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## Data Structures and Algorithms

John E. Ullman, Stanford University, Stanford,. This book presents the data structures and algorithms that underpin much of today's computer programming. The basis of this book is the material contained in the first six chapters of our earlier work, The Design and Analysis of Computer Algorithms. We have expanded that coverage and have added material on algorithms for external storage and memory management.

As a consequence, this book should be suitable as a text for a first course on data structures and algorithms. The only prerequisite we assume is familiarity with some high-level programming language such as Pascal. We have attempted to cover data structures and algorithms in the broader context of solving problems using computers. We use abstract data types informally in the description and implementation of algorithms. Although abstract data types are only starting to appear in widely available programming languages, we feel they are a useful tool in designing programs, no matter what the language.

We also introduce the ideas of step counting and time complexity as an integral part of the problem solving process. This decision reflects our longheld belief that programmers are going to continue to tackle problems of progressively larger size as machines get faster, and that consequently the time complexity of algorithms will become of even greater importance, rather than of less importance, as new.

Initially we present several of our algorithms both abstractly and as Pascal programs, because we feel it is. The algorithms we present, however, can be readily. Chapter 1 contains introductory remarks, including an explanation of our view of the problem-to-program process and the role of abstract data types in that process.

Also appearing is an introduction to step counting and "big-oh" and "big-omega" notation. The third chapter introduces trees and the basic data structures that can be used to support various operations on trees efficiently. Chapters 4 and 5 introduce a number of important abstract data types that are based on the mathematical model of a set.

Dictionaries and priority queues are covered in depth. Standard implementations for these concepts, including hash tables, binary search trees, partially ordered trees, tries, and trees are covered, with the more advanced material clustered in Chapter 5. Chapters 6 and 7 cover graphs, with directed graphs in Chapter 6 and undirected graphs in 7.

These chapters begin a section of the book devoted more to issues of algorithms than data structures, although we do discuss the basics of data structures suitable for representing graphs. A number of important graph algorithms are. Chapter 8 is devoted to the principal internal sorting algorithms: quicksort, heapsort, binsort, and the simpler, less efficient methods such as insertion sort.

In this chapter we also cover the linear-time algorithms for finding medians and other order statistics. Chapter 9 discusses the asymptotic analysis of recursive procedures, including, of course, recurrence relations and techniques for solving them. Chapter 10 outlines the important techniques for designing algorithms, including divide-and-conquer, dynamic programming, local search algorithms, and various forms of organized tree searching. The last two chapters are devoted to external storage organization and memory management.

Chapter 11 covers external sorting and large-scale storage. Chapter 12 contains material on memory management, divided into four subareas, depending on whether allocations involve fixed or varying sized blocks, and whether the freeing of blocks takes place by explicit program action or implicitly when garbage collection occurs. A number of exercises of varying degrees of difficulty are found at the end of each chapter.

Many of these are fairly straightforward tests of the mastery of the material of the chapter. Some exercises require more thought, and these have been singly starred.

Doubly starred exercises are harder still, and are suitable for more advanced courses. The bibliographic notes at the end of each chapter provide references for additional reading. Many of our colleagues have read various portions of the manuscript and have given us valuable comments and advice. In particular, we would like to thank Ed Beckham, Jon. Finally, we would like to give our warmest thanks to Mrs.

Claire Metzger for her expert assistance in helping prepare the manuscript for typesetting. There are many steps involved in writing a computer program to solve a given problem. The steps go from problem formulation and specification, to design of the solution, to implementation, testing and documentation, and finally to evaluation of the solution. This chapter outlines our approach to these steps.

Subsequent chapters discuss the algorithms and data structures that are the building blocks of most computer programs. In fact, certain problems, such as creating a "gourmet" recipe or preserving world peace, may be impossible to formulate in terms that admit of a computer solution. Even if we suspect our problem can be solved on a computer, there is usually considerable latitude in several problem parameters.

Often it is only by experimentation that reasonable values for these parameters can be found. If certain aspects of a problem can be expressed in terms of a formal model, it is usually beneficial to do so, for once a problem is formalized, we can look for solutions in terms of a precise model and determine whether a program already exists to solve that problem. Even if there is no existing program, at least we can discover what is known about this model and use the properties of the model to help construct a good solution.

Almost any branch of mathematics or science can be called into service to help model some problem domain. Problems essentially numerical in nature can be modeled by such common mathematical concepts as simultaneous linear equations e.

Symbol and text processing problems can be modeled by character strings and formal grammars. Problems of this nature include compilation the translation of programs written in a programming language into machine language and information retrieval tasks such as recognizing particular words in lists of titles owned by a library.

Once we have a suitable mathematical model for our problem, we can attempt to find a solution in terms of that model. Our initial goal is to find a solution in the form of an. In an algorithm instructions can be executed any number of times, provided the instructions themselves indicate the repetition.

However, we require that, no matter what the input values may be, an algorithm terminate after executing a finite number of instructions. Thus, a program is an algorithm as long as it never enters an. There is one aspect of this definition of an algorithm that needs some clarification. We said each instruction of an algorithm must have a "clear meaning" and must be executable with a "finite amount of effort. It is often difficult as well to prove that on any input, a sequence of instructions terminates, even if we understand clearly what each instruction means.

By argument and counterargument, however, agreement can usually be reached as to whether a sequence of instructions constitutes an algorithm. The burden of proof lies with the person claiming to have an algorithm. In Section 1. In addition to using Pascal programs as algorithms, we shall often present algorithms using a pseudo-language that is a combination of the constructs of a programming language together with informal English statements.

We shall use Pascal as the programming. The following example illustrates many of the steps in our approach to writing a computer program. We shall then associate a phase of the traffic light with each group in the partition. By finding a partition with the smallest number of groups, we can construct a traffic light with the smallest number of phases.

For example, the intersection shown in Fig. Roads C and E are oneway, the others two way. There are 13 turns one might make at this intersection. Some pairs of turns, like AB from A to B and. We can model this problem with a mathematical structure known as a graph.

A graph consists of a set of points called vertices, and lines connecting the points, called edges. For the traffic intersection problem we can draw a graph whose vertices represent turns and whose edges connect pairs of vertices whose turns cannot be performed simultaneously. For the intersection of Fig.

The graph can aid us in solving the traffic light design problem. A coloring of a graph is an assignment of a color to each vertex of the graph so that no two vertices connected by an edge have the same color. It is not hard to see that our problem is one of coloring the graph of incompatible turns using as few colors as possible. The problem of coloring graphs has been studied for many decades, and the theory of algorithms tells us a lot about this problem.

Unfortunately, coloring an arbitrary graph with as few colors as possible is one of a large class of problems called "NP-complete. With care, we can be a little speedier than this, but it is generally. We are now confronted with the possibility that finding an optimal solution for the problem at hand is computationally very expensive. We can adopt. If the graph is small, we might attempt to find an optimal solution exhaustively, trying all possibilities.

This approach, however, becomes prohibitively. A second approach would be to look for additional information about the problem at hand. It may turn out that the graph has some special properties, which make it unnecessary to try all possibilities in finding an optimal solution.

The third approach is to change the problem a little and look for a good but not necessarily optimal solution. We might be happy with a solution that gets close to the minimum number of colors on small graphs, and works quickly, since most intersections are not even as complex as Fig. An algorithm that quickly produces good but not necessarily optimal solutions is called a heuristic. One reasonable heuristic for graph coloring is the following "greedy" algorithm.

Initially we try to color as many vertices as possible with the first color, then as many as possible of the uncolored vertices with the second color, and so on. To color vertices with a new color, we perform the following steps. Scan the list of uncolored vertices. For each uncolored vertex, determine whether it has an edge to any vertex already colored with the new color. If there is no such edge, color the present vertex with the new color. This approach is called "greedy" because it colors a vertex whenever it can, without considering the potential drawbacks inherent in making such a move.

There are situations where we could color more vertices with one color if we were less "greedy" and skipped some vertex we could legally color.

## Data Structures and Algorithms Alfred V Aho pdf

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## Data Structures and Algorithms Alfred V Aho pdf

Data Structures and AlgorithmsAlfred V. PrefaceThis book presents the data structures and algorithms that underpin much of today's computer programming. The basis of this book is the material contained in the first six chapters of our earlier work, The Design and Analysis of Computer Algorithms.

Data Structures and AlgorithmsAlfred V. PrefaceThis book presents the data structures and algorithms that underpin much of today's computer programming. The basis of this book is the material contained in the first six chapters of our earlier work, The Design and Analysis of Computer Algorithms.

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