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

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Write a program that can input and display a person's family tree. P-7.6 Implement the tree ADT using the binary tree representation described in Exercise C-7.12. You may reuse the LinkedBinaryTree implementation of a binary tree. P-7.7 A slicing floorplan divides a rectangle with horizontal and vertical sides using horizontal and vertical cuts. (See Figure 7.23a.) A slicing floorplan can be represented by a proper binary tree, called a slicing tree, whose internal nodes represent the cuts, and whose external nodes represent the basic rectangles into which the floorplan is decomposed by the cuts. (See Figure 7.23b.) The compaction problem for a slicing floorplan is defined as follows. Assume that each basic rectangle of a slicing floorplan is assigned a minimum width w and a minimum height h. The compaction problem is to find the smallest possible height and width for each rectangle of the slicing floorplan that is compatible with the minimum dimensions of the basic rectangles. Namely, this problem requires the assignment of values h(v) and w(v) to each node v of the slicing tree such that: Design a data structure for slicing floorplans that supports the operations: • Create a floorplan consisting of a single basic rectangle. • Decompose a basic rectangle by means of a horizontal cut. • Decompose a basic rectangle by means of a vertical cut. • Assign minimum height and width to a basic rectangle. • Draw the slicing tree associated with the floorplan. • Compact and draw the floorplan. Figure 7.23: (a) Slicing floorplan; (b) slicing tree associated with the floorplan. 444
P-7.8 Write a program that can play tic-tac-toe effectively. (See Section 3.1.5.) To do this, you will need to create a game tree T, which is a tree where each node corresponds to a game configuration, which, in this case, is a representation of the tic-tac-toe board. The root node corresponds to the initial configuration. For each internal node v in T, the children of v correspond to the game states we can reach from v's game state in a single legal move for the appropriate player, A (the first player) or B (the second player). Nodes at even depths correspond to moves for A and nodes at odd depths correspond to moves for B. External nodes are either final game states or are at a depth beyond which we don't want to explore. We score each external node with a value that indicates how good this state is for player A. In large games, like chess, we have to use a heuristic scoring function, but for small games, like tic-tac-toe, we can construct the entire game tree and score external nodes as + 1, 0, − 1, indicating whether player A has a win, draw, or lose in that configuration. A good algorithm for choosing moves is minimax. In this algorithm, we assign a score to each internal node v in T, such that if v represents A's turn, we compute v's score as the maximum of the scores of v's children (which corresponds to A's optimal play from v). If an internal node v represents B's turn, then we compute v's score as the minimum of the scores of v's children (which corresponds to B's optimal play from v). P-7.9 Write a program that takes as input a fully parenthesized, arithmetic expression and converts it to a binary expression tree. Your program should display the tree in some way and also print the value associated with the root. For an additional challenge, allow for the leaves to store variables of the form x 1 , x 2 , x 3 , and so on, which are initially 0 and which can be updated interactively by your program, with the corresponding update in the printed value of the root of the expression tree. 445
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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
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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
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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 ).
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Describe a method for finding both
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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
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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
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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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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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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
Page 393 and 394: The postorder traversal method is u
Page 395 and 396: Although the preorder and postorder
Page 397 and 398: A Recursive Binary Tree Definition
Page 399 and 400: same depth d as the level dof T. In
Page 401 and 402: Figure 7.13: Operation that removes
Page 403 and 404: We use a Java interface BTPosition
Page 405 and 406: addition to the Binary Tree interfa
Page 407 and 408: Code Fragment 7.17: Portions of the
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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
Page 417 and 418: 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
Page 423 and 424: The inorder traversal can also be a
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Page 427 and 428: • Calls auxiliary method visitRig
Page 429 and 430: ExpressionTerm at each node. Class
Page 431 and 432: Code Fragment 7.31: Class EvaluateE
Page 433 and 434: Let T be an n-node binary tree that
Page 435 and 436: c. Let T be a proper binary tree wi
Page 437 and 438: 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
Page 443: Describe a nonrecursive method for
Page 447 and 448: Chapter 8 Priority Queues Contents
Page 449 and 450: The Heap Data Structure............
Page 451 and 452: As in the examples above from an ai
Page 453 and 454: package includes a similar Entry in
Page 455 and 456: Code Fragment 8.3: Class representi
Page 457 and 458: insert(5,A) e 1 [=(5,A)] {(5,A)} in
Page 459 and 460: size() size() isEmpty() isEmpty() i
Page 461 and 462: at the time the method is executed.
Page 463 and 464: Code Fragment 8.7: Java class Defau
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Page 467 and 468: Insertion-Sort If we implement the
Page 469 and 470: An efficient realization of a prior
Page 471 and 472: Proposition 8.5: A heap T storing n
Page 473 and 474: The Array List Representation of a
Page 475 and 476: java.util.ArrayList. (Continues in
Page 477 and 478: Code Fragment 8.12: Class ArrayList
Page 479 and 480: 8.3.3 Implementing a Priority Queue
Page 481 and 482: The upward movement of the newly in
Page 483 and 484: Analysis Table 8.3 shows the runnin
Page 485 and 486: We conclude that the heap data stru
Page 487 and 488: Code Fragment 8.15: Remaining auxil
Page 489 and 490: As we have previously observed, rea
Page 491 and 492: 8.3.6 Bottom-Up Heap Construction
Page 493 and 494: 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), (
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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)
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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
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insertion of 5, which causes an ove
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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
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Performance of Red-Black Trees Tabl
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Thus, a red-black tree achieves log
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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
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C-10.8 Let D be an ordered dictiona
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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
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for a child invocation to return. (
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Proposition 11.1: The merge-sort tr
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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
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the case when n is a power of 2. (W
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each node of the quick-sort tree. T
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eturn). Note the divide steps perfo
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Performing Quick-Sort on Arrays and
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E has at least one element (the piv
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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
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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
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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
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Performance of the Sequence Impleme
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With this partition data structure,
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The Log-Star and Inverse Ackermann
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This may come as a small surprise,
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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
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threads of b. Describe and analyze
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Design and implement a stable versi
Page 741 and 742:
Contents 12.1 String Operations ...
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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
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
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A Java implementation of the BM pat
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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
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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
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trie. In the next section, we prese
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if such a path can be traced. The d
Page 773 and 774:
Let m be the size of pattern P and
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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
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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
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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
Page 831 and 832:
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
Page 839 and 840:
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
Page 845 and 846:
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
Page 859 and 860:
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.
Page 895 and 896:
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
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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