# FUNDAMENTALS OF DATA STRUCTURES IN C PDF

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 instruktsiya.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.

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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:\instruktsiya.info Principles of Data Structures.

On the contrary, the more expressive our languages are, the more we can accomplish easily. 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. 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. A variable or a constant is a simple form of an expression. The expr may be omitted in which case a return is made to the calling procedure. The execution of an end at the end of procedure implies a return.

A procedure may be invoked by using a call statement call NAME parameter list Procedures may call themselves, direct recursion, or there may be a sequence resulting in indirect recursion.

Though recursion often carries with it a severe penalty at execution time, it remains all elegant way to describe many computing processes. This penalty will not deter us from using recursion. Many such programs are easily translatable so that the recursion is removed and efficiency achieved. All procedures are treated as external, which means that the only means for communication between them is via parameters. This may be somewhat restrictive in practice, but for the purpose of exposition it helps to list all variables explicitly, as either local or parameter.

The association of actual to formal parameters will be handled using the call by reference rule. This means that at run time the address of each parameter is passed to the called procedure. Parameters which are constants or values of expressions are stored into internally generated words whose addresses are then passed to the procedure. We avoid the problem of defining a "format" statement as we will need only the simplest form of input and output. The command stop halts execution of the currently executing procedure.

Comments may appear anywhere on a line enclosed by double slashes, e. An n-dimensional array A with lower and upper bounds li, ui, 1 i n may be declared by using the syntax declare A l1:u1, We have avoided introducing the record or structure concept. These are often useful features and when available they should be used. However, we will persist in building up a structure from the more elementary array concept.

Since most of the SPARKS programs will be read many more times than they will be executed, we have tried to make the code readable. This is a goal which should be aimed at by everyone who writes programs. The SPARKS language is rich enough so that one can create a good looking program by applying some simple rules of style. 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. This method uses the philosophy: write something down and then try to get it working. Surprisingly, this method is in wide use today, with the result that an average programmer on an average job turns out only between five to ten lines of correct code per day.

We hope your productivity will be greater. But to improve requires that you apply some discipline to the process of creating programs. To understand this process better, we consider it as broken up into five phases: requirements, design, analysis, coding, and verification. Make sure you understand the information you are given the input and what results you are to produce the output.

Try to write down a rigorous description of the input and output which covers all cases. You are now ready to proceed to the design phase. Designing an algorithm is a task which can be done independently of the programming language you eventually plan to use. In fact, this is desirable because it means you can postpone questions concerning how to represent your data and what a particular statement looks like and concentrate on the order of processing.

You may have several data objects such as a maze, a polynomial, or a list of names. For each object there will be some basic operations to perform on it such as print the maze, add two polynomials, or find a name in the list.

Assume that these operations already exist in the form of procedures and write an algorithm which solves the problem according to the requirements. Use a notation which is natural to the way you wish to describe the order of processing. Can you think of another algorithm? If so, write it down. Next, try to compare these two methods. It may already be possible to tell if one will be more desirable than the other.

If you can't distinguish between the two, choose one to work on for now and we will return to the second version later. You must now choose representations for your data objects a maze as a two dimensional array of zeros and ones, a polynomial as a one dimensional array of degree and coefficients, a list of names possibly as an array and write algorithms for each of the operations on these objects.

The order in which you do this may be crucial, because once you choose a representation, the resulting algorithms may be inefficient. Modern pedagogy suggests that all processing which is independent of the data representation be written out first.

By postponing the choice of how the data is stored we can try to isolate what operations depend upon the choice of data representation. You should consider alternatives, note them down and review them later. Finally you produce a complete version of your first program. It is often at this point that one realizes that a much better program could have been built. Perhaps you should have chosen the second design alternative or perhaps you have spoken to a friend who has done it better.

This happens to industrial programmers as well. If you have been careful about keeping track of your previous work it may not be too difficult to make changes. It is usually hard to decide whether to sacrifice this first attempt and begin again or just continue to get the first version working.

Different situations call for different decisions, but we suggest you eliminate the idea of working on both at the same time. If you do decide to scrap your work and begin again, you can take comfort in the fact that it will probably be easier the second time.

In fact you may save as much debugging time later on by doing a new version now. This is a phenomenon which has been observed in practice. The graph in figure 1. For each compiler there is the time they estimated it would take them and the time it actually took.

For each subsequent compiler their estimates became closer to the truth, but in every case they underestimated. Unwarrented optimism is a familiar disease in computing. But prior experience is definitely helpful and the time to build the third compiler was less than one fifth that for the first one. Verification consists of three distinct aspects: program proving, testing and debugging.

Each of these is an art in itself. Before executing your program you should attempt to prove it is correct. Proofs about programs are really no different from any other kinds of proofs, only the subject matter is different.

If a correct proof can be obtained, then one is assured that for all possible combinations of inputs, the program and its specification agree. Testing is the art of creating sample data upon which to run your program.

If the program fails to respond correctly then debugging is needed to determine what went wrong and how to correct it. One proof tells us more than any finite amount of testing, but proofs can be hard to obtain.

Many times during the proving process errors are discovered in the code. The proof can't be completed until these are changed. This is another use of program proving, namely as a methodology for discovering errors. Finally there may be tools available at your computing center to aid in the testing process. One such tool instruments your source code and then tells you for every data set: i the number of times a statement was executed, ii the number of times a branch was taken, iii the smallest and largest values of all variables.

As a minimal requirement, the test data you construct should force every statement to execute and every condition to assume the value true and false at least once. One thing you have forgotten to do is to document. First of all. Thus we say that integers are represented by bit strings. In the previous example The set of axioms describes the semantics of the operations. We would rather not have any individual rule us out simply because he did not know or.

But at the first stage a data structure should be designed so that we know what it does. Instead we choose to use a language which is tailored to describing the algorithms we want to write. The triple denotes the data structure d and it will usually be abbreviated by writing d.

This division of tasks. This mapping specifies how every object of d is to be represented by the objects of e.. In current parlance the triple is referred to as an abstract data type. We might begin by considering using some existing language. Expressions can be either arithmetic. Figure 1. In order to produce these values. In addition to the assignment statement. Most importantly. Several such statements can be combined on a single line if they are separated by a semi-colon.

Several cute ideas have been suggested. In the boolean case there can be only one of two values.. The way to assign values is by the assignment statement variable expression. 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.

So we will provide other statements such as a second iteration statement. If S1 or S2 contains more than one statement. On the contrary. Brackets must be used to show how each else corresponds to one if. This result was obtained by Bohm and Jacopini. Though this is very interesting from a theoretical viewpoint. The meaning of this statement is given by the flow charts: We will assume that conditional expressions are evaluated in "short circuit" mode..

One of them is while cond do S end where cond is as before. To accomplish iteration. 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.

Another iteration statement is loop S forever which has the meaning As it stands. This looping statement may be a while. It has the form where the Si. The semantics is easily described by the file: A variable or a constant is a simple form of an expression. This means that at run time the address of each parameter is passed to the called procedure. The execution of an end at the end of procedure implies a return.

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This may be somewhat restrictive in practice. This penalty will not deter us from using recursion. The else clause is optional. The association of actual to formal parameters will be handled using the call by reference rule. A procedure may be invoked by using a call statement call NAME parameter list Procedures may call themselves.

Though recursion often carries with it a severe penalty at execution time. Parameters which are constants or values of expressions are stored into internally generated words whose addresses are then passed to the procedure. All procedures are treated as external. The expr may be omitted in which case a return is made to the calling procedure.

Many such programs are easily translatable so that the recursion is removed and efficiency achieved. An n-dimensional array A with lower and upper bounds li. The SPARKS language is rich enough so that one can create a good looking program by applying some simple rules of style. 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.

See the book The Elements of Programming Style by Kernighan and Plauger for more examples of good rules of programming. Avoid sentences like ''i is increased by one. We have avoided introducing the record or structure concept. These are often useful features and when available they should be used.. The command stop halts execution of the currently executing procedure. By postponing the choice of how the data is stored we can try to isolate what operations depend upon the choice of data representation..

Modern pedagogy suggests that all processing which is independent of the data representation be written out first. This method uses the philosophy: But to improve requires that you apply some discipline to the process of creating programs. We hope your productivity will be greater. You must now choose representations for your data objects a maze as a two dimensional array of zeros and ones.

To understand this process better. One of the criteria of a good design is file: You are now ready to proceed to the design phase. Make sure you understand the information you are given the input and what results you are to produce the output.

Finally you produce a complete version of your first program. This happens to industrial programmers as well. You may have several data objects such as a maze. If you can't distinguish between the two. Designing an algorithm is a task which can be done independently of the programming language you eventually plan to use.

For each object there will be some basic operations to perform on it such as print the maze. Try to write down a rigorous description of the input and output which covers all cases. Use a notation which is natural to the way you wish to describe the order of processing. Assume that these operations already exist in the form of procedures and write an algorithm which solves the problem according to the requirements.

If you have been careful about keeping track of your previous work it may not be too difficult to make changes. Can you think of another algorithm?

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If so. In fact. It is often at this point that one realizes that a much better program could have been built. Perhaps you should have chosen the second design alternative or perhaps you have spoken to a friend who has done it better.

It may already be possible to tell if one will be more desirable than the other. The order in which you do this may be crucial. You should consider alternatives. Finally there may be tools available at your computing center to aid in the testing process. This is a phenomenon which has been observed in practice.

B and C. The graph in figure 1. The previous discussion applies to the construction of a single procedure as well as to the writing of a large software system.

Each of these is an art in itself. One proof tells us more than any finite amount of testing. As a minimal requirement. 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. Verification consists of three distinct aspects: For each compiler there is the time they estimated it would take them and the time it actually took. It is usually hard to decide whether to sacrifice this first attempt and begin again or just continue to get the first version working..

Let us concentrate for a while on the question of developing a single procedure which solves a specific task. Before executing your program you should attempt to prove it is correct. Many times during the proving process errors are discovered in the code. If you have written more than a few procedures. Different situations call for different decisions. But prior experience is definitely helpful and the time to build the third compiler was less than one fifth that for the first one. If a correct proof can be obtained.

The larger the software. Testing is the art of creating sample data upon which to run your program. One such tool instruments your source code and then tells you for every data set: This shifts our emphasis away from the management and integration of the file: If you note them down with the code. This is another use of program proving. For each subsequent compiler their estimates became closer to the truth.

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In fact you may save as much debugging time later on by doing a new version now. Proofs about programs are really no different from any other kinds of proofs.

One thing you have forgotten to do is to document. If you do decide to scrap your work and begin again. The proof can't be completed until these are changed. 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. This is referred to as the bottom-up approach.. Experience suggests that the top-down approach should be followed when creating a program.

This method of design is called the top-down approach. 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. One of the simplest solutions is given by the following "from those integers which remain unsorted.

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.

Suppose we devise a program for sorting a set of n 1 distinct integers. There now remain two clearly defined subtasks: Each subtask is similarly decomposed until all tasks are expressed within a programming language. Let us examine two examples of top-down program development. The initial solution may be expressed in English or some form of mathematical notation. Underlying all of these strategies is the assumption that a language exists for adequately describing the processing of data at several abstract levels.

The design process consists essentially of taking a proposed solution and successively refining it until an executable program is achieved. A look ahead to problems which may arise later is often useful.

This latter problem can be solved by the code file: We are now ready to give a second refinement of the solution: If possible the designer attempts to partition the solution into logical subtasks. We observe at this point that the upper limit of the for-loop in line 1 can be changed to n. A j t The first subtask can be solved by assuming the minimum is A i..

From the standpoint of readability we can ask if this program is good.. We first note that for any i. Eventually A n is compared to the current minimum and we are done.. 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.. There are three possibilities. By making use of the fact that the set is sorted we conceive of the following efficient method: Let us develop another program..

We assume that we have n 1 distinct integers which are already sorted and stored in the array A 1: Continue in this way by keeping two pointers.

Part of the freedom comes from the initialization step. Below is one complete version. In fact there are at least six different binary search programs that can be produced which are all correct. Note how at each stage the number of elements in the remaining set is decreased by about one half. This method is referred to as binary search. There are many more that we might produce which would be incorrect. For instance we could replace the while loop by a repeat-until statement with the same English condition.

Whichever version we choose. Given a set of instructions which perform a logical operation. The procedure name and its parameters file: As we enter this loop and as long as x is not found the following holds: Actually one of the most useful syntactical features for accomplishing this is the procedure. 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.

Recursion We have tried to emphasize the need to structure a program to make it easier to achieve the goals of readability and correctness. This view of the procedure implies that it is invoked.

These recursive mechanisms are extremely powerful. For these reasons we introduce recursion here. This is unfortunate because any program that can be written using assignment..

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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.

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. Given the input-output specifications of a procedure.

Factorial fits this category. Of course..

When is recursion an appropriate mechanism for algorithm exposition? One instance is when the problem itself is recursively defined.

It implies that we can solve the problem for a set with n elements if we had an algorithm which worked on n. The answer is obtained by printing i a followed by all permutations of b. It is easy to see that given n elements there are n! A is a character string e. 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 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.

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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?

## Data Structure and Algorithms Tutorial

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. Data structures can be used to organize the storage and retrieval of information stored in both main memory and secondary memory. Thus, the array and record data structures are based on computing the addresses of data items with arithmetic operations , while the linked data structures are based on storing addresses of data items within the structure itself. Many data structures use both principles, sometimes combined in non-trivial ways as in XOR linking.

The efficiency of a data structure cannot be analyzed separately from those operations. This observation motivates the theoretical concept of an abstract data type , a data structure that is defined indirectly by the operations that may be performed on it, and the mathematical properties of those operations including their space and time cost.

Elements are accessed using an integer index to specify which element is required. Typical implementations allocate contiguous memory words for the elements of arrays but this is not always a necessity. Arrays may be fixed-length or resizable. A linked list also just called list is a linear collection of data elements of any type, called nodes, where each node has itself a value, and points to the next node in the linked list.

The principal advantage of a linked list over an array, is that values can always be efficiently inserted and removed without relocating the rest of the list. Certain other operations, such as random access to a certain element, are however slower on lists than on arrays.Which criteria do they violate?

Examples of nonlinear data structure include trees and graphs. J is located near one of the edges of the board. But at the first stage a data structure should be designed so that we know what it does. The rules on line 8 tell us exactly how the addition operation works. There is an intimate connection between the structuring of data, and the synthesis of algorithms.

When every square has been entered at least once..

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