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

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For example, the code for "a" is 010, and the code for "f" is 1100. 12.4.1 The Huffman Coding Algorithm The Huffman coding algorithm begins with each of the d distinct characters of the string X to encode being the root node of a single-node binary tree. The algorithm proceeds in a series of rounds. In each round, the algorithm takes the two binary trees with the smallest frequencies and merges them into a single binary tree. It repeats this process until only one tree is left. (See Code Fragment 12.8.) Each iteration of the while loop in Huffman's algorithm can be implemented in O(logd) time using a priority queue represented with a heap. In addition, each iteration takes two nodes out of Q and adds one in, a process that will be repeated d − 1 times before exactly one node is left in Q. Thus, this algorithm runs in O(n + dlogd) time. Although a full justification of this algorithm's correctness is beyond our scope here, we note that its intuition comes from a simple idea—any optimal code can be converted into an optimal code in which the code words for the two lowest-frequency characters, a and b, differ only in their last bit. Repeating the argument for a string with a and b replaced by a character c, gives the following: Proposition 12.10: Huffman's algorithm constructs an optimal prefix code for a string of length n with d distinct characters in O(n + dlogd) time. Code Fragment 12.8: Huffman coding algorithm. 776
12.4.2 The Greedy Method Huffman's algorithm for building an optimal encoding is an example application of an algorithmic design pattern called the greedy method. This design pattern is applied to optimization problems, where we are trying to construct some structure while minimizing or maximizing some property of that structure. The general formula for the greedy method pattern is almost as simple as that for the brute-force method. In order to solve a given optimization problem using the greedy method, we proceed by a sequence of choices. The sequence starts from some well-understood starting condition, and computes the cost for that initial condition. The pattern then asks that we iteratively make additional choices by identifying the decision that achieves the best cost improvement from all of the choices that are currently possible. This approach does not always lead to an optimal solution. But there are several problems that it does work for, and such problems are said to possess the greedy-choice property. This is the property that a global optimal condition can be reached by a series of locally optimal choices (that is, choices that are each the current best from among the possibilities available at the time), starting from a well-defined starting condition. The problem of computing an optimal variable-length prefix code is just one example of a problem that possesses the greedy-choice property. 777
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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
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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), (
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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
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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
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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,
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
Page 747 and 748: 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: transmit our text. Likewise, text c
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
Page 785 and 786: 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
Page 791 and 792: 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
Page 797 and 798: are communication connections betwe
Page 799 and 800: vertices and m edges, an edge list
Page 801 and 802: The main feature of the edge list s
Page 803 and 804: • The edge object for an edge e w
Page 805 and 806: insertVertex, insertEdge, removeEdg
Page 807 and 808: not. The optimal performance of met
Page 809 and 810: came from to get to v, and repeat t
Page 811 and 812: We can visualize a DFS traversal by
Page 813 and 814: • Computing a cycle in G, or repo
Page 815 and 816: depend on the implementation of the
Page 817 and 818: Code Fragment 13.5: The main templa
Page 819 and 820: Using the Template Method Pattern f
Page 821 and 822: Class FindCycleDFS (Code Fragment 1
Page 823 and 824: levels. BFS can also be thought of
Page 825 and 826: 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
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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.
Page 895 and 896:
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
Page 911 and 912:
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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