Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

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into a bowl of water to see if they both displaced the same amount. This discovery was unfortunate for the goldsmith, however, for when Archimedes did his analysis, the crown displaced more water than an equal-weight lump of pure gold, indicating that the crown was not, in fact, pure gold. In this book, we are interested in the design of "good" data structures and algorithms. Simply put, a data structure is a systematic way of organizing and accessing data, and an algorithm is a step-by-step procedure for performing some task in a finite amount of time. These concepts are central to computing, but to be able to classify some data structures and algorithms as "good," we must have precise ways of analyzing them. The primary analysis tool we will use in this book involves characterizing the running times of algorithms and data structure operations, with space usage also being of interest. Running time is a natural measure of "goodness," since time is a precious resource—computer solutions should run as fast as possible. In general, the running time of an algorithm or data structure method increases with the input size, although it may also vary for different inputs of the same size. Also, the running time is affected by the hardware environment (as reflected in the processor, clock rate, memory, disk, etc.) and software environment (as reflected in the operating system, programming language, compiler, interpreter, etc.) in which the algorithm is implemented, compiled, and executed. All other factors being equal, the running time of the same algorithm on the same input data will be smaller if the computer has, say, a much faster processor or if the implementation is done in a program compiled into native machine code instead of an interpreted implementation run on a virtual machine. Nevertheless, in spite of the possible variations that come from different environmental factors, we would like to focus on the relationship between the running time of an algorithm and the size of its input. We are interested in characterizing an algorithm's running time as a function of the input size. But what is the proper way of measuring it? 4.2.1 Experimental Studies if an algorithm has been implemented, we can study its running time by executing it on various test inputs and recording the actual time spent in each execution. Fortunately, such measurements can be taken in an accurate manner by using system calls that are built into the language or operating system (for example, by using the System.current Time Millis () method or calling the run-time environment with profiling enabled). Such tests assign a specific running time to a specific input size, but we are interested in determining the general dependence of running time on the size of the input. In order to determine this dependence, we should perform several experiments on many different test inputs of various sizes. Then we can visualize the results of such experiments by plotting the performance of each run of the algorithm as a point with x-coordinate equal to the input size, n, 224
and y-coordinate equal to the running time, t. (See Figure 4.3.) From this visualization and the data that supports it, we can perform a statistical analysis that seeks to fit the best function of the input size to the experimental data. To be meaningful, this analysis requires that we choose good sample inputs and test enough of them to be able to make sound statistical claims about the algorithm's running time. Figure 4.3: Results of an experimental study on the running time of an algorithm. A dot with coordinates (n, t) indicates that on an input of size n, the running time of the algorithm is t milliseconds (ms). While experimental studies of running times are useful, they have three major limitations: • Experiments can be done only on a limited set of test inputs; hence, they leave out the running times of inputs not included in the experiment (and these inputs may be important). 225
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Data Structures and Algorithms in J
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To Isabel -Roberto Tamassia Preface
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The Java code implementing fundamen
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PL6. Object-Oriented Programming Ch
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14. Memory A. Useful Mathematical F
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Sun, Nikos Triandopoulos, Luca Vism
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Getting Started: Classes, Types, an
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39 1.7 An Example Program..........
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The main "actors" in a Java program
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else interface switch boolean exten
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Class modifiers are optional keywor
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Comments Note the use of comments i
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Calling the new operator on a class
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n.longValue( ) float Float n = new
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The Dot Operator Every object refer
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object (assuming we are allowed acc
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Note the uses of instance variables
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1.2 Methods Methods in Java are con
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Abstract methods may only appear wi
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is intended to be used to initializ
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} int i = 512; double e = 2.71828;
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% the modulo operator This last ope
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zeros >>> shift bits right, filling
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3 add./subt. + − 4 shift > >>> 5
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double d1 = 3.2; double d2 = 3.9999
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String s = " " + 4.5; // correct, b
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System.out.println("This is tough."
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Defining a For Loop Here is the syn
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that input and inform the user that
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if (a[i][j] == 0) { foundFlag = tru
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the array, we will sometimes refer
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The cells of this new array, "a," a
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ways to reference such an object. W
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float height = s.nextFloat(); Syste
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We show the CreditCard class in Cod
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Code Fragment 1.7: Output from the
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a text editor class may wish to def
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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
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Chapter 3 Arrays, Linked Lists, and
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3.4 Circularly Linked Lists and Lin
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A Class for High Scores Suppose we
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Note that we include a method, toSt
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Once we have identified the place i
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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
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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
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The reverse operation of inserting
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Header and Trailer Sentinels To sim
Page 173 and 174: Likewise, we can easily perform an
Page 175 and 176: Figure 3.17: Adding a new node afte
Page 177 and 178: Code Fragment 3.23: Java class DLis
Page 179 and 180: We make the following observations
Page 181 and 182: advance(): Advance the cursor to th
Page 183 and 184: Some Observations about the CircleL
Page 185 and 186: Some Sample Output 185
Page 187 and 188: We show in Code Fragment 3.27 thein
Page 189 and 190: The factorial function can be defin
Page 191 and 192: Each multiple of 1 inch also has a
Page 193 and 194: The trace for drawTicks is more com
Page 195 and 196: The simplest form of recursion is l
Page 197 and 198: nonrecursive part of each call. Mor
Page 199 and 200: 3.5.2 Binary Recursion When an algo
Page 201 and 202: Unfortunately, in spite of the Fibo
Page 203 and 204: To solve such a puzzle, we need to
Page 205 and 206: 3.6 Exercises For source code and h
Page 207 and 208: C-3.2 Let A be an array of size n
Page 209 and 210: Write a recursive Java program that
Page 211 and 212: Chapter Notes The fundamental data
Page 213 and 214: Primitive Operations . . . . . . .
Page 215 and 216: 4.1.2 The Logarithm function One of
Page 217 and 218: 4.1.5 The Quadratic function Anothe
Page 219 and 220: which assigns to an input value n t
Page 221 and 222: For example, we have the following:
Page 223: primarily as slopes. Even so, the e
Page 227 and 228: Counting Primitive Operations Speci
Page 229 and 230: Simplifying the Analysis Further In
Page 231 and 232: Proposition 4.7: The Algorithmarray
Page 233 and 234: is O(4n 2 + 6n log n), although thi
Page 235 and 236: 256 16 4 16 64 256 4,096 65,536 32
Page 237 and 238: Running Maximum Problem Size (n) Ti
Page 239 and 240: should the constant factors they "h
Page 241 and 242: • There are two nested for loops,
Page 243 and 244: This definition leads immediately t
Page 245 and 246: Besides showing a use of the contra
Page 247 and 248: We may sometimes feel overwhelmed b
Page 249 and 250: The number of operations executed b
Page 251 and 252: R-4.22 Show that if d(n) is O(f(n))
Page 253 and 254: 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
Page 259 and 260: Stacks.............................
Page 261 and 262: down on the stack to become the new
Page 263 and 264: 7 (5) top() 5 (5) pop() 5 () pop()
Page 265 and 266: In addition, we must define excepti
Page 267 and 268: 5.1.2 A Simple Array-Based Stack Im
Page 269 and 270: we indicate that the stack no longe
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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
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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
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Code Fragment 5.18: Class NodeDeque
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5.4 Exercises For source code and h
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Suppose Alice has picked three dist
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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
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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
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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
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Code Fragment 6.11: Portions of the
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An iterator is a software design pa
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Since looping through the elements
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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
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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
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accesses several Web pages and then
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6.6 Exercises For source code and h
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R-6.16 Describe how to create an it
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Think about how to extend the circu
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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
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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
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Figure 7.13: Operation that removes
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We use a Java interface BTPosition
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addition to the Binary Tree interfa
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Binary Tree interface. (Continues i
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• Methods size() and isEmpty() us
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The level numbering function p sugg
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Time size, isEmpty O(1) iterator, p
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As is the case for general trees, t
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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
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C-7.10 Two ordered trees T ′ and
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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
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A group of children want to play a
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C-8.13 We can represent a path from
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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
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In either case, we use a key as a u
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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), (
Page 549 and 550:
Analysis of a List-Based Dictionary
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for entries with duplicate keys. As
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Figure 9.7: Realization of a dictio
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Considering the running time of bin
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Method List Hash Table Search Table
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An example of a skip list is shown
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We give a pseudo-code description o
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the new entry are drawn with thick
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can go into what is almost an infin
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of entries in the dictionary at the
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O(n) O(logn) (expected) 9.5.2 The O
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lastKey() last().getValue() get(las
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Formally, we say a price-performanc
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What is the worst-case running time
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Describe how to use a map to implem
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in memory, describe a method runnin
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please see the various research pap
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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
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The implementation of the remove(k)
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O(h) time, where h is the height of
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Class BinarySearchTree uses locatio
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Code Fragment 10.4: Class BinarySea
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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
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We restore the balance of the nodes
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Figure 10.9: Schematic illustration
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find another, we perform a restruct
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In Code Fragments 10.7 and 10.8, we
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Code Fragment 10.8: Auxiliary metho
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10.3 Splay Trees Another way we can
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We perform a zig-zig or a zig-zag w
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Figure 10.16: Example of splaying a
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10.3.2 When to Splay The rules that
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10.3.3 Amortized Analysis of Splayi
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When we perform a splaying, we pay
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≤ r ′(x)−r(x) ≤ 3(r ′(x)
Page 627 and 628:
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
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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
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collection returned by findAll. The
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Figure 10.27: Correspondence betwee
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• Take node z, its parent v, and
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Figures 10.30 and 10.31 show a sequ
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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
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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
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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
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early approaches to balancing trees
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11.5 Comparison of Sorting Algorith
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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
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merge-sort, are implemented by eith
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Code Fragment 11.3: Methods mergeSo
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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
Page 707 and 708:
Unfortunately, the implementation a
Page 709 and 710:
algorithm to output the elements of
Page 711 and 712:
itself is a sequence (of entries wi
Page 713 and 714:
first component, we guarantee that
Page 715 and 716:
Bucket-Sort and Radix-Sort Finally,
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generic merging takes time proporti
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• 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 ..
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finite, we usually assume that the
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Return and update S to be the strin
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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
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In Figure 12.5, we illustrate the e
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A Java implementation of the KMP pa
Page 763 and 764:
12.3 Tries The pattern matching alg
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The worst case for the number of no
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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
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There are few algorithmic technique
Page 781 and 782:
Turning this definition of L[i, j]
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Reinforcement R-12.1 List the prefi
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C-12.5 Say that a pattern P of leng
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suffix of length j in y. Design an
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pattern matching. The reader intere
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Depth-First Search Traversing a Dig
Page 793 and 794:
Example 13.2: We can associate with
Page 795 and 796:
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
Page 843 and 844:
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
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identify and mark each live object.
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Caches and Disks Specifically, the
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are addressed. Virtual memory does
Page 899 and 900:
The Random strategy is one of the e
Page 901 and 902:
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
Page 907 and 908:
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
Page 923 and 924:
[69] B. Liskov and J. Guttag, Abstr
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