Fundamentals: APPENDIX A: SPARKS . APPENDIX C: ALGORITHM INDEX BY CHAPTER. .. DATA REPRESENTATIONS FOR STRINGS. Fundamentals of Data Structures in C Horowitz PDF Fundamentals Of Data Structures In C Author: Ellis Horowitz, Anderson-Freed, Sahni other link other link . Throughout this book we will assume a knowledge of C [10]. The C Programming Language, 2d ed mk:@MSITStore:K:\lesforgesdessalles.info Principles of Data Structures.

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PDF generated using the open source mwlib toolkit. Fundamental Data algorithms, to classify and evaluate data structures, and to formally describe the type .. As an example, here is an implementation of the stack ADT above in the C. Fundamentals of Data Structures - Ellis Horowitz, Sartaj lesforgesdessalles.info - Ebook download as PDF file:///C|/E%20Drive%20Data/My%20Books/Algorithm/DrDob . Fundamentals of Data Structures - Ellis Horowitz & Sartaj Sahni - Ebook download as PDF File .pdf), Text File .txt) or read Download as PDF, TXT or read online from Scribd . file:///C|/E%20Drive%20Data/My%20Books/Algorithm/ DrDob.

In each iteration of the while loop of lines either the value of i or j or of both increases by 1 or i and col are reset. Many of these are based on the strange "L-shaped" move of the knight. Write a recursive procedure to compute powerset S. We now have a matrix transpose algorithm which we believe is correct and which has a computing time of O nt. These axioms are valuable in that they describe the meaning of each operation concisely and without implying an implementation. A maximum of

One often distinguishes between two phases of this area: 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.

Abstract models of computers are devised so that these properties can be studied. This was realized as far back as 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. An algorithm is a finite set of instructions which, if followed, accomplish a particular task. In addition every algorithm must satisfy the following criteria: 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.

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: Figure 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 , 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, 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 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. 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.

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. 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. 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. To be more precise lets examine a modest example.

The following notation can be used: SUCC stands for successor. 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: 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.

A data structure is a set of domains , a designated domain , a set of functions and a end. Instead we choose to use a language which is tailored to describing the algorithms we want to write. 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.

This division of tasks. This mapping specifies how every object of d is to be represented by the objects of e. We might begin by considering using some existing language. Thus we would have to make pretense to build up a capability which already exists. In current parlance the triple is referred to as an abstract data type. An implementation of a data structure d is a mapping from d to a set of other data structures e.

It is called abstract precisely because the axioms do not imply a form of representation. Furthermore it is not really necessary to write programs in a language for which a compiler exists. We would rather not have any individual rule us out simply because he did not know or.

Our goal here is to write the axioms in a representation independent way. Though some of these are more preferable than others.. First of all. The triple denotes the data structure d and it will usually be abbreviated by writing In the previous example The set of axioms describes the semantics of the operations. Thus we say that integers are represented by bit strings. But at the first stage a data structure should be designed so that we know what it does.

The form in which we choose to write the axioms is important. The way to assign values is by the assignment statement variable expression. In the boolean case there can be only one of two values. Expressions can be either arithmetic. In order to produce these values. Several such statements can be combined on a single line if they are separated by a semi-colon. Most importantly. In addition to the assignment statement. Several cute ideas have been suggested.

To accomplish iteration. Brackets must be used to show how each else corresponds to one if. The meaning of this statement is given by the flow charts: One of them is while cond do S end where cond is as before. So we will provide other statements such as a second iteration statement. If S1 or S2 contains more than one statement. Though this is very interesting from a theoretical viewpoint.

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.. This result was obtained by Bohm and Jacopini. On the contrary. Another iteration statement is loop S forever which has the meaning As it stands.

This looping statement may be a while. One way of exiting such a loop is by using a go to label statement which transfers control to "label. 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.. The semantics is easily described by the file: It has the form where the Si. A variable or a constant is a simple form of an expression.. We can write the meaning of this statement in SPARKS as vble fin incr start finish increment 0 do start to finish by increment do while vble.

This may be somewhat restrictive in practice.. This penalty will not deter us from using recursion. Many such programs are easily translatable so that the recursion is removed and efficiency achieved. The execution of an end at the end of procedure implies a return. The expr may be omitted in which case a return is made to the calling procedure. All procedures are treated as external.

The else clause is optional.. Parameters which are constants or values of expressions are stored into internally generated words whose addresses are then passed to the procedure.

The association of actual to formal parameters will be handled using the call by reference rule. Though recursion often carries with it a severe penalty at execution time. A procedure may be invoked by using a call statement call NAME parameter list Procedures may call themselves. This means that at run time the address of each parameter is passed to the called procedure.

The SPARKS language is rich enough so that one can create a good looking program by applying some simple rules of style. The command stop halts execution of the currently executing procedure. These are often useful features and when available they should be used. This is a goal which should be aimed at by everyone who writes programs. Comments may appear anywhere on a line enclosed by double slashes. We avoid the problem of defining a "format" statement as we will need only the simplest form of input and output.

An n-dimensional array A with lower and upper bounds li. We have avoided introducing the record or structure concept. Avoid sentences like ''i is increased by one. See the book The Elements of Programming Style by Kernighan and Plauger for more examples of good rules of programming. Try to write down a rigorous description of the input and output which covers all cases. To understand this process better. For each object there will be some basic operations to perform on it such as print the maze.

Finally you produce a complete version of your first program. You must now choose representations for your data objects a maze as a two dimensional array of zeros and ones. You are now ready to proceed to the design phase.. It is often at this point that one realizes that a much better program could have been built. In fact. But to improve requires that you apply some discipline to the process of creating programs.. We hope your productivity will be greater. Designing an algorithm is a task which can be done independently of the programming language you eventually plan to use.

If you have been careful about keeping track of your previous work it may not be too difficult to make changes. This method uses the philosophy: Modern pedagogy suggests that all processing which is independent of the data representation be written out first.

You may have several data objects such as a maze. Make sure you understand the information you are given the input and what results you are to produce the output. Assume that these operations already exist in the form of procedures and write an algorithm which solves the problem according to the requirements.

You should consider alternatives. The order in which you do this may be crucial. Use a notation which is natural to the way you wish to describe the order of processing. One of the criteria of a good design is file: Perhaps you should have chosen the second design alternative or perhaps you have spoken to a friend who has done it better.

By postponing the choice of how the data is stored we can try to isolate what operations depend upon the choice of data representation. It may already be possible to tell if one will be more desirable than the other. If you can't distinguish between the two. Can you think of another algorithm? If so.

This happens to industrial programmers as well. B and C. As a minimal requirement. The proof can't be completed until these are changed. Finally there may be tools available at your computing center to aid in the testing process.

Testing is the art of creating sample data upon which to run your program. This is a phenomenon which has been observed in practice. Many times during the proving process errors are discovered in the code. One thing you have forgotten to do is to document. In fact you may save as much debugging time later on by doing a new version now. Different situations call for different decisions. If a correct proof can be obtained. One proof tells us more than any finite amount of testing.

It is usually hard to decide whether to sacrifice this first attempt and begin again or just continue to get the first version working. For each subsequent compiler their estimates became closer to the truth. Figure Before executing your program you should attempt to prove it is correct. This is another use of program proving. But prior experience is definitely helpful and the time to build the third compiler was less than one fifth that for the first one.

The graph in figure 1. If the program fails to respond correctly then debugging is needed to determine what went wrong and how to correct it. Unwarrented optimism is a familiar disease in computing. Each of these is an art in itself. For each compiler there is the time they estimated it would take them and the time it actually took. The larger the software. If you have written more than a few procedures.

The previous discussion applies to the construction of a single procedure as well as to the writing of a large software system. If you do decide to scrap your work and begin again. This shifts our emphasis away from the management and integration of the file: But why bother to document until the program is entirely finished and correct?

Because for each procedure you made some assumptions about its input and output. Proofs about programs are really no different from any other kinds of proofs. Verification consists of three distinct aspects: If you note them down with the code. Let us concentrate for a while on the question of developing a single procedure which solves a specific task. One such tool instruments your source code and then tells you for every data set: If possible the designer attempts to partition the solution into logical subtasks.

Suppose we devise a program for sorting a set of n given by the following 1 distinct integers. One solution is to store the values in an array in such a way that the i-th integer is stored in the i-th array position.

At this level the formulation is said to be abstract because it contains no details regarding how the objects will be represented and manipulated in a computer. The initial solution may be expressed in English or some form of mathematical notation. The design process consists essentially of taking a proposed solution and successively refining it until an executable program is achieved. This is referred to as the bottom-up approach.

One of the simplest solutions is "from those integers which remain unsorted. There now remain two clearly defined subtasks: Let us examine two examples of top-down program development. This method of design is called the top-down approach. We are now ready to give a second refinement of the solution: Underlying all of these strategies is the assumption that a language exists for adequately describing the processing of data at several abstract levels.

A look ahead to problems which may arise later is often useful. Each subtask is similarly decomposed until all tasks are expressed within a programming language. Experience suggests that the top-down approach should be followed when creating a program. This latter problem can be solved by the code file: Eventually A n is compared to the current minimum and we are done. We first note that for any i. From the standpoint of readability we can ask if this program is good. A j t The first subtask can be solved by assuming the minimum is A i.

We observe at this point that the upper limit of the for-loop in line 1 can be changed to n. Is there a more concise way of describing this algorithm which will still be as easy to comprehend?

Substituting while statements for the for loops doesn't significantly change anything.. Continue in this way by keeping two pointers. We assume that we have n 1 distinct integers which are already sorted and stored in the array A 1: Let us develop another program.

By making use of the fact that the set is sorted we conceive of the following efficient method: There are three possibilities. There are many more that we might produce which would be incorrect. Part of the freedom comes from the initialization step. In fact there are at least six different binary search programs that can be produced which are all correct.

Whichever version we choose.. Note how at each stage the number of elements in the remaining set is decreased by about one half. Below is one complete version. This method is referred to as binary search. For instance we could replace the while loop by a repeat-until statement with the same English condition. The procedure name and its parameters file: Recursion We have tried to emphasize the need to structure a program to make it easier to achieve the goals of readability and correctness.

Actually one of the most useful syntactical features for accomplishing this is the procedure. As we enter this loop and as long as x is not found the following holds: Given a set of instructions which perform a logical operation. Unfortunately a complete proof takes us beyond our scope but for those who wish to pursue program proving they should consult our references at the end of this chapter.

These recursive mechanisms are extremely powerful.

This is unfortunate because any program that can be written using assignment. For these reasons we introduce recursion here. Given the input-output specifications of a procedure..

What this fails to stress is the fact that procedures may call themselves direct recursion before they are done or they may call other procedures which again invoke the calling procedure indirect recursion.. Most students of computer science view recursion as a somewhat mystical technique which only is useful for some very special class of problems such as computing factorials or Ackermann's function.

Factorial fits this category. When is recursion an appropriate mechanism for algorithm exposition? One instance is when the problem itself is recursively defined. This view of the procedure implies that it is invoked.. Of course. A simple algorithm can be achieved by looking at the case of four elements a. The answer is obtained by printing i a followed by all permutations of b. Then try to do one or more of the exercises at the end of this chapter which ask for recursive procedures.

Given a set of n 1 elements the problem is to print all possible permutations of this set. A is a character string e. B file: It is easy to see that given n elements there are n! It implies that we can solve the problem for a set with n elements if we had an algorithm which worked on n. Another instance when recursion is invaluable is when we want to describe a backtracking procedure.

Every place where a ''go to label'' appears. Suppose we start with the sorting algorithm presented in this section. This may sound strange.. The main purpose is to make one more familiar with the execution of a recursive procedure. But for now we will content ourselves with examining some simple. This gives us the following set of three procedures. We will see several important examples of such structures. To rewrite it recursively the first thing we do is to remove the for loops and express the algorithm using assignment.

Procedure MAXL2 is also directly reculsive. These two procedures use eleven lines while the original iterative version was expressed in nine lines. The effect of increasing k by one and restarting the procedure has essentially the same effect as the for loop.. Notice how in MAXL2 the fourth parameter k is being changed. Thus a recursive call of a file: Now let us trace the action of these procedures as they sort a set of five integers When a procedure is invoked an implicit branch to its beginning is made.

There are many criteria upon which we can judge a program. We would like to determine two numbers for this statement. Rules are also given there for eliminating recursion. Hopefully this more subtle approach will gradually infect your own program writing habits so that you will automatically strive to achieve these goals. The product of these numbers will be the total time taken by this statement.

The parameter mechanism of the procedure is a form of assignment. The first is the amount of time a single execution will take. Though we will not be discussing how to reach these goals. Also in that section are several recursive procedures. Both of these are equally important. Performance evaluation can be loosely divided into 2 major phases: These have to do with computing time and storage requirements of the algorithms.

First consider a priori estimation. In section 4. There are other criteria for judging programs which have a more direct relationship to performance. The above criteria are all vitally important when it comes to writing software. The second statistic is called the frequency count. Parallelism will not be considered.

One of the hardest tasks in estimating frequency counts is to choose adequate samples of data. Neither of these alternatives seems attractive. It is possible to determine these figures by choosing a real machine and an existing compiler. Consider the three examples of Figure 1.

It is impossible to determine exactly how much time it takes to execute any command unless we have the following information: All these considerations lead us to limit our goals for an a priori analysis.

In both cases the exact times we would determine would not apply to many machines or to any machine. The anomalies of machine configuration and language will be lumped together when we do our experimental studies. Another approach would be to define a hypothetical machine with imaginary execution times. In our analysis of execution we will be concerned chiefly with determining the order of magnitude of an algorithm.

Then its frequency count is one. This means determining those statements which may have the greatest frequency count. In general To clarify some of these ideas. Each new term is obtained by taking the sum of the two previous terms. To determine the order of magnitude. The Fibonacci sequence starts as 0.

In the program segment of figure 1. Three simple programs for frequency counting. The program on the following page takes any non-negative integer n and prints the value Fn. In program b the same statement will be executed n times and in program c n2 times assuming n 1. Now 1. A complete set would include four cases: Below is a table which summarizes the frequency counts for the first three cases.

These may have different execution counts. None of them exercises the program very much. We can summarize all of this with a table. Though 2 to n is only n.

Step Frequency Step Frequency 2 3 4 5 6 7 1 1 1 0 1 0 1 9 10 11 12 13 14 15 2 n n-1 n-1 n-1 n-1 1 file: At this point the for loop will actually be entered. Both commands in step 9 are executed once. Steps If an algorithm takes time O log n it is faster. If we have two algorithms which perform the same task.

O log n. For example. O n2 is called quadratic. For example n might be the number of inputs or the number of outputs or their sum or the magnitude of one of them. Execution Count for Computing Fn Each statement is counted once. This notation means that the order of magnitude is proportional to n. The for statement is really a combination of several statements. We will often write this as O n.

These seven computing times.. O n3 is called cubic. O n is called linear. We write O 1 to mean a computing time which is a constant. When we say that the computing time of an algorithm is O g n we mean that its execution takes no more than a constant times g n. The reason for this is that as n increases the time for the second algorithm will get far worse than the time for the first.

O n log n is better than O n2 but not as good as O n. Often one can trade space for time. For large data sets.. In practice these constants depend on many factors. Notice how the times O n and O n log n grow much more slowly than the others.

Another valid performance measure of an algorithm is the space it requires. An algorithm which is exponential will work only for very small inputs. Figures On the other hand.

Using big-oh notation. For exponential algorithms. This shows why we choose the algorithm with the smaller order of magnitude. Given an algorithm. For small data sets. Then a performance profile can be gathered using real time calculation. We will see cases of this in subsequent chapters. Coxeter has given a simple rule for generating a magic square: When n is odd H. For example.. A magic square is an n x n matrix of the integers 1 to n2 such that the sum of every row. The statement i.

The file: It emphasizes that the variables are thought of as pairs and are changed as a unit. For a discussion of tools and procedures for developing very large software systems see Practical Strategies for Developing Large Software Systems. Academic Press. The magic square is represented using a two dimensional array having n rows and n column. Fundamental Algorithms. The while loop is governed by the variable key which is an integer variable initialized to 2 and increased by one each time through the loop.

For this application it is convenient to number the rows and columns from zero to n. Thus each statement within the while loop will be executed no more than n2.

The Elements of Programming Style by B. Kernighan and P. Special Issue: Since there are n2 positions in which the algorithm must place a number. For a discussion of the more abstract formulation of data structures see "Toward an understanding of data structures" by J. For a further discussion of program proving see file: ACM Computing Surveys. For a discussion of good programming techniques see Structured Programming by O.

Both do not satisfy one of the five criteria of an algorithm. Describe the flowchart in figure 1. Can you think of a clever meaning for S. Concentrate on the letter K first. How would you handle people with the same last name. Consider the two statements: Look up the word algorithm or its older form algorism in the dictionary. Discuss how you would actually represent the list of name and telephone number pairs in a real machine.

American Mathematical Society. Can you do this without using the go to? Now make it into an algorithm. Which criteria do they violate?

For instance.. String x is unchanged. If x occurs The rule is: Try writing this without using the go to statement.

What is the computing time of your method? Given n boolean variables x Determine when the second becomes larger than the first. Determine how many times each statement is executed. Strings x and y remain unchanged. Implement these procedures using the array facility.

NOT X:: List as many rules of style in programming that you can think of that you would be willing to follow yourself. Take any version of binary search. Prove by induction: Represent your answer in the array ANS 1: Trace the action of the procedure below on the elements 2. Using the notation introduced at the end of section 1. Write a recursive procedure which prints the sequence of moves which accomplish this task. Write a recursive procedure for computing this function.

Ackermann's function A m. Given n. Then write a nonrecursive algorithm for computing Ackermann's function. The pigeon hole principle states that if a function f has n distinct inputs but less than n distinct outputs then there exists two inputs a. The disks are in order of decreasing diameter as one scans up the tower. If S is a set of n elements the powerset of S is the set of all possible subsets of S.

Write a recursive procedure to compute powerset S. Tower of Hanoi There are three towers and sixty four disks of different diameters placed on the first tower. Analyze the time and space requirements of your algorithm. Analyze the computing time of procedure SORT as given in section 1. Give an algorithm which finds the values a.

Write a recursive procedure for computing the binomial coefficient where. Monks were reputedly supposed to move the disks from tower 1 to tower 3 obeying the rules: For each index which is defined. For arrays this means we are concerned with only two operations which retrieve and store values. This is unfortunate because it clearly reveals a common point of confusion. STORE is used to enter new index-value pairs.

The array is often the only means for structuring data which is provided in a programming language. Therefore it deserves a significant amount of attention. If one asks a group of programmers to define an array.

It is true that arrays are almost always implemented by using consecutive memory. In mathematical terms we call this a correspondence or a mapping. Using our notation this object can be defined as: If we restrict the index values to be integers. If we interpret the indices to be n-dimensional.. There are a variety of operations that are performed on these lists. Ace or the floors of a building basement. In section 2. If we consider an ordered list more abstractly.. Notice how the axioms are independent of any representation scheme.

These operations include: This we will refer to as a sequential mapping. It is only operations v and vi which require real effort The problem calls for building a set of subroutines which allow for the manipulation of symbolic polynomials. In the study of data structures we are interested in ways of representing ordered lists so that these operations can be carried out efficiently.. Let us jump right into a problem requiring ordered lists which we will solve by using one dimensional arrays.

We can access the list element values in either direction by changing the subscript values in a controlled way. This problem has become the classical example for motivating the use of list processing techniques which we will see in later chapters It is precisely this overhead which leads us to consider nonsequential mappings of ordered lists into arrays in Chapter 4. By "symbolic. It is not always necessary to be able to perform all of these operations.

Perhaps the most common way to represent an ordered list is by an array where we associate the list element ai with the array index i. This gives us the ability to retrieve or modify the values of random elements in the list in a constant amount of time.. Insertion and deletion using sequential allocation forces us to move some of the remaining elements so the sequential mapping is preserved in its proper form.

MULT poly. We will also need input and output routines and some suitable format for preparing polynomials as input. For a mathematician a polynomial is a sum of terms where each term has the form axe. A complete specification of the data structure polynomial is now given. When defining a data object one must decide what functions will be available. The first step is to consider how to define polynomials as a computer structure.

However this is not an appropriate definition for our purposes. These assumptions are decisions of representation. Notice the absense of any assumptions about the order of exponents. Then we would write REM P. Suppose we wish to remove from P those terms having exponent one. Now we can make some representation decisions. COEF B. EXP B.

Note how trivial the addition and multiplication operations have become. Exponents should be unique and in decreasing order is a very reasonable first decision. EXP B file: These axioms are valuable in that they describe the meaning of each operation concisely and without implying an implementation.

Now assuming a new function EXP poly exp which returns the leading exponent of poly. EXP A. This representation leads to very simple algorithms for addition and multiplication. COEF A.

EXP B: C A end end insert any remaining terms in A or B into C The basic loop of this algorithm consists of merging the terms of the two polynomials. The case statement determines how the exponents are related and performs the proper action.

We have avoided the need to explicitly store the exponent of each term and instead we can deduce its value by knowing our position in the list and the degree. B REM B.

COEF A EXP A.. Since the tests within the case statement require two terms. But are there any disadvantages to this representation? Hopefully you have already guessed the worst one.

Given a set of n 1 elements the problem is to print all possible permutations of this set. A simple algorithm can be achieved by looking at the case of four elements a. Another instance when recursion is invaluable is when we want to describe a backtracking procedure. Suppose we start with the sorting algorithm presented in this section.

The main purpose is to make one more familiar with the execution of a recursive procedure. We will see several important examples of such structures. This may sound strange. To rewrite it recursively the first thing we do is to remove the for loops and express the algorithm using assignment. This gives us the following set of three procedures.

Every place where a ''go to label'' appears. But for now we will content ourselves with examining some simple. The effect of increasing k by one and restarting the procedure has essentially the same effect as the for loop.

Procedure MAXL2 is also directly reculsive. Notice how in MAXL2 the fourth parameter k is being changed. Thus a recursive call of a file: These two procedures use eleven lines while the original iterative version was expressed in nine lines.

Now let us trace the action of these procedures as they sort a set of five integers When a procedure is invoked an implicit branch to its beginning is made. These have to do with computing time and storage requirements of the algorithms. Performance evaluation can be loosely divided into 2 major phases: Though we will not be discussing how to reach these goals. The second statistic is called the frequency count. We would like to determine two numbers for this statement.

There are many criteria upon which we can judge a program. First consider a priori estimation. Rules are also given there for eliminating recursion. Also in that section are several recursive procedures. The parameter mechanism of the procedure is a form of assignment. The first is the amount of time a single execution will take.

The product of these numbers will be the total time taken by this statement. There are other criteria for judging programs which have a more direct relationship to performance. Hopefully this more subtle approach will gradually infect your own program writing habits so that you will automatically strive to achieve these goals.

In section 4. Both of these are equally important. The above criteria are all vitally important when it comes to writing software.

It is impossible to determine exactly how much time it takes to execute any command unless we have the following information: Parallelism will not be considered. One of the hardest tasks in estimating frequency counts is to choose adequate samples of data. All these considerations lead us to limit our goals for an a priori analysis.

It is possible to determine these figures by choosing a real machine and an existing compiler. The anomalies of machine configuration and language will be lumped together when we do our experimental studies.

Neither of these alternatives seems attractive. Consider the three examples of Figure 1. Another approach would be to define a hypothetical machine with imaginary execution times.

In both cases the exact times we would determine would not apply to many machines or to any machine.. Each new term is obtained by taking the sum of the two previous terms. Then its frequency count is one.. In the program segment of figure 1.

In program b the same statement will be executed n times and in program c n2 times assuming n 1. The program on the following page takes any non-negative integer n and prints the value Fn. In our analysis of execution we will be concerned chiefly with determining the order of magnitude of an algorithm.

Three simple programs for frequency counting. Now 1. To determine the order of magnitude. This means determining those statements which may have the greatest frequency count.

The Fibonacci sequence starts as 0. In general To clarify some of these ideas. Below is a table which summarizes the frequency counts for the first three cases.

A complete set would include four cases: None of them exercises the program very much. Both commands in step 9 are executed once. We can summarize all of this with a table. At this point the for loop will actually be entered. These may have different execution counts. Step Frequency Step Frequency 1 1 9 2 2 1 10 n 3 1 11 n-1 4 0 12 n-1 5 1 13 n-1 6 0 14 n-1 7 1 15 1 file: Though 2 to n is only n. Steps 1. Execution Count for Computing Fn Each statement is counted once.

O n2 is called quadratic. If we have two algorithms which perform the same task. The reason for this is that as n increases the time for the second algorithm will get far worse than the time for the first. When we say that the computing time of an algorithm is O g n we mean that its execution takes no more than a constant times g n. If an algorithm takes time O log n it is faster. For example n might be the number of inputs or the number of outputs or their sum or the magnitude of one of them.

O n is called linear. O log n. For example. We will often write this as O n. O n3 is called cubic. The for statement is really a combination of several statements. We write O 1 to mean a computing time which is a constant.

These seven computing times. This notation means that the order of magnitude is proportional to n. O n log n is better than O n2 but not as good as O n. Given an algorithm. For small data sets. Using big-oh notation. We will see cases of this in subsequent chapters. Then a performance profile can be gathered using real time calculation. For exponential algorithms. Another valid performance measure of an algorithm is the space it requires.

Often one can trade space for time. For large data sets. Figures 1. On the other hand. This shows why we choose the algorithm with the smaller order of magnitude. Notice how the times O n and O n log n grow much more slowly than the others.. An algorithm which is exponential will work only for very small inputs. In practice these constants depend on many factors. When n is odd H.. A magic square is an n x n matrix of the integers 1 to n2 such that the sum of every row.

Coxeter has given a simple rule for generating a magic square: The statement i. The file: It emphasizes that the variables are thought of as pairs and are changed as a unit. ACM Computing Surveys. For a discussion of tools and procedures for developing very large software systems see Practical Strategies for Developing Large Software Systems. The Elements of Programming Style by B. Since there are n2 positions in which the algorithm must place a number.

For a discussion of the more abstract formulation of data structures see "Toward an understanding of data structures" by J. Thus each statement within the while loop will be executed no more than n2. The magic square is represented using a two dimensional array having n rows and n column. For this application it is convenient to number the rows and columns from zero to n.

For a discussion of good programming techniques see Structured Programming by O. Academic Press. Kernighan and P. For a further discussion of program proving see file: Fundamental Algorithms. Special Issue: The while loop is governed by the variable key which is an integer variable initialized to 2 and increased by one each time through the loop.

Both do not satisfy one of the five criteria of an algorithm. Describe the flowchart in figure 1. Can you think of a clever meaning for S. Concentrate on the letter K first. American Mathematical Society. Discuss how you would actually represent the list of name and telephone number pairs in a real machine. Consider the two statements: Which criteria do they violate? Look up the word algorithm or its older form algorism in the dictionary.

Can you do this without using the go to? Now make it into an algorithm. How would you handle people with the same last name. Determine how many times each statement is executed. Determine when the second becomes larger than the first.. If x occurs. For instance. Given n boolean variables x1. Try writing this without using the go to statement. Implement these procedures using the array facility The rule is: String x is unchanged.

What is the computing time of your method? Strings x and y remain unchanged. NOT X:: Prove by induction: Trace the action of the procedure below on the elements 2. Using the notation introduced at the end of section 1. List as many rules of style in programming that you can think of that you would be willing to follow yourself.

Represent your answer in the array ANS 1: Take any version of binary search. If S is a set of n elements the powerset of S is the set of all possible subsets of S. This function is studied because it grows very fast for small values of m and n.

Write a recursive procedure for computing this function. Write a recursive procedure to compute powerset S. Tower of Hanoi There are three towers and sixty four disks of different diameters placed on the first tower.

Monks were reputedly supposed to move the disks from tower 1 to tower 3 obeying the rules: Write a recursive procedure for computing the binomial coefficient as defined in section 1. Analyze the time and space requirements of your algorithm.. Given n. Ackermann's function A m. The pigeon hole principle states that if a function f has n distinct inputs but less than n distinct outputs then there exists two inputs a. Analyze the computing time of procedure SORT as given in section 1.

Then write a nonrecursive algorithm for computing Ackermann's function. The disks are in order of decreasing diameter as one scans up the tower. Give an algorithm which finds the values a.

Write a recursive procedure which prints the sequence of moves which accomplish this task. Therefore it deserves a significant amount of attention. It is true that arrays are almost always implemented by using consecutive memory. The array is often the only means for structuring data which is provided in a programming language.

This is unfortunate because it clearly reveals a common point of confusion. For each index which is defined. If one asks a group of programmers to define an array. In mathematical terms we call this a correspondence or a mapping. For arrays this means we are concerned with only two operations which retrieve and store values. Using our notation this object can be defined as: STORE is used to enter new index-value pairs.

In section ARRAYS second axiom is read as "to retrieve the j-th item where x has already been stored at index i in A is equivalent to checking if i and j are equal and if so. There are a variety of operations that are performed on these lists. Notice how the axioms are independent of any representation scheme. If we restrict the index values to be integers. These operations include: Ace or the floors of a building basement.

If we consider an ordered list more abstractly. If we interpret the indices to be n-dimensional. It is only operations v and vi which require real effort. It is not always necessary to be able to perform all of these operations. In the study of data structures we are interested in ways of representing ordered lists so that these operations can be carried out efficiently. By "symbolic. Let us jump right into a problem requiring ordered lists which we will solve by using one dimensional arrays.

This problem has become the classical example for motivating the use of list processing techniques which we will see in later chapters. This we will refer to as a sequential mapping..

We can access the list element values in either direction by changing the subscript values in a controlled way The problem calls for building a set of subroutines which allow for the manipulation of symbolic polynomials.

See exercise 24 for a set of axioms which uses these operations to abstractly define an ordered list.. Insertion and deletion using sequential allocation forces us to move some of the remaining elements so the sequential mapping is preserved in its proper form.. Perhaps the most common way to represent an ordered list is by an array where we associate the list element ai with the array index i. It is precisely this overhead which leads us to consider nonsequential mappings of ordered lists into arrays in Chapter 4.

This gives us the ability to retrieve or modify the values of random elements in the list in a constant amount of time. A complete specification of the data structure polynomial is now given. We will also need input and output routines and some suitable format for preparing polynomials as input. When defining a data object one must decide what functions will be available. However this is not an appropriate definition for our purposes.. For a mathematician a polynomial is a sum of terms where each term has the form axe.

The first step is to consider how to define polynomials as a computer structure. MULT poly. Then we would write REM P. Notice the absense of any assumptions about the order of exponents.. Suppose we wish to remove from P those terms having exponent one. These assumptions are decisions of representation.

COEF B. These axioms are valuable in that they describe the meaning of each operation concisely and without implying an implementation. Now we can make some representation decisions. Exponents should be unique and in decreasing order is a very reasonable first decision.

Note how trivial the addition and multiplication operations have become. EXP B. Now assuming a new function EXP poly exp which returns the leading exponent of poly.

EXP B file: B REM B. We have avoided the need to explicitly store the exponent of each term and instead we can deduce its value by knowing our position in the list and the degree. But are there any disadvantages to this representation? Hopefully you have already guessed the worst one. The case statement determines how the exponents are related and performs the proper action..

With these insights. EXP B: EXP A. EXP A end end insert any remaining terms in A or B into C The basic loop of this algorithm consists of merging the terms of the two polynomials. Since the tests within the case statement require two terms.

COEF A. EXP A.. This representation leads to very simple algorithms for addition and multiplication.

But scheme 1 could be much more wasteful. The first entry is the number of nonzero terms. It will require a vector of length In the worst case. As for storage. Then for each term there are two entries representing an exponent-coefficient pair. Is this method any better than the first scheme? In general. Suppose we take the polynomial A x above and keep only its nonzero coefficients. Basic algorithms will need to be more complex because we must check each exponent before we handle its coefficient.

If all of A's coefficients are nonzero. The assignments of lines 1 and 2 are made only once and hence contribute O 1 to the overall computing time. This is a practice you should adopt in your own coding. The procedure has parameters which are polynomial or array names. The code is indented to reinforce readability and to reveal more clearly the scope of reserved words.

Statement two is a shorthand way of writing r Notice how closely the actual program matches with the original design. Comments appear to the right delimited by double slashes. Three pointers p. Blocks of statements are grouped together using square brackets. The basic iteration step is governed by a while loop.

It is natural to carry out this analysis in terms of m and n. To make this problem more concrete. Returning to the abstract object--the ordered list--for a moment. These are defined by the recurrence relation file: A two dimensional array could be a poor way to represent these lists because we would have to declare it as A m.

This hypothetical user may have many polynomials he wants to compute and he may not know their sizes.. This worst case is achieved. Suppose in addition to PADD. Consider the main routine our mythical user might write if he wanted to compute the Fibonacci polynomials. In this main program he needs to declare arrays for all of his polynomials which is reasonable and to declare the maximum size that every polynomial might achieve which is harder and less reasonable.

Taking the sum of all of these steps. He would include these subroutines along with a main procedure he writes himself. This example shows the array as a useful representational form for ordered lists Each iteration of this while loop requires O 1 time. In particular we now have the m lists a If he declares the arrays too large. Instead we might store them in a one dimensional array and include a front i and rear i pointer for the beginning and end of each list.

Since the iteration terminates when either p or q exceeds 2m or 2n respectively. We are making these four procedures available to any user who wants to manipulate polynomials. At each iteration.

For example F 2. Suppose the programmer decides to use a two dimensional array to store the Fibonacci polynomials. Then the following program is produced. If we made a call to our addition routine. Let's pursue the idea of storing all polynomials in a single array called POLY. Exponents and coefficients are really different sorts of numbers. If the result has k terms. Also we need a pointer to tell us where the next free location is. The array is usually a homogeneous collection of data which will not allow us to intermix data of different types.

Then by storing all polynomials in a single array. A much greater saving could be achieved if Fi x were printed as soon as it was computed in the first loop. Different types of data cannot be accommodated within the usual array concept.. This example reveals other limitations of the array as a means for data representation.

When m is equal to n. We could write a subroutine which would compact the remaining polynomials. It is very natural to store a matrix in a two dimensional array. A general matrix consists of m rows and n columns of numbers as in figure 2.

Such a matrix is called sparse.. We might then store a matrix as a list of 3-tuples of the form i. There may be several such polynomials whose space can be reused. On most computers today it would be impossible to store a full X matrix in the memory at once.

There is no precise definition of when a matrix is sparse and when it is not.

Then we can work with any element by writing A i. This comes about because in practice many of the matrices we want to deal with are large. As computer scientists. A sparse matrix requires us to consider an alternate form of representation. Even worse. As we create polynomials. Such a matrix has mn elements. Now if we look at the second matrix of figure 2. Now we have localized all storage to one array. Example of 2 matrices The first matrix has five rows and three columns. This demands a sophisticated compacting routine coupled with a disciplined use of names for polynomials.

When this happens must we quit? We must unless there are some polynomials which are no longer needed. Figure 2. But this may require much data movement. In Chapter 4 we will see an elegant solution to these problems. The alternative representation will explicitly store only the nonzero elements.

Each element of a matrix is uniquely characterized by its row and column position. Sparse matrix stored as triples The elements A 0. This is where we move the elements so that the element in the i. The transpose of the example matrix looks like 1. Another way of saying this is that we are interchanging rows and columns.

We can go one step farther and require that all the 3-tuples of any row be stored so that the columns are increasing.. The elements on the diagonal will remain unchanged. Now what are some of the operations we might want to perform on these matrices? One operation is to compute the transpose matrix. If we just place them consecutively.

In our example of figure 2. We can avoid this data movement by finding the elements in the order we want them. Since the rows are originally in order. Let us write out the algorithm in full. Since the rows of B are the columns of A.

The variable q always gives us the position in B where the next term in the transpose is to be inserted. The assignment in lines takes place exactly t times as there are only t nonzero terms in the sparse matrix being generated.

How about the computing time of this algorithm! For each iteration of the loop of lines On the first iteration of the for loop of lines all terms from column 1 of A are collected.

Lines take a constant amount of time. This is precisely what is being done in lines The total time for the algorithm is therefore O nt. Since the number of iterations of the loop of lines is n. This computing time is a little disturbing since we know that in case the matrices had been represented as two dimensional arrays.

In addition to the space needed for A and B.. The terms in B are generated by rows. The algorithm for this takes the form: The statement a. We now have a matrix transpose algorithm which we believe is correct and which has a computing time of O nt. It is not too difficult to see that the algorithm is correct. This is worse than the O nm time using arrays. This gives us the number of elements in each row of B.

We can now move the elements of A one by one into their correct position in B. This algorithm. From this information. Each iteration of the loops takes only a constant amount of time. T j is maintained so that it is always the position in B where the next element in row j is to be inserted..

The computation of S and T is carried out in lines Hence in this representation. When t is sufficiently small compared to its maximum of nm. In lines the elements of A are examined one by one starting from the first and successively moving to the t-th element. This is the same as when two dimensional arrays were in use. MACH m 6. Associated with each machine that the company produces.

If we try the algorithm on the sparse matrix of figure 2. Suppose now you are working for a machine manufacturer who is using a computer to do inventory control T i points to the position in the transpose where the next element of row i is to be stored.

Each part is itself composed of smaller parts called microparts. The product of two sparse matrices may no longer be sparse. Regarding these tables as matrices this application leads to the general definition of matrix product: We want to determine the number of microparts that are necessary to make up each machine. Once the elements in row i of A and column j of B have been located. Before we write a matrix multiplication procedure. To compute the elements of C row-wise so we can store them in their proper place without moving previously computed elements.

To avoid this. Consider an algorithm which computes the product of two sparse matrices represented as an ordered list instead of an array. This sum is more conveniently written as If we compute these sums for each machine and each micropart then we will have a total of mp values which we might store in a third table MACHSUM m. An alternative approach is explored in the exercises. Its i. This enables us to handle end conditions i. C and some simple variables. In addition to the space needed for A.

We leave the correctness proof of this algorithm as an exercise. The total maximum increments in i is therefore pdr. It makes use of variables i. In each iteration of the while loop of lines either the value of i or j or of both increases by 1 or i and col are reset. In addition to all this. The while loop of lines is executed at most m times once for each row of A. When this happens Let us examine its complexity. The variable r is the row of A that is currently being multiplied with the columns of B.

The maximum total increment in j over the whole loop is t2. The maximum number of iterations of the while loop of lines file: At the same time col is advanced to the next column.

If dr is the number of terms in row r of A then the value of i can increase at most dr times before i moves to the next row of A. Since the number of terms in a sparse matrix is variable. It introduces some new concepts in algorithm analysis and you should make sure you understand the analysis. As in the case of polynomials. The classical multiplication algorithm is: Once again. Lines take only O dr time. There are.

MMULT will be slower by a constant factor. A and B are sparse. MMULT will outperform the above multiplication algorithm for arrays.. This would enable us to make efficient utilization of space. Since t1 nm and t2 np. These difficulties also arise with the polynomial representation of the previous section and will become apparent when we study a similar representation for multiple stacks and queues section 3.

If an array is declared A l1: If we have the declaration A 4: Recall that memory may be regarded as one dimensional with words numbered from 1 to m. This is necessary since programs using arrays may. We see that the subscript at the right moves the fastest. While many representations might seem plausible. In addition to being able to retrieve array elements easily. Assuming that each array element requires only one word of memory.

Then using row major order these elements will be stored as A Then A 4. To simplify the discussion we shall assume that the lower bounds on each dimension li are 1. Another synonym for row major order is lexicographic order. Knowing the address of A i.

The general case when li can be any integer is discussed in the exercises. Sequential representation of A u1. Suppose A 4. In general.. To begin with. A u1 address: This formula makes use of only the starting address of the array plus the declared dimensions.

Sequential representation of A 1: In a row major representation. From the compiler's point of view If is the address of A 1. Before obtaining a formula for the case of an n-dimensional array. These two addresses are easy to guess.. Repeating in this way the address for A i This array is interpreted as u1 2 dimensional arrays of dimension u2 x u3.

Generalizing on the preceding discussion. From this and the formula for addressing a 2 dimensional array. An alternative scheme for array representation.. The address for A i Each 2-dimensional array is represented as in Figure 2. By using a sequential mapping which associates ai of a1. To review. The address of A i1. To locate A i. However several problems have been raised. If is the address for A 1. In all cases we have been able to move the values around.

Assume that n lists.. The i-th list should be maintained as sequentially stored. For these polynomials determine the exact number of times each statement will be executed..

The functions to be performed on these lists are insertion and deletion. What is the computing time of your procedure? How much space is actually needed to hold the Fibonacci polynomials F0. Write a procedure which returns Assume you can compare atoms ai and bj. What can you say about the existence of an even faster algorithm? Try to minimize the number of operations. The band includes a. Obtain an addressing formula for elements aij in the lower triangle if this lower triangle is stored by rows in an array B 1: Another kind of sparse matrix that arises often in numerical analysis is the tridiagonal matrix.

What is the relationship between i and j for elements in the zero part of A? Let A and B be two lower triangular matrices. Devise a scheme to represent both the triangles in an array C 1: What is the computing time of your algorithm? Tridiagonal matrix A If the elements in the band formed by these three diagonals are represented rowwise in an array. When all the elements either above or below the main diagonal of a square matrix are zero.

For large n it would be worthwhile to save the space taken by the zero entries in the upper triangle. In this square matrix. Define a square band matrix An. How much time does it take to locate an arbitrary element A i. B which determines the value of element aij in the matrix An.

A variation of the scheme discussed in section 2. Thus A4. A generalized band matrix An. The band of An. How much time does your algorithm take?

Assume a row major representation of the array with one word per element and the address of A l1.

Do exercise 20 assuming a column major representation. Obtain an addressing formula for the element A i1. Consider space and time requirements for such operations as random access. B where A and B contain real values. How many values can be held by an array with dimensions A 0: The figure below illustrates the representation for the sparse matrix of figure 2.

In this representation. In addition. Write a program which computes the product of two complex valued matrices A.. A complex-valued matrix X is represented by a pair of matrices A.

An m X n matrix is said to have a saddle point if some entry A i.. Use a minimal amount of storage. Given an array A 1: One possible set of axioms for an ordered list comes from the six operations of section 2. The bug wanders possibly in search of an aspirin randomly from tile to tile throughout the room.

Hard as this problem may be to solve by pure probability theory techniques. One such problem may be stated as follows: A drunken cockroach is placed on a given square in the middle of a tile floor in a rectangular room of size n x m tiles. All the cells of this array are initialized to zero. The position of the bug on the floor is represented by the coordinates IBUG.

There are a number of problems. JBUG and is initialized by a data card. Assuming that he may move from his present tile to any of the eight tiles surrounding him unless he is against a wall with equal probability.

The problem may be simulated using the following method: The technique for doing so is called "simulation" and is of wide-scale use in industry to predict traffic-flow.

All but the most simple of these are extremely difficult to solve and for the most part they remain largely unsolved. Of course the bug cannot move outside the room. Each time a square is entered. Many of these are based on the strange "L-shaped" move of the knight. Your program MUST: Chess provides the setting for many fascinating diversions which are quite independent of the game itself.

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