Fundamentals: Table of Contents Fundamentals of Data Structures by Ellis Horowitz and Sartaj Sahni PREFACE CHAPTER 1: INTRODUCTION CHAPTER 2: ARRAYS CHAPTER 3: STACKS AND QUEUES CHAPTER 4: LINKED LISTS CHAPTER 5: TREES CHAPTER 6: GRAPHS CHAPTER 7: INTERNAL SORTING CHAPTER 8: EXTERNAL SORTING CHAPTER 9: SYMBOL TABLES CHAPTER 10: FILES APPENDIX A: SPARKS APPENDIX B: ETHICAL CODE IN INFORMATION PROCESSING APPENDIX C: ALGORITHM INDEX BY CHAPTER file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDobbs_Books_Algorithms_Collection2ed/books/book1/toc.htm7/3/2004 3:56:06 PM www.itdevelopteam.com Fundamentals: PREFACE PREFACE For many years a data structures course has been taught in computer science programs. Often it is regarded as a central course of the curriculum. It is fascinating and instructive to trace the history of how the subject matter for this course has changed. Back in the middle1960's the course was not entitled Data Structures but perhaps List Processing Languages. The major subjects were systems such as SLIP (by J. Weizenbaum), IPL-V (by A. Newell, C. Shaw, and H. Simon), LISP 1.5 (by J. McCarthy) and SNOBOL (by D. Farber, R. Griswold, and I. Polonsky). Then, in 1968, volume I of the Art of Computer Programming by D. Knuth appeared. His thesis was that list processing was not a magical thing that could only be accomplished within a specially designed system. Instead, he argued that the same techniques could be carried out in almost any language and he shifted the emphasis to efficient algorithm design. SLIP and IPL-V faded from the scene, while LISP and SNOBOL moved to the programming languages course. The new strategy was to explicitly construct a representation (such as linked lists) within a set of consecutive storage locations and to describe the algorithms by using English plus assembly language. Progress in the study of data structures and algorithm design has continued. Out of this recent work has come many good ideas which we believe should be presented to students of computer science. It is our purpose in writing this book to emphasize those trends which we see as especially valuable and long lasting. The most important of these new concepts is the need to distinguish between the specification of a data structure and its realization within an available programming language. This distinction has been mostly blurred in previous books where the primary emphasis has either been on a programming language or on representational techniques. Our attempt here has been to separate out the specification of the data structure from its realization and to show how both of these processes can be successfully accomplished. The specification stage requires one to concentrate on describing the functioning of the data structure without concern for its implementation. This can be done using English and mathematical notation, but here we introduce a programming notation called axioms. The resulting implementation independent specifications valuable in two ways: (i) to help prove that a program which uses this data structure is correct and (ii) to prove that a particular implementation of the data structure is correct. To describe a data structure in a representation independent way one needs a syntax. This can be seen at the end of section 1.1 where we also precisely define the notions of data object and data structure. This book also seeks to teach the art of analyzing algorithms but not at the cost of undue mathematical sophistication. The value of an implementation ultimately relies on its resource utilization: time and space. This implies that the student needs to be capable of analyzing these factors. A great many analyses have appeared in the literature, yet from our perspective most students don't attempt to rigorously analyze their programs. The data structures course comes at an opportune time in their training to advance and promote these ideas. For every algorithm that is given here we supply a simple, yet rigorous worst case analysis of its behavior. In some cases the average computing time is also file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDob...Books_Algorithms_Collection2ed/books/book1/preface.htm (1 of 4)7/3/2004 3:56:18 PM www.itdevelopteam.com Fundamentals: PREFACE derived. The growth of data base systems has put a new requirement on data structures courses, namely to cover the organization of large files. Also, many instructors like to treat sorting and searching because of the richness of its examples of data structures and its practical application. The choice of our later chapters reflects this growing interest. One especially important consideration is the choice of an algorithm description language. Such a choice is often complicated by the practical matters of student background and language availability. Our decision was to use a syntax which is particularly close to ALGOL, but not to restrict ourselves to a specific language. This gives us the ability to write very readable programs but at the same time we are not tied to the idiosyncracies of a fixed language. Wherever it seemed advisable we interspersed English descriptions so as not to obscure the main pointof an algorithm. For people who have not been exposed to the IF-THEN-ELSE, WHILE, REPEAT- UNTIL and a few other basic statements, section 1.2 defines their semantics via flowcharts. For those who have only FORTRAN available, the algorithms are directly translatable by the rules given in the appendix and a translator can be obtained (see appendix A). On the other hand, we have resisted the temptation to use language features which automatically provide sophisticated data structuring facilities. We have done so on several grounds. One reason is the need to commit oneself to a syntax which makes the book especially hard to read by those as yet uninitiated. Even more importantly, these automatic featules cover up the implementation detail whose mastery remains a cornerstone of the course. The basic audience for this book is either the computer science major with at least one year of courses or a beginning graduate student with prior training in a field other than computer science. This book contains more than one semester's worth of material and several of its chapters may be skipped without harm. The following are two scenarios which may help in deciding what chapters should be covered. The first author has used this book with sophomores who have had one semester of PL/I and one semester of assembly language. He would cover chapters one through five skipping sections 2.2, 2.3, 3.2, 4.7, 4.11, and 5.8. Then, in whatever time was left chapter seven on sorting was covered. The second author has taught the material to juniors who have had one quarter of FORTRAN or PASCAL and two quarters of introductory courses which themselves contain a potpourri of topics. In the first quarter's data structure course, chapters one through three are lightly covered and chapters four through six are completely covered. The second quarter starts with chapter seven which provides an excellent survey of the techniques which were covered in the previous quarter. Then the material on external sorting, symbol tables and files is sufficient for the remaining time. Note that the material in chapter 2 is largely mathematical and can be skipped without harm. The paradigm of class presentation that we have used is to begin each new topic with a problem, usually chosen from the computer science arena. Once defined, a high level design of its solution is made and each data structure is axiomatically specified. A tentative analysis is done to determine which operations are critical. Implementations of the data structures are then given followed by an attempt at verifying file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDob...Books_Algorithms_Collection2ed/books/book1/preface.htm (2 of 4)7/3/2004 3:56:18 PM www.itdevelopteam.com Fundamentals: PREFACE that the representation and specifications are consistent. The finishedalgorithm in the book is examined followed by an argument concerning its correctness. Then an analysis is done by determining the relevant parameters and applying some straightforward rules to obtain the correct computing time formula. In summary, as instructors we have tried to emphasize the following notions to our students: (i) the ability to define at a sufficiently high level of abstraction the data structures and algorithms that are needed; (ii) the ability to devise alternative implementations of a data structure; (iii) the ability to synthesize a correct algorithm; and (iv) the abilityto analyze the computing time of the resultant program. In addition there are two underlying currents which, though not explicitly emphasized are covered throughout. The first is the notion of writing nicely structured programs. For all of the programs contained herein we have tried our best to structure them appropriately. We hope that by reading programs with good style the students will pick up good writing habits. A nudge on the instructor's part will also prove useful. The second current is the choice of examples. We have tried to use those examples which prove a point well, have application to computer programming, and exhibit some of the brightest accomplishments in computer science. At the close of each chapter there is a list of references and selected readings. These are not meant to be exhaustive. They are a subset of those books and papers that we found to be the most useful. Otherwise, they are either historically significant or develop the material in the text somewhat further. Many people have contributed their time and energy to improve this book. For this we would like to thank them. We wish to thank Arvind [sic], T. Gonzalez, L. Landweber, J. Misra, and D. Wilczynski, who used the book in their own classes and gave us detailed reactions. Thanks are also due to A. Agrawal, M. Cohen, A. Howells, R. Istre, D. Ledbetter, D. Musser and to our students in CS 202, CSci 5121 and 5122 who provided many insights. For administrative and secretarial help we thank M. Eul, G. Lum, J. Matheson, S. Moody, K. Pendleton, and L. Templet. To the referees for their pungent yet favorable comments we thank S. Gerhart, T. Standish, and J. Ullman. Finally, we would like to thank our institutions, the University of Southern California and the University of Minnesota, for encouraging in every way our efforts to produce this book. Ellis Horowitz Sartaj Sahni Preface to the Ninth Printing We would like to acknowledge collectively all of the individuals who have sent us comments and corrections since the book first appeared. For this printing we have made many corrections and improvements. October 198l file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDob...Books_Algorithms_Collection2ed/books/book1/preface.htm (3 of 4)7/3/2004 3:56:18 PM www.itdevelopteam.com Fundamentals: PREFACE Ellis Horowitz Sartaj Sahni file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDob...Books_Algorithms_Collection2ed/books/book1/preface.htm (4 of 4)7/3/2004 3:56:18 PM www.itdevelopteam.com Fundamentals: CHAPTER 1: INTRODUCTION CHAPTER 1: INTRODUCTION 1.1 OVERVIEW The field of computer science is so new that one feels obliged to furnish a definition before proceeding with this book. One often quoted definition views computer science as the study of algorithms. This study encompasses four distinct areas: (i) machines for executing algorithms--this area includes everything from the smallest pocket calculator to the largest general purpose digital computer. The goal is to study various forms of machine fabrication and organization so that algorithms can be effectively carried out. (ii) languages for describing algorithms--these languages can be placed on a continuum. At one end are the languages which are closest to the physical machine and at the other end are languages designed for sophisticated problem solving. One often distinguishes between two phases of this area: language design and translation. The first calls for methods for specifying the syntax and semantics of a language. The second requires a means for translation into a more basic set of commands. (iii) foundations of algorithms--here people ask and try to answer such questions as: is a particular task accomplishable by a computing device; or what is the minimum number of operations necessary for any algorithm which performs a certain function? Abstract models of computers are devised so that these properties can be studied. (iv) analysis of algorithms--whenever an algorithm can be specified it makes sense to wonder about its behavior. This was realized as far back as 1830 by Charles Babbage, the father of computers. An algorithm's behavior pattern or performance profile is measured in terms of the computing time and space that are consumed while the algorithm is processing. Questions such as the worst and average time and how often they occur are typical. We see that in this definition of computer science, "algorithm" is a fundamental notion. Thus it deserves a precise definition. The dictionary's definition "any mechanical or recursive computational procedure" is not entirely satisfying since these terms are not basic enough. Definition: An algorithm is a finite set of instructions which, if followed, accomplish a particular task. In addition every algorithm must satisfy the following criteria: (i) input: there are zero or more quantities which are externally supplied; (ii) output: at least one quantity is produced; file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDo...Books_Algorithms_Collection2ed/books/book1/chap01.htm (1 of 38)7/3/2004 3:56:36 PM www.itdevelopteam.com Fundamentals: CHAPTER 1: INTRODUCTION (iii) definiteness: each instruction must be clear and unambiguous; (iv) finiteness: if we trace out the instructions of an algorithm, then for all cases the algorithm will terminate after a finite number of steps; (v) effectiveness: every instruction must be sufficiently basic that it can in principle be carried out by a person using only pencil and paper. It is not enough that each operation be definite as in (iii), but it must also be feasible. In formal computer science, one distinguishes between an algorithm, and a program. A program does not necessarily satisfy condition (iv). One important example of such a program for a computer is its operating system which never terminates (except for system crashes) but continues in a wait loop until more jobs are entered. In this book we will deal strictly with programs that always terminate. Hence, we will use these terms interchangeably. An algorithm can be described in many ways. A natural language such as English can be used but we must be very careful that the resulting instructions are definite (condition iii). An improvement over English is to couple its use with a graphical form of notation such as flowcharts. This form places each processing step in a "box" and uses arrows to indicate the next step. Different shaped boxes stand for different kinds of operations. All this can be seen in figure 1.1 where a flowchart is given for obtaining a Coca-Cola from a vending machine. The point is that algorithms can be devised for many common activities. Have you studied the flowchart? Then you probably have realized that it isn't an algorithm at all! Which properties does it lack? Returning to our earlier definition of computer science, we find it extremely unsatisfying as it gives us no insight as to why the computer is revolutionizing our society nor why it has made us re-examine certain basic assumptions about our own role in the universe. While this may be an unrealistic demand on a definition even from a technical point of view it is unsatisfying. The definition places great emphasis on the concept of algorithm, but never mentions the word "data". If a computer is merely a means to an end, then the means may be an algorithm but the end is the transformation of data. That is why we often hear a computer referred to as a data processing machine. Raw data is input and algorithms are used to transform it into refined data. So, instead of saying that computer science is the study of algorithms, alternatively, we might say that computer science is the study of data: (i) machines that hold data; (ii) languages for describing data manipulation; (iii) foundations which describe what kinds of refined data can be produced from raw data; file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDo...Books_Algorithms_Collection2ed/books/book1/chap01.htm (2 of 38)7/3/2004 3:56:36 PM www.itdevelopteam.com Fundamentals: CHAPTER 1: INTRODUCTION (iv) structures for representing data. Figure 1.1: Flowchart for obtaining a Coca-Cola There is an intimate connection between the structuring of data, and the synthesis of algorithms. In fact, a data structure and an algorithm should be thought of as a unit, neither one making sense without the other. For instance, suppose we have a list of n pairs of names and phone numbers (a1,b1)(a2,b2), ..., (an, bn), and we want to write a program which when given any name, prints that person's phone number. This task is called searching. Just how we would write such an algorithm critically depends upon how the names and phone numbers are stored or structured. One algorithm might just forge ahead and examine names, a1,a2,a3, ... etc., until the correct name was found. This might be fine in Oshkosh, but in Los Angeles, with hundreds of thousands of names, it would not be practical. If, however, we knew that the data was structured so that the names were in alphabetical order, then we could do much better. We could make up a second list which told us for each letter in the alphabet, where the first name with that letter appeared. For a name beginning with, say, S, we would avoid having to look at names beginning with other letters. So because of this new structure, a very different algorithm is possible. Other ideas for algorithms become possible when we realize that we can organize the data as we wish. We will discuss many more searching strategies in Chapters 7 and 9. Therefore, computer science can be defined as the study of data, its representation and transformation by a digital computer. The goal of this book is to explore many different kinds of data objects. For each object, we consider the class of operations to be performed and then the way to represent this object so that these operations may be efficiently carried out. This implies a mastery of two techniques: the ability to devise alternative forms of data representation, and the ability to analyze the algorithm which operates on that structure . The pedagogical style we have chosen is to consider problems which have arisen often in computer applications. For each problem we will specify the data object or objects and what is to be accomplished. After we have decided upon a representation of the objects, we will give a complete algorithm and analyze its computing time. After reading through several of these examples you should be confident enough to try one on your own. There are several terms we need to define carefully before we proceed. These include data structure, data object, data type and data representation. These four terms have no standard meaning in computer science circles, and they are often used interchangeably. A data type is a term which refers to the kinds of data that variables may "hold" in a programming language. In FORTRAN the data types are INTEGER, REAL, LOGICAL, COMPLEX, and DOUBLE PRECISION. In PL/I there is the data type CHARACTER. The fundamental data type of SNOBOL is the character string and in LISP it is the list (or S-expression). With every programming language there is a set of built-in data types. This means that the language allows variables to name data of that type and file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDo...Books_Algorithms_Collection2ed/books/book1/chap01.htm (3 of 38)7/3/2004 3:56:36 PM www.itdevelopteam.com Fundamentals: CHAPTER 1: INTRODUCTION provides a set of operations which meaningfully manipulates these variables. Some data types are easy to provide because they are already built into the computer's machine language instruction set. Integer and real arithmetic are examples of this. Other data types require considerably more effort to implement. In some languages, there are features which allow one to construct combinations of the built-in types. In COBOL and PL/I this feature is called a STRUCTURE while in PASCAL it is called a RECORD. However, it is not necessary to have such a mechanism. All of the data structures we will see here can be reasonably built within a conventional programming language. Data object is a term referring to a set of elements, say D. For example the data object integers refers to D = {0, 1, 2, ...}. The data object alphabetic character strings of length less than thirty one implies D = {",'A','B', ...,'Z','AA', ...}. Thus, D may be finite or infinite and if D is very large we may need to devise special ways of representing its elements in our computer. The notion of a data structure as distinguished from a data object is that we want to describe not only the set of objects, but the way they are related. Saying this another way, we want to describe the set of operations which may legally be applied to elements of the data object. This implies that we must specify the set of operations and show how they work. For integers we would have the arithmetic operations +, -, *, / and perhaps many others such as mod, ceil, floor, greater than, less than, etc. The data object integers plus a description of how +, -, *, /, etc. behave constitutes a data structure definition. To be more precise lets examine a modest example. Suppose we want to define the data structure natural number (abbreviated natno) where natno = {0,1,2,3, ...} with the three operations being a test for zero addition and equality. The following notation can be used: structure NATNO 1 declare ZERO( ) natno 2 ISZERO(natno) boolean 3 SUCC(natno) natno 4 ADD(natno, natno) natno 5 EQ(natno, natno) boolean 6 for all x, y natno let 7 ISZERO(ZERO) ::= true; ISZERO(SUCC(x)) ::= false file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDo...Books_Algorithms_Collection2ed/books/book1/chap01.htm (4 of 38)7/3/2004 3:56:36 PM www.itdevelopteam.com Fundamentals: CHAPTER 1: INTRODUCTION 8 ADD(ZERO, y) :: = y, ADD(SUCC(x), y) :: = SUCC(ADD(x, y)) 9 EQ(x, ZERO) :: = if ISZERO(x) then true else false 10 EQ(ZERO, SUCC(y)) :: = false EQ(SUCC(x), SUCC(y)) :: = EQ(x, y) 11 end end NATNO In the declare statement five functions are defined by giving their names, inputs and outputs. ZERO is a constant function which means it takes no input arguments and its result is the natural number zero, written as ZERO. ISZERO is a boolean function whose result is either true or false. SUCC stands for successor. Using ZERO and SUCC we can define all of the natural numbers as: ZERO, l = SUCC (ZERO), 2 = SUCC(SUCC(ZERO)), 3 = SUCC(SUCC(SUCC(ZERO))), ... etc. The rules on line 8 tell us exactly how the addition operation works. For example if we wanted to add two and three we would get the following sequence of expressions: ADD(SUCC(SUCC(ZERO)),SUCC(SUCC(SUCC(ZERO)))) which, by line 8 equals SUCC(ADD(SUCC(ZERO),SUCC(SUCC(SUCC(ZERO))))) which, by line 8 equals SUCC(SUCC(ADD(ZERO,SUCC(SUCC(SUCC(ZERO)))))) which by line 8 equals SUCC(SUCC(SUCC(SUCC(SUCC(ZERO))))) Of course, this is not the way to implement addition. In practice we use bit strings which is a data structure that is usually provided on our computers. But however the ADD operation is implemented, it must obey these rules. Hopefully, this motivates the following definition. Definition: A data structure is a set of domains , a designated domain , a set of functions and a file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDo...Books_Algorithms_Collection2ed/books/book1/chap01.htm (5 of 38)7/3/2004 3:56:36 PM www.itdevelopteam.com Fundamentals: CHAPTER 1: INTRODUCTION set of axioms . The triple denotes the data structure d and it will usually be abbreviated by writing d. In the previous example The set of axioms describes the semantics of the operations. The form in which we choose to write the axioms is important. Our goal here is to write the axioms in a representation independent way. Then, we discuss ways of implementing the functions using a conventional programming language. An implementation of a data structure d is a mapping from d to a set of other data structures e. This mapping specifies how every object of d is to be represented by the objects of e. Secondly, it requires that every function of d must be written using the functions of the implementing data structures e. Thus we say that integers are represented by bit strings, boolean is represented by zero and one, an array is represented by a set of consecutive words in memory. In current parlance the triple is referred to as an abstract data type. It is called abstract precisely because the axioms do not imply a form of representation. Another way of viewing the implementation of a data structure is that it is the process of refining an abstract data type until all of the operations are expressible in terms of directly executable functions. But at the first stage a data structure should be designed so that we know what it does, but not necessarily how it will do it. This division of tasks, called specification and implementation, is useful because it helps to control the complexity of the entire process. 1.2 SPARKS The choice of an algorithm description language must be carefully made because it plays such an important role throughout the book. We might begin by considering using some existing language; some names which come immediately to mind are ALGOL, ALGOL-W, APL, COBOL, FORTRAN, LISP, PASCAL, PL/I, SNOBOL. Though some of these are more preferable than others, the choice of a specific language leaves us with many difficulties. First of all, we wish to be able to write our algorithms without dwelling on the idiosyncracies of a given language. Secondly, some languages have already provided the mechanisms we wish to discuss. Thus we would have to make pretense to build up a capability which already exists. Finally, each language has its followers and its detractors. We would rather not have any individual rule us out simply because he did not know or, more particularly, disliked to use the language X. Furthermore it is not really necessary to write programs in a language for which a compiler exists. Instead we choose to use a language which is tailored to describing the algorithms we want to write. file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDo...Books_Algorithms_Collection2ed/books/book1/chap01.htm (6 of 38)7/3/2004 3:56:36 PM www.itdevelopteam.com Fundamentals: CHAPTER 1: INTRODUCTION Using it we will not have to define many aspects of a language that we will never use here. Most importantly, the language we use will be close enough to many of the languages mentioned before so that a hand translation will be relatively easy to accomplish. Moreover, one can easily program a translator using some existing, but more primitive higher level language as the output (see Appendix A). We call our language SPARKS. Figure 1.2 shows how a SPARKS program could be executed on any machine. Figure 1.2: Translation of SPARKS Many language designers choose a name which is an acronym. But SPARKS was not devised in that way; it just appeared one day as Athena sprang from the head of Zeus. Nevertheless, computerniks still try to attach a meaning. Several cute ideas have been suggested, such as Structured Programming: A Reasonably Komplete Set or Smart Programmers Are Required To Know SPARKS. SPARKS contains facilities to manipulate numbers, boolean values and characters. The way to assign values is by the assignment statement variable expression. In addition to the assignment statement, SPARKS includes statements for conditional testing, iteration, input-output, etc. Several such statements can be combined on a single line if they are separated by a semi-colon. Expressions can be either arithmetic, boolean or of character type. In the boolean case there can be only one of two values, true or false. In order to produce these values, the logical operators and, or, not are provided, plus the relational operators file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDo...Books_Algorithms_Collection2ed/books/book1/chap01.htm (7 of 38)7/3/2004 3:56:36 PM www.itdevelopteam.com Fundamentals: CHAPTER 1: INTRODUCTION A conditional statement has the form if cond then S1 if cond then S1 or else S2 where cond is a boolean expression and S1, S2 are arbitrary groups of SPARKS statements. If S1 or S2 contains more than one statement, these will be enclosed in square brackets. Brackets must be used to show how each else corresponds to one if. The meaning of this statement is given by the flow charts: We will assume that conditional expressions are evaluated in "short circuit" mode; given the boolean expression (cond1 or cond2), if condl is true then cond2 is not evaluated; or, given (condl and cond2), if cond1 is false then cond2 is not evaluated. To accomplish iteration, several statements are available. One of them is while cond do S end where cond is as before, S is as S1 before and the meaning is given by It is well known that all "proper" programs can be written using only the assignment, conditional and while statements. This result was obtained by Bohm and Jacopini. Though this is very interesting from a theoretical viewpoint, we should not take it to mean that this is the way to program. On the contrary, the more expressive our languages are, the more we can accomplish easily. So we will provide other statements such as a second iteration statement, the repeat-until, repeat S file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDo...Books_Algorithms_Collection2ed/books/book1/chap01.htm (8 of 38)7/3/2004 3:56:36 PM www.itdevelopteam.com Fundamentals: CHAPTER 1: INTRODUCTION until cond which has the meaning In contrast to the while statement, the repeat-until guarantees that the statements of S will be executed at least once. Another iteration statement is loop S forever which has the meaning As it stands, this describes an infinite loop! However, we assume that this statement is used in conjunction with some test within S which will cause an exit. One way of exiting such a loop is by using a go to label statement which transfers control to "label." Label may be anywhere in the procedure. A more restricted form of the go to is the command exit which will cause a transfer of control to the first statement after the innermost loop which contains it. This looping statement may be a while, repeat, for or a loop-forever. exit can be used either conditionally or unconditionally, for instance loop S1 if cond then exit S2 file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDo...Books_Algorithms_Collection2ed/books/book1/chap01.htm (9 of 38)7/3/2004 3:56:36 PM www.itdevelopteam.com Fundamentals: CHAPTER 1: INTRODUCTION forever which will execute as The last statement for iteration is called the for-loop, which has the form for vble start to finish by increment do S end Vble is a variable, while start, finish and increment are arithmetic expressions. A variable or a constant is a simple form of an expression. The clause "by increment" is optional and taken as +1 if it does not occur. We can write the meaning of this statement in SPARKS as vble start fin finish incr increment while (vble - fin) * incr 0 do S vble vble + incr end Another statement within SPARKS is the case, which allows one to distinguish easily between several alternatives without using multiple if-then-else statements. It has the form where the Si, 1 i n + 1 are groups of SPARKS statements. The semantics is easily described by the file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrD...ooks_Algorithms_Collection2ed/books/book1/chap01.htm (10 of 38)7/3/2004 3:56:36 PM www.itdevelopteam.com