FUNDAMENTALS OF ALGORITHMICS PDF
P- 2 ELEMENTARYALGORITHMICs Int lurti 7. Prºblems and instances The efficiency of algorithms Average and worst-case analyses. Fundamentals of Algorithmics - Brassard, Bratley - Ebook download as PDF File ( .pdf), Text File .txt) or read book online. Nice book. Gilles Brassard and Paul Bartley, Fundamental of Algorithmics. Pages · Horowitz and Sahani, Fundamentals of Computer Algorithms, 2ND Edition.
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FUNDAMENTALS. OF ALGORITHMICS. Gilles Brassard and Paul Bratley. Departement d'informatique et de recherche operationeile. Universite de Montreal. Fundamentals of algorithmics pdf. Fundamentals of Algorithmics Gilles Brassard, Paul Bratley Publisher: Pearson Release Date: ISBN. You can't. Free PDFs do not exist. Seriously though you may want to check a university library. Solution manuals are there to help get you on the right track.
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Fundamentals of Algorithmics - Brassard, Bratley
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Brassard & Bratley, ―Fundamentals of Algorithmics
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Upcoming SlideShare. Like this document? Why not share! When this happens, we are obliged, if we must have some kind of solution to the problem, to look for a set of rules that we believe give us a good approximation to the correct answer, and that we can execute in a reasonable time.
If we can prove that the answer computed by this set of rules is not too badly in error, so much the better. Sometimes even this is not possible, and we can rely only on good luck. This kind of procedure, based largely on optimism and often with minimal theoretical support, is called a heuristic algorithm, or simply a heuristic.
Notice one crucial difference between approximate algorithms and heuristics: with the former we can specify the error we are willing to accept; with the latter, we cannot control the error, but we may be able to estimate how large it is. Section 1. In the first twelve chapters of this book, unless the context clearly indicates the contrary, we assume that an algorithm is a set of rules for calculating the correct answer to some problem. Chapter 13, on the other hand, deals entirely with approximate algorithms and heuristics.
Algorithmics can now be defined simply as the study of algorithms. When we set out to solve a problem, there may be a choice of algorithms available. In this case it is important to decide which one to use.
Depending on our priorities and on the limits of the equipment available to us, we may want to choose the algorithm that takes the least time, or that uses least storage, or that is easiest to program, and so on.
The answer can depend on many factors, such as the numbers involved, the way the problem is presented, or the speed and storage capacity of the available computing equipment. It may be that none of the available algorithms is entirely suitable so that we have to design a new algorithm of our own. Algorithmics is the science that lets us evaluate the effect of these various external factors on the available algorithms so that we can choose the one that best suits our particular circumstances; it is also the science that tells us how to design a new algorithm for a particular task.
Take elementary arithmetic as an example. Suppose you have to multiply two positive integers using only pencil and paper. If you were raised in North America, the chances are that you will multiply the multiplicand successively by each figure of the multiplier taken from right to left, that you will write these intermediate results one beneath the other shifting each line one place left, and that finally you will add all these rows to obtain your answer.
Thus to multiply by you would produce an arrangement like that of Figure 1. If, on the other hand, you went to school in England, you would be more likely to work from left to right, producing the arrangement shown in Figure 1.
Multiplication a American b English These two algorithms for multiplication are very similar: so similar, in fact, that we shall refer to them as the "classic" multiplication algorithm, without worrying precisely which one we mean. A third, different algorithm for doing the same thing is illustrated in Figure 1.
Write the multiplicand and the multiplier side by side. Make two columns, one under each operand, by repeating the following rule until the number in the left-hand column is 1: divide the number in the left-hand column by 2, ignoring Preliminaries Chapter 1 Figure 1. Multiplication i la russe any fractions, and double the number in the right-hand column by adding it to itself. Next cross out each row where the number in the left-hand column is even, and finally add up the numbers that remain in the right-hand column.
The figure illustrates how to multiply by This algorithm, sometimes called multiplication a la russe, resembles the one used in the hardware of a binary computer. It has the advantage that there is no need to memorize any multiplication tables.
All we need to know is how to add up, and how to divide a number by 2. Although it is not the algorithm usually taught in school, it certainly offers an alternative paper-and-pencil method for multiplying two positive integers. Still another algorithm for multiplying two positive integers is illustrated in Figures 1. Again we illustrate the method by multiplying by For this algorithm, however, we require that both the multiplicand and the multiplier have the same number of figures, and furthermore that this number be a power of 2, such as 1, 2, 4, 8, 16, etc.
This is easily fixed by adding zeros on the left if necessary: in our example, we add just one 0 on the left of the multiplicand, making it into , so that both operands have four figures. Multiply i ii iii iv 09 09 81 81 12 34 12 34 Shift 4 2 2 0 Result Multiplying by by divide-and-conquer Now to multiply by we first multiply the left half of the multiplicand 09 by the left half of the multiplier 12 , and write the result shifted left as Section 1.
Next we multiply the left half of the multiplicand 09 by the right half of the multiplier 34 , and write the result shifted left by half as many places as there are figures in the multiplier: two, in this case.
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Thirdly we multiply the right half of the multiplicand 81 by the left half of the multiplier 12 , and write the result also shifted left by half as many places as there are figures in the multiplier; and fourthly we multiply the right half of the multiplicand 81 by the right half of the multiplier 34 and write the result , not shifted at all.
Finally we add up the four intermediate results as shown in Figure 1. Multiplying 09 by 12 by divide-and-conquer If you have followed the working of the algorithm so far, you will have seen that we have reduced the multiplication of two four-figure numbers to four multiplications of two-figure numbers 09 x 12, 09 x 34, 81 x 12 and 81 x 34 together with a certain number of shifts and a final addition.
The trick is to observe that each of these multiplications of two-figure numbers can be carried out in exactly the same way, except that each multiplication of two-figure numbers requires four multiplications of one-figure numbers, some shifts, and an addition.
For instance, Figure 1. Finally we add these intermediate results to obtain the answer Using these ideas the whole of our calculation can be carried out in such a way that the multiplications involve only one-figure operands.
Although we described Figure 1. Of course we have to do the four multiplications of two-figure numbers first, since we use the values thus calculated when we do the multiplication of the four-figure numbers. This unusual algorithm is an example of the technique called "divide-andconquer", which we shall study in Chapter 7. If you think it unlikely that it could outperform the classic algorithm, you are perfectly right. However we shall see in Chapter 7 that it is possible to reduce the multiplication of two large numbers to three, and not four, multiplications of numbers roughly half the size, together with a certain number of shifts and additions.
If you are stimulated by challenges, try to figure out how to do this!
With this improvement, the divide-and-conquer multiplication algorithm runs faster on a computer than any of the preceding methods, provided the numbers to be multiplied are sufficiently large. Still faster methods are known for very large operands. It is not absolutely necessary for the length Preliminaries Chapter 1 of the operands to be a power of two, nor that they have the same length. Problem 1. The point of all these examples is that, even in an area as simple as elementary arithmetic, there may be several algorithms available to us for carrying out the necessary operations.
One may appeal by its familiarity, a second because of the elementary nature of the intermediate calculations involved, or a third by its speed on a machine. It is by making a more formal study of the properties of the algorithms-by using algorithmics, in other words-that we can make a wise choice of the technique to use in any given situation. As we shall see, a good choice can save both money and time; in some cases, it can make all the difference between success and failure when solving a large, hard problem.
The aim of our book is to teach you how to make such choices. If we try to explain them in English, we rapidly discover that natural languages are not at all suited to this kind of thing. To avoid confusion, we shall in future specify our algorithms by giving a corresponding program. We assume that the reader is familiar with at least one well-structured programming language such as Pascal.
However, we shall not confine ourselves strictly to any particular programming language: in this way, the essential points of an algorithm will not be obscured by relatively unimportant programming details, and it does not really matter which well-structured language the reader prefers.
A few aspects of our notation for programs deserve special attention. We use phrases in English in our programs whenever this makes for simplicity and clarity.
Similarly, we use mathematical language, such as that of algebra and set theory, whenever appropriate-including symbols such as. As a consequence, a single "instruction" in our programs may have to be translated into several instructions-perhaps a while loop-if the algorithm is to be implemented in a conventional programming language.
Therefore, you should not expect to be able to run the algorithms we give directly: you will always be obliged to make the necessary effort to transcribe them into a "real" programming language. Nevertheless, this approach best serves our primary purpose, to present as clearly as possible the basic concepts underlying our algorithms. To simplify our programs further, we usually omit declarations of scalar quantities integer, real, or Boolean.
In cases where it matters-as in recursive functions and procedures-all variables used are implicitly understood to be local variables, unless the context makes it clear otherwise.
In the same spirit of simplification, proliferation of begin and end statements, that plague programs written in Pascal, is avoided: the range of statements such as if, while, or for, as well as that of declarations such as procedure, function, or record, is shown by indenting the statements affected. The statement return marks the dynamic end of a procedure or a function, and in the latter case it also supplies the value of the function.
Scalar parameters are passed by value, which means they are treated as local variables within the procedure or function, unless they are declared to be var parameters, in which case they can be used to return a value to the calling program. In contrast, array parameters are passed by reference, which means that any modifications made within the procedure or function are reflected in the array actually passed in the calling statement.
Finally, we assume that the reader is familiar with the concepts of recursion, record, and pointer. The last two are denoted exactly as in Pascal, except for the omission of begin and end in records. In particular, pointers are denoted by the symbol " I ".
To wrap up this section, here is a program for multiplication a la russe. Here denotes integer division: any fraction in the answer is discarded. Compare this program to the informal English description of the same algorithm in Section 1.Multiplication i la russe any fractions, and double the number in the right-hand column by adding it to itself.
Since the basis cases contain only 0 and 1. There are two cases. On the other hand, if it includes such vague notions as "add salt to taste" or "cook until tender" then we would no longer call it an algorithm. In this context the following approximations are useful.
Parts of this book were written while Paul Bratley was on sabbatical at the Universities of Strasbourg.
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