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for a handy companion book in a curriculum that allows for either a Java or C++ track in the introductory courses. • M.T. Goodrich and R. Tamassia, Algorithm Design: Foundations, Analysis, and Internet Examples, John Wiley & Sons, Inc., 2002. This is a textbook for a more advanced algorithms and data structures course, such as CS210 (T/W/C/S versions) in the IEEE/ACM 2001 curriculum. Use as a Textbook The design and analysis of efficient data structures has long been recognized as a vital subject in computing, for the study of data structures is part of the core of every collegiate computer science and computer engineering major program we are familiar with. Typically, the introductory courses are presented as a two- or threecourse sequence. Elementary data structures are often briefly introduced in the first programming or introduction to computer science course and this is followed by a more in-depth introduction to data structures in the following course(s). Furthermore, this course sequence is typically followed at a later point in the curriculum by a more in-depth study of data structures and algorithms. We feel that the central role of data structure design and analysis in the curriculum is fully justified, given the importance of efficient data structures in most software systems, including the Web, operating systems, databases, compilers, and scientific simulation systems. With the emergence of the object-oriented paradigm as the framework of choice for building robust and reusable software, we have tried to take a consistent objectoriented viewpoint throughout this text. One of the main ideas of the objectoriented approach is that data should be presented as being encapsulated with the methods that access and modify them. That is, rather than simply viewing data as a collection of bytes and addresses, we think of data as instances of an abstract data type (ADT) that include a repertory of methods for performing operations on the data. Likewise, object-oriented solutions are often organized utilizing common design patterns, which facilitate software reuse and robustness. Thus, we present each data structure using ADTs and their respective implementations and we introduce important design patterns as means to organize those implementations into classes, methods, and objects. For each ADT presented in this book, we provide an associated Java interface. Also, concrete data structures realizing the ADTs are provided as Java classes implementing the interfaces above. We also give Java implementations of fundamental algorithms (such as sorting and graph traversals) and of sample applications of data structures (such as HTML tag matching and a photo album). Due to space limitations, we sometimes show only code fragments in the book and make additional source code available on the companion Web site, http://java.datastructures.net. 4
The Java code implementing fundamental data structures in this book is organized in a single Java package, net.datastructures. This package forms a coherent library of data structures and algorithms in Java specifically designed for educational purposes in a way that is complementary with the Java Collections Framework. Web Added-Value Education This book is accompanied by an extensive Web site: http://java.datastructures.net. Students are encouraged to use this site along with the book, to help with exercises and increase understanding of the subject. Instructors are likewise welcome to use the site to help plan, organize, and present their course materials. For the Student for all readers, and specifically for students, we include: • All the Java source code presented in this book. • The student version of the net.datastructures package. • Slide handouts (four-per-page) in PDF format. • A database of hints to all exercises, indexed by problem number. • Java animations and interactive applets for data structures and algorithms. • Hyperlinks to other data structures and algorithms resources. We feel that the Java animations and interactive applets should be of particular interest, since they allow readers to interactively "play" with different data structures, which leads to better understanding of the different ADTs. In addition, the hints should be of considerable use to anyone needing a little help getting started on certain exercises. For the Instructor For instructors using this book, we include the following additional teaching aids: • Solutions to over two hundred of the book's exercises. • A keyword-searchable database of additional exercises. • The complete net.datastructures package. 5
Page 1 and 2: Data Structures and Algorithms in J
Page 3: To Isabel -Roberto Tamassia Preface
Page 7 and 8: PL6. Object-Oriented Programming Ch
Page 9 and 10: 14. Memory A. Useful Mathematical F
Page 11 and 12: Sun, Nikos Triandopoulos, Luca Vism
Page 13 and 14: Getting Started: Classes, Types, an
Page 15 and 16: 39 1.7 An Example Program..........
Page 17 and 18: The main "actors" in a Java program
Page 19 and 20: else interface switch boolean exten
Page 21 and 22: Class modifiers are optional keywor
Page 23 and 24: Comments Note the use of comments i
Page 25 and 26: Calling the new operator on a class
Page 27 and 28: n.longValue( ) float Float n = new
Page 29 and 30: The Dot Operator Every object refer
Page 31 and 32: object (assuming we are allowed acc
Page 33 and 34: Note the uses of instance variables
Page 35 and 36: 1.2 Methods Methods in Java are con
Page 37 and 38: Abstract methods may only appear wi
Page 39 and 40: is intended to be used to initializ
Page 41 and 42: } int i = 512; double e = 2.71828;
Page 43 and 44: % the modulo operator This last ope
Page 45 and 46: zeros >>> shift bits right, filling
Page 47 and 48: 3 add./subt. + − 4 shift > >>> 5
Page 49 and 50: double d1 = 3.2; double d2 = 3.9999
Page 51 and 52: String s = " " + 4.5; // correct, b
Page 53 and 54: System.out.println("This is tough."
Page 55 and 56:
Defining a For Loop Here is the syn
Page 57 and 58:
that input and inform the user that
Page 59 and 60:
if (a[i][j] == 0) { foundFlag = tru
Page 61 and 62:
the array, we will sometimes refer
Page 63 and 64:
The cells of this new array, "a," a
Page 65 and 66:
ways to reference such an object. W
Page 67 and 68:
float height = s.nextFloat(); Syste
Page 69 and 70:
We show the CreditCard class in Cod
Page 71 and 72:
Code Fragment 1.7: Output from the
Page 73 and 74:
a text editor class may wish to def
Page 75 and 76:
The process of writing a Java progr
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low-level implementation details. A
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variable declaration, or method def
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• @exception exception-name descr
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A careful testing plan is an essent
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Modify the CreditCard class from Co
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A common punishment for school chil
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Object-Oriented Design Principles .
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java.datastructures.net 2.1 Goals,
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Abstraction The notion of abstracti
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2.1.3 Design Patterns One of the ad
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Example 2.1: Consider a class S tha
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S's a() method when asked for o.a()
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Sometimes, in a Java class, it is c
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103
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Code Fragment 2.3: Class for arithm
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A Fibonacci Progression Class As a
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Figure 2.5 : Inheritance diagram fo
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2.3 Exceptions Exceptions are unexp
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goShopping(); // I don't have to tr
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try problem… // This code might h
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instance, wish to identify some of
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The class BoxedItem shows another f
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comparability feature to a class (i
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• T and S are class types and S i
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We show in Code Fragment 2.13 a cla
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Starting with 5.0, Java includes a
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public void insert (P person, P oth
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• Class Goat extends Object and a
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((Place) usa).printMe(); } } class
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C-2.5 Write a program that consists
Page 137 and 138:
Chapter 3 Arrays, Linked Lists, and
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3.4 Circularly Linked Lists and Lin
Page 141 and 142:
A Class for High Scores Suppose we
Page 143 and 144:
Note that we include a method, toSt
Page 145 and 146:
Once we have identified the place i
Page 147 and 148:
Object Removal Suppose some hot sho
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These methods for adding and removi
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This description is much closer to
Page 153 and 154:
there is no swap needed, and return
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num = [10,12,19,28,33,38,41,46,48,5
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In Code Fragment 3.9, we give a sim
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Many computer games, be they strate
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We give a complete Java class for m
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Code Fragment 3.11: A simple, compl
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Like an array, a singly linked list
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Code Fragment 3.14: Inserting a new
Page 169 and 170:
The reverse operation of inserting
Page 171 and 172:
Header and Trailer Sentinels To sim
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Likewise, we can easily perform an
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Figure 3.17: Adding a new node afte
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Code Fragment 3.23: Java class DLis
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We make the following observations
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advance(): Advance the cursor to th
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Some Observations about the CircleL
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Some Sample Output 185
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We show in Code Fragment 3.27 thein
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The factorial function can be defin
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Each multiple of 1 inch also has a
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The trace for drawTicks is more com
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The simplest form of recursion is l
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nonrecursive part of each call. Mor
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3.5.2 Binary Recursion When an algo
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Unfortunately, in spite of the Fibo
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To solve such a puzzle, we need to
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3.6 Exercises For source code and h
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C-3.2 Let A be an array of size n
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Write a recursive Java program that
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Chapter Notes The fundamental data
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Primitive Operations . . . . . . .
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4.1.2 The Logarithm function One of
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4.1.5 The Quadratic function Anothe
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which assigns to an input value n t
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For example, we have the following:
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primarily as slopes. Even so, the e
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and y-coordinate equal to the runni
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Counting Primitive Operations Speci
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Simplifying the Analysis Further In
Page 231 and 232:
Proposition 4.7: The Algorithmarray
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is O(4n 2 + 6n log n), although thi
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256 16 4 16 64 256 4,096 65,536 32
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Running Maximum Problem Size (n) Ti
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should the constant factors they "h
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• There are two nested for loops,
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This definition leads immediately t
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Besides showing a use of the contra
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We may sometimes feel overwhelmed b
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The number of operations executed b
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R-4.22 Show that if d(n) is O(f(n))
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R-4.29 Show that 2 n+1 is O(2 n ).
Page 255 and 256:
Describe a method for finding both
Page 257 and 258:
Describe, in pseudo-code a method f
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Stacks.............................
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down on the stack to become the new
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7 (5) top() 5 (5) pop() 5 () pop()
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In addition, we must define excepti
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5.1.2 A Simple Array-Based Stack Im
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we indicate that the stack no longe
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271
Page 273 and 274:
273
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The array implementation of a stack
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of the list and return its element.
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5.1.5 Matching Parentheses and HTML
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Matching Tags in an HTML Document A
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Code Fragment 5.12: Java program fo
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Another fundamental data structure
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true ( ) enqueue(9) - (9) enqueue(7
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5.2.2 A Simple Array-Based Queue Im
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Using the Modulo Operator to Implem
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which gives the correct result both
Page 295 and 296:
Each of the methods of the singly l
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potato is equivalent to dequeuing a
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- (3) addFirst(5) - (5,3) removeFir
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Thus, a doubly linked list can be u
Page 303 and 304:
Code Fragment 5.18: Class NodeDeque
Page 305 and 306:
Page 307 and 308:
Suppose Alice has picked three dist
Page 309 and 310:
Implement a program that can input
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223 6.1.3 A Simple Array-Based Impl
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Sequences....................... 25
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- (7) add(0,4) - (4,7) get(1) 7 (4,
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get(size() −1) addFirst(e) add(0,
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have to be shifted forward. A simil
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Moreover, the java.util.ArrayList c
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An Amortized Analysis of Extendable
Page 325 and 326:
Proposition 6.2: Let S be an array
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linked list without revealing this
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The node list ADT allows us to view
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(8,3,5) set(p 3 ,7) 3 (8,7,5) addAf
Page 333 and 334:
Deque Method Realization with Node-
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new node v to hold the element e, l
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In conclusion, using a doubly linke
Page 339 and 340:
Code Fragment 6.11: Portions of the
Page 341 and 342:
An iterator is a software design pa
Page 343 and 344:
Since looping through the elements
Page 345 and 346:
Code Fragment 6.15: The iterator()
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the last element. That is, it uses
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isEmpty() isEmpty() O(1) time get(i
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deletion is at cursor; A is O(n),L
Page 353 and 354:
where n is the number of elements i
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Efficiency Trade-Offs with an Array
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357
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The previous implementation of a fa
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decreasing order and then pops up a
Page 363 and 364:
accesses several Web pages and then
Page 365 and 366:
Page 367 and 368:
R-6.16 Describe how to create an it
Page 369 and 370:
Think about how to extend the circu
Page 371 and 372:
C-6.24 Isabel has an interesting wa
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Chapter 7 Trees Contents 7.1 Genera
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7.3.3 Properties of Binary Trees ..
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Electronics R'Us. The children of t
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Edges and Paths in Trees An edge of
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There are also a number of generic
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Figure 7.5: The linked structure fo
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Code Fragment 7.3: Method depth wri
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Code Fragment 7.6: Algorithm height
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Example 7.6: The preorder traversal
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Code Fragment 7.10: Algorithm paren
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The postorder traversal method is u
Page 395 and 396:
Although the preorder and postorder
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A Recursive Binary Tree Definition
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same depth d as the level dof T. In
Page 401 and 402:
Figure 7.13: Operation that removes
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We use a Java interface BTPosition
Page 405 and 406:
addition to the Binary Tree interfa
Page 407 and 408:
Page 409 and 410:
Page 411 and 412:
Binary Tree interface. (Continues i
Page 413 and 414:
• Methods size() and isEmpty() us
Page 415 and 416:
The level numbering function p sugg
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Time size, isEmpty O(1) iterator, p
Page 419 and 420:
As is the case for general trees, t
Page 421 and 422:
The inorder traversal of a binary t
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The inorder traversal can also be a
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start an Euler tour by initializing
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• Calls auxiliary method visitRig
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ExpressionTerm at each node. Class
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Code Fragment 7.31: Class EvaluateE
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Let T be an n-node binary tree that
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c. Let T be a proper binary tree wi
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Creativity C-7.1 For each node v in
Page 439 and 440:
C-7.10 Two ordered trees T ′ and
Page 441 and 442:
postorderDraw is similar to preorde
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Describe a nonrecursive method for
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P-7.8 Write a program that can play
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Chapter 8 Priority Queues Contents
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The Heap Data Structure............
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As in the examples above from an ai
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package includes a similar Entry in
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Code Fragment 8.3: Class representi
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insert(5,A) e 1 [=(5,A)] {(5,A)} in
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size() size() isEmpty() isEmpty() i
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at the time the method is executed.
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Code Fragment 8.7: Java class Defau
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465
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Insertion-Sort If we implement the
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An efficient realization of a prior
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Proposition 8.5: A heap T storing n
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The Array List Representation of a
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java.util.ArrayList. (Continues in
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Code Fragment 8.12: Class ArrayList
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8.3.3 Implementing a Priority Queue
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The upward movement of the newly in
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Analysis Table 8.3 shows the runnin
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We conclude that the heap data stru
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Code Fragment 8.15: Remaining auxil
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As we have previously observed, rea
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8.3.6 Bottom-Up Heap Construction
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Recursive Bottom-Up Heap Constructi
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Figure 8.11: Visual justification o
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insert(3,B) e 2 {(3,B), (5,A)} inse
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The performance of an adaptable pri
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Code Fragment 8.18: An adaptable pr
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503
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Illustrate the execution of the sel
Page 507 and 508:
A group of children want to play a
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C-8.13 We can represent a path from
Page 511 and 512:
P-8.1 Give a Java implementation of
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Chapter 9 Maps and Dictionaries Con
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9.3.3 Ordered Search Tables and Bin
Page 517 and 518:
In either case, we use a key as a u
Page 519 and 520:
null {(5,A), (7,B)} put(2,C) null {
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get(k) put(k,v) put(k,v) remove(k)
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time of map operations in an n-entr
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Hash Codes in Java The generic Obje
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of the word lists provided in two v
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6 6 2 7 14 2 8 105 2 9 18 2 10 277
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however, for if there is a repeated
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only the one entry. We leave the de
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key k will skip over cells containi
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537
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539
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9.2.7 Load Factors and Rehashing In
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example, when pundits study politic
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for the keys, and we provide additi
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findAll(2) {(2,C), (2,E)} {(5,A), (
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Analysis of a List-Based Dictionary
Page 551 and 552:
for entries with duplicate keys. As
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Figure 9.7: Realization of a dictio
Page 555 and 556:
Considering the running time of bin
Page 557 and 558:
Method List Hash Table Search Table
Page 559 and 560:
An example of a skip list is shown
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We give a pseudo-code description o
Page 563 and 564:
the new entry are drawn with thick
Page 565 and 566:
can go into what is almost an infin
Page 567 and 568:
of entries in the dictionary at the
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O(n) O(logn) (expected) 9.5.2 The O
Page 571 and 572:
lastKey() last().getValue() get(las
Page 573 and 574:
Formally, we say a price-performanc
Page 575 and 576:
What is the worst-case running time
Page 577 and 578:
Describe how to use a map to implem
Page 579 and 580:
in memory, describe a method runnin
Page 581 and 582:
please see the various research pap
Page 583 and 584:
Amortized Analysis of Splaying ★.
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10.1.1 Searching To perform operati
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We can also show that a variation o
Page 589 and 590:
The implementation of the remove(k)
Page 591 and 592:
O(h) time, where h is the height of
Page 593 and 594:
Class BinarySearchTree uses locatio
Page 595 and 596:
Code Fragment 10.4: Class BinarySea
Page 597 and 598:
Code Fragment 10.5: Class BinarySea
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10.2 AVL Trees In the previous sect
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(10.3) > 2 i · n(h − 2i). That i
Page 603 and 604:
We restore the balance of the nodes
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Figure 10.9: Schematic illustration
Page 607 and 608:
find another, we perform a restruct
Page 609 and 610:
In Code Fragments 10.7 and 10.8, we
Page 611 and 612:
Code Fragment 10.8: Auxiliary metho
Page 613 and 614:
10.3 Splay Trees Another way we can
Page 615 and 616:
We perform a zig-zig or a zig-zag w
Page 617 and 618:
Figure 10.16: Example of splaying a
Page 619 and 620:
10.3.2 When to Splay The rules that
Page 621 and 622:
10.3.3 Amortized Analysis of Splayi
Page 623 and 624:
When we perform a splaying, we pay
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≤ r ′(x)−r(x) ≤ 3(r ′(x)
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Proposition 10.6: Consider a sequen
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629
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The dictionary we store at each nod
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h≤log(n+1) (10.10) are true. To j
Page 635 and 636:
insertion of 5, which causes an ove
Page 637 and 638:
A split operation affects a constan
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operation; (f) after the fusion ope
Page 641 and 642:
collection returned by findAll. The
Page 643 and 644:
Figure 10.27: Correspondence betwee
Page 645 and 646:
• Take node z, its parent v, and
Page 647 and 648:
Figures 10.30 and 10.31 show a sequ
Page 649 and 650:
The cases for insertion imply an in
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and (b) and for node b in part (c)
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Case 2: The Sibling y of r is Black
Page 655 and 656:
Case 3: The Sibling y of r is Red.
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constant amount of work (in a restr
Page 659 and 660:
Performance of Red-Black Trees Tabl
Page 661 and 662:
Thus, a red-black tree achieves log
Page 663 and 664:
Methods insert (Code Fragment 10.10
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665
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Are the rotations in Figures 10.8 a
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T is a (2,4) tree. c. T is a red-bl
Page 671 and 672:
C-10.8 Let D be an ordered dictiona
Page 673 and 674:
Projects P-10.1 N-body simulations
Page 675 and 676:
early approaches to balancing trees
Page 677 and 678:
11.5 Comparison of Sorting Algorith
Page 679 and 680:
elements of S; that is, S 1 contain
Page 681 and 682:
for a child invocation to return. (
Page 683 and 684:
Proposition 11.1: The merge-sort tr
Page 685 and 686:
idea is to iteratively remove the s
Page 687 and 688:
merge-sort, are implemented by eith
Page 689 and 690:
Code Fragment 11.3: Methods mergeSo
Page 691 and 692:
A Nonrecursive Array-Based Implemen
Page 693 and 694:
the case when n is a power of 2. (W
Page 695 and 696:
each node of the quick-sort tree. T
Page 697 and 698:
eturn). Note the divide steps perfo
Page 699 and 700:
Performing Quick-Sort on Arrays and
Page 701 and 702:
E has at least one element (the piv
Page 703 and 704:
the size of the input for this call
Page 705 and 706:
In-place quick-sort modifies the in
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Unfortunately, the implementation a
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algorithm to output the elements of
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itself is a sequence (of entries wi
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first component, we guarantee that
Page 715 and 716:
Bucket-Sort and Radix-Sort Finally,
Page 717 and 718:
generic merging takes time proporti
Page 719 and 720:
• In class Union Merge, merge cop
Page 721 and 722:
Performance of the Sequence Impleme
Page 723 and 724:
With this partition data structure,
Page 725 and 726:
The Log-Star and Inverse Ackermann
Page 727 and 728:
This may come as a small surprise,
Page 729 and 730:
is the same as the expected number
Page 731 and 732:
Show that the probability that any
Page 733 and 734:
Describe what additional informatio
Page 735 and 736:
Argue why the previous claim proves
Page 737 and 738:
threads of b. Describe and analyze
Page 739 and 740:
Design and implement a stable versi
Page 741 and 742:
Contents 12.1 String Operations ...
Page 743 and 744:
567 12.5 Text Similarity Testing ..
Page 745 and 746:
finite, we usually assume that the
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Return and update S to be the strin
Page 749 and 750:
Performance The brute-force pattern
Page 751 and 752:
We will formalize these heuristics
Page 753 and 754:
In Figure 12.3, we illustrate the e
Page 755 and 756:
A Java implementation of the BM pat
Page 757 and 758:
using an alternative shift heuristi
Page 759 and 760:
In Figure 12.5, we illustrate the e
Page 761 and 762:
A Java implementation of the KMP pa
Page 763 and 764:
12.3 Tries The pattern matching alg
Page 765 and 766:
The worst case for the number of no
Page 767 and 768:
A compressed trie is similar to a s
Page 769 and 770:
trie. In the next section, we prese
Page 771 and 772:
if such a path can be traced. The d
Page 773 and 774:
Let m be the size of pattern P and
Page 775 and 776:
transmit our text. Likewise, text c
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12.4.2 The Greedy Method Huffman's
Page 779 and 780:
There are few algorithmic technique
Page 781 and 782:
Turning this definition of L[i, j]
Page 783 and 784:
Reinforcement R-12.1 List the prefi
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C-12.5 Say that a pattern P of leng
Page 787 and 788:
suffix of length j in y. Design an
Page 789 and 790:
pattern matching. The reader intere
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Depth-First Search Traversing a Dig
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Example 13.2: We can associate with
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In the propositions that follow, we
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are communication connections betwe
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vertices and m edges, an edge list
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The main feature of the edge list s
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• The edge object for an edge e w
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insertVertex, insertEdge, removeEdg
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not. The optimal performance of met
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came from to get to v, and repeat t
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We can visualize a DFS traversal by
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• Computing a cycle in G, or repo
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depend on the implementation of the
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Code Fragment 13.5: The main templa
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Using the Template Method Pattern f
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Class FindCycleDFS (Code Fragment 1
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levels. BFS can also be thought of
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825
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• Given a start vertex s of G, co
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Interesting problems that deal with
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A DFS on a digraph partitions the e
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Directed Breadth-First Search As wi
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edges (vi,vk) and (vk,vj) with dash
Page 837 and 838:
ordering such that any directed pat
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Justification: The initial computat
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computers. In this case, it is prob
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their distances from v. Thus, in ea
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edge for pulling in u with a solid
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Proposition 13.23: In Dijkstra's al
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Referring back to Code Fragment 13.
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The main computation of Dijkstra's
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13.7 Minimum Spanning Trees Suppose
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13.7.1 Kruskal's Algorithm The reas
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Figure 13.19: An example of an exec
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execution of Kruskal's algorithm. (
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Analyzing the Prim-Jarn ık Algorit
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13.8 Exercises For source code and
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LA22: (none) • LA31: LA15 • LA3
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The graph has 10,000 vertices and 2
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265 175 215 180 185 155 3- - - - 26
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_ _ _ _ Find which bridges to build
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determine if it is constructed corr
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An Euler tour of a directed graph t
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C-13.21 Computer networks should av
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c. Now suppose that is weighted and
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P-13.9 Design an experimental compa
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Chapter 14 Memory Contents 14.1 Mem
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Multi-way Merging..................
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Keeping Track of the Program Counte
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In Section 7.3.6 we described how t
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Memory Allocation Algorithms The Ja
Page 893 and 894:
identify and mark each live object.
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Caches and Disks Specifically, the
Page 897 and 898:
are addressed. Virtual memory does
Page 899 and 900:
The Random strategy is one of the e
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each would require to perform the s
Page 903 and 904:
describe this structure, however, l
Page 905 and 906:
are too large to fit entirely into
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Reinforcement R-14.1 Describe, in d
Page 909 and 910:
Describe how to use a B-tree to imp
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Appendix A Useful Mathematical Fact
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The factorial function is defined a
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for any real number 0 < a ≠ 1. Pr
Page 917 and 918:
E(X + Y) = E(X) + E(Y) and E(cX) =
Page 919 and 920:
Bibliography [1] G. M. Adelson-Vels
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[28] S. A. Demurjian, Sr., Software
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[69] B. Liskov and J. Guttag, Abstr
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