted to overheaddepends on whether the tree is full. If the tree is not full, then conceivablythere might only be one leaf node at the end of a series of internal nodes. Thus,the overhead can be an arbitrarily high percentage for non-full binary trees. Theoverhead fraction drops as the tree becomes closer to full, being lowest when thetree is truly full. In this case, about one half of the nodes are internal.Great savings can be had by eliminating the pointers from leaf nodes in full binarytrees. Again assume the tree stores a reference to the data field. Because abouthalf of the nodes are leaves and half internal nodes, and because only internal nodesnow have child pointers, the overhead fraction in this case will be approximatelyn2n2 (2P ) P=(2P ) + Dn P + D .If P = D, the overhead drops to about one half of the total space. However, if onlyleaf nodes store useful information, the overhead fraction for this implementation isactually three quarters of the total space, because half of the “data” space is unused.If a full binary tree needs to store data only at the leaf nodes, a better implementationwould have the internal nodes store two pointers and no data field whilethe leaf nodes store only a reference to the data field. This implementation requires
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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. 2.8 Further Reading 45one inch
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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. 3.11 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 107curr...23 10 12(a
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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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214 Chap. 6 Non-Binary Treestwo chi
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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. 7.6 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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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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312 Chap. 9 Searchingand the total
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314 Chap. 9 SearchingThis method of
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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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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. 12.3 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. 13.2 Balanced Trees 437that is
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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. 14.4 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 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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