Discrete Mathematics and Mathematical Algorithms - Research Paper Example

DISCRETE MATHEMATICS Mathematical Algorithms Table of Contents Introduction 3 Mathematical Algorithms Overview 3 Evolution 5 Relation of Computer Science and Mathematical Algorithms 6 Algorithm-Supported Mathematical Theory 7 Mathematical Algorithm Analysis 8 Impotence of Mathematical Algorithms 9 More Real-world Examples 10 Conclusion 10 Bibliography 11 Introduction Discrete mathematics is a section or element of mathematics that is concerned with the objects which are capable of assuming just divided, distinctive values. The concept of discrete mathematics is thus applied in distinction with continuous mathematics, that is the subdivision of mathematics concerned with the objects which have the competence to deviate quite easily (as well as that includ2e for instance, calculus). While distinct objects are often competent enough to be exemplified through integers and therefore these continuous objects should be in the format of real numbers. The comprehensive study of how discrete objects come together through one another and the likelihoods of a diversity of outcomes is acknowledged as combinatory. Further areas of mathematics which are known to be component of discrete mathematics comprise graph theory as well as the theory of computation. In addition, subjects that are related to number theory like that resemblances and recurrence relations are as well renowned as a division of discrete mathematics (Wolfram Research, Inc. , 2010). This paper presents a detailed analysis of the existing one of most famous paradigms of discrete mathematics which is known as “Mathematical Algorithms”. Mathematical Algorithms are presently one of the hot areas in discrete mathematics because of their extensive and effective use in the field of computer and information technology. This paper will provide comprehensive analysis of main features and areas of “Mathematical Algorithms”. Mathematical Algorithms Overview The comprehensive study and areas of interest in discrete mathematics generally comprise the analysis and systematic study of algorithms, their efficiencies and implementations in various fields of life. Discrete mathematics is also acknowledged as the language of mathematics used in conjunction with computer science, as well as its significance has increased noticeably in recent few decades (John & Weisstein, 2010). In a broad sense, an algorithm can be thought as instruction (or a set of commands or course of actions) according to that a specific procedure has to take place. We can say that a computer program is an illustration or an accomplishment of an algorithm. This command/instruction has to be organized in such an accurate manner so that it is proficient to be executed by an operator that is capable to comprehend as well as efficiently run the algorithms commands. Several cases of consideration of algorithms are the strategies planned for summing up and finding difference between two or more numbers, preparing food as per stated instructions, playing board games etc (Peschl, 2010). Usually, an algorithm gives effective directions to the administrative tool (usually a computer, a person or some other mechanism which is capable of understanding and executing commands) gradually or which accomplishments to acquire. So considering it in broad-spectrum, we can say that an algorithm generally provide solution to a collection of associated problems. And the collection of a detailed problem is evaluated through the features that carry out as participation planned for the algorithm. Also, an algorithm has to be foretelling that is, the comparable early conditions as well as the input of same nature should have to provide the same output or outcomes all the times (or we can say that it would be a random algorithm). Furthermore, an algorithm must terminate, that is, the running of the algorithm has to finish by producing an outcome within the fixed time (Peschl, 2010). Generally we will try to evaluate algorithms in terms of computer science, and then we will be able to know that, an algorithm describes a step-by-step of how input numbers or figures are converted into output figures in order to provide solution to a specific problem under consideration. The successive situations have to be fulfilled for an algorithm to be efficient and proficient (Peschl, 2010): An algorithm needs to have specific clarification of the situation or problem which requires to be resolved or represented in definite way (Peschl, 2010). An algorithm has to identify the methodologies of providing solution to the problem in “small” as well as definite parts (An example is the typical use of information of a programming language) (Peschl, 2010). An algorithm must exist and be proficient to noticeably understanding as well as running the instructions mentioned into that algorithm (Peschl, 2010). Evolution When computer science started to come out like a topic of interest at worldwide institutions and universities in the 60s and 70s, it revealed a lot of amounts of perplexity from the practitioners of widely known walks of everyday life. Definitely, it is not basically understandable that why computer science should be taken like an unusual knowledge area of educational levels. The world flourishes extensively by means of new and innovative technologies, however we do not usually provide a independent field about each one; somewhat, we are inclined to observe them as derivatives of existing branches of science as well as engineering. Taken in meditation, like that discussions has made prominent a significant concern: computer science is not providing a great deal about the computer like specific part of tools as it is regarding more inclusive observable fact of calculation itself, the design of techniques and processes that indicate and operate on information. Similar methods turn out to fulfill with their own defined inherent rules and principles, as well as they are accomplishing their tasks not just using computers however through people, by means of several organizations, as well as via systems that take place in this natural world. We can definitely say that these structured computational methods known as algorithms (Kleinberg, 2003). This perspective of algorithms is universal and sufficient to obtain both the means a computer generally processes the data and the means a person carries out calculations via hand. For instance, the strategy planned for adding and multiplying numbers that have been taught in our childhood are algorithms; the strategy used by the corporation of an Airline which is planned for scheduling comprise an algorithm; and the plan used by a search engine similar to Google which is designed for grading different web pages forms an algorithm. Moreover, the researchers have concluded that the policy employed by the human brain to identify different things existing in this universe from the visual perspective comprise a kind of mathematical algorithm, although we are presently too far away from distinguishing and knowing what this algorithm seems like or how it is operating on our nervous system (Kleinberg, 2003). Relation of Computer Science and Mathematical Algorithms An extensive idea here is that one can clarify regarding all of these algorithms exclusive of having an option to classify programming languages associated with the computers or computing devices, in its position providing communication between them by means of best usage of language of mathematics. Actually, the basic concept of an algorithm like we currently consider was formalized in massive division through the untiring hard work of logicians of mathematics in the 1930s, as well as the study of algorithms is integrated in the past various thousand years of mathematical progress. (For instance equation-solving techniques and tactics have constantly inclined to have a powerful algorithmic and the algebraic framework of the earliest Greeks were also essentially algorithmic.) (Knuth, 1980; Chauvin et al., 2002) Nowadays, the mathematical analysis of algorithms engages a central position in field of computer science; analysis concerning algorithms independently of the specific devices on which they execute can produce an effective insight into worldwide standards of design and primary limitations on computation (Knuth, 1980; Chauvin et al., 2002; Fathallah-Shaykh et al., 2004). At the similar time, the research efforts carried out in computer science to keep two diverse observations in focus: this is an extra conceptual vision which creates algorithms mathematically, plus the more realistic analysis that the community normally acquaintances by means of the field, the one that looks for to develop applications like that banking systems based on electronic technology, Internet search engines, medical imaging and capturing software as well as the host of other formations we have approached to expect from computer technology. The hassle between these two assessment means that the fields mathematical formulations are persistently being experienced beside their establishment in practice; it provides new ways designed for mathematical thoughts to extensively employed applications; as well as it is occasionally intended for modern mathematical problems which are aggravated with the use of these applications (Fathallah-Shaykh et al., 2004) Algorithm-Supported Mathematical Theory Theories which are supported by mathematics are set of formulas in a number of recognized languages of sense (like that in predicate logic). Additionally, the analysis of mathematical assumption progresses through execution, under the course of a human user, a diversity of algorithmic motives planned for making or deriving some new and novel formulae from known ones and tactics at strengthening (huge) mathematical knowledge fundamentals in a disciplined, consistent, reclaimable, coherent and workable means. In addition, algorithm which is supported by mathematical assumption analysis can as well be realistic like the consistent essence of the existing field of MKM (mathematical knowledge management). During the earlier few decades, an exciting diversity of outcomes has been achieved in the area of algorithm-supported judgment collectively in terms of logical, mathematical potency and also in terms of software systems (Buchberger, 2004). Mathematical Algorithm Analysis A large number of algorithms might work well to carry out the similar operation; however a number of algorithms rely upon the more memory consumption as well as require rather longer execution as compare to others. As well, how well do we come to know that how fine algorithms operate and do tasks in an ordinary way, specified variations among computers as well as data inputs? One typical technique to evaluate an algorithm is to execute a computer program to discover that how efficiently it works. We may experience a difficulty by using this technique that is it only gives us information about the efficient working of an algorithm on a particular computer as well as to inputs collectively. The reason behind conducting analysis of algorithm is to investigate and then explain conclusions regarding how effectively a particular algorithm works in general conditions. So it would be tremendously hard as well as time consuming procedure to perform on different computer systems; therefore researchers have created models of computer performance to have better analysis and assessment of algorithms. Considering broadly, an algorithm analysis is mainly associated with learning that how a great deal of time a program generally requires to execute, as well as how a great deal of memory space it requires to run the program efficiently. Specifically, computer researchers and scientists utilize the details gained from algorithm analysis to find out how the data given into a program manipulates and changes its overall execution time, how a great deal of memory space the computer needs planned for storing program data, how a great deal of space the code of the program require in the computer, if an algorithm provides exact and speedy calculations, how complicated a program is, as well as how effectively it deals and handles the outcomes unpredicted (Fine, 2010). Impotence of Mathematical Algorithms The primary step towards recognizing why the assessment and information of algorithms are extremely useful and significant is to mention precisely that what we want to represent with the implementation of an algorithm. We can consider that an algorithm is a well-defined computational procedure which gets numerous figures, or collection of figures, as input as well as creates several figures, or collection of figures as a result." Explaining in a different way, algorithms can be regarded as a blueprint designed for completing a certain, well-defined task. Consequently, a part of code that finds out the terms of the Fibonacci sequence is an accomplishment of some specific algorithm. However an easy job proposed for adding two numbers or digits is an algorithm in a perception, although this job is rather a straightforward one. A large number of algorithms similar to those which calculate the Fibonacci sequences, are instinctive in nature and can be naturally integrated into our rational thoughts plus expertise of solving various problems. Despite the fact that among majority of us, complicated algorithms provides a means of most excellent knowledge so we are competent enough to employ them as a key element used for disciplined and well-defined solution of diverse consistent problem in prospect. Actually, we can be immediately amazed to know that how much complicated algorithms people practice in their everyday life such as the e-mail validation or music enjoyment on the Personal computers (Strom, 2010). More Real-world Examples Many other cases of real-world problems that require solutions necessitate higher progress of algorithms. Approximately all the tasks and functions that we perform with the help of a computer are dependent on a mathematical algorithm in a number of ways that someone has worked extremely hard to understand. So far the simplest computer application on present PCs would not be possible without algorithms being employed behind the analysis to efficiently handle and manage memory as well as load data as of the hard drive. There are lots of computer applications formulated on the basis of complex algorithms (Strom, 2010). The special algorithms that people learn are as much as the problems that they resolve. Though, prospect is superb that the problem you are finding difficult to resolve is associated to another problem in a number of perspectives (Strom, 2010). Conclusion In this paper I have presented a deep analysis of some of the main characteristics of the discrete mathematics and mathematical algorithms. This paper has summarized some of the main features of mathematical algorithms. Through developing a high-class understanding of a large array of algorithms, we will be competent enough to decide the accurate one for a complicated problem as well as apply it in a right way. Also, solutions of problems similar to those established in contests of Top Coder will definitely help us to furnish our skills in this respect. A lot of the problems, though we don’t think them reasonable, necessitate the similar collection of algorithmic knowledge that advances every day in the real world. Bibliography Bondy, J. & Murty, U., 2010. Graph Theory with Applications. [Online] Available at: http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html [Accessed 05 October 2010]. Buchberger, B., 2004. Algorithm-Supported Mathematical Theory Exploration: A Personal View and Strategy. [Online] Available at: http://webcache.googleusercontent.com/search?q=cache:R2lnU20bbFEJ:citeseerx.ist.psu.edu/viewdoc/download?doi%3D10.1.1.58.7867%26rep%3Drep1%26type%3Dpdf+Algorithmic+and+mathematical+relation&hl=en&gl=pk [Accessed 05 October 2010]. Chauvin, B., Flajolet, P., Gardy, D. & Mokkadem, A., 2002. Mathematics and Computer Science II Algorithms, Trees, Combinatorics and Probabilities. Boston. Fathallah-Shaykh, H.M., He, B., Zhao, L.-J. & Badruddin, A., 2004. Mathematical algorithm for discovering states of expression from direct genetic comparison by microarrays. Nucleic Acids Res, 32(13), pp.3807-14. Fine, M., 2010. What is Algorithm Analysis? [Online] Available at: http://www.wisegeek.com/what-is-algorithm-analysis.htm [Accessed 04 October 2010]. John, R. & Weisstein, E.W., 2010. Discrete Mathematics." From MathWorld--A Wolfram Web Resource. [Online] Available at: http://mathworld.wolfram.com/DiscreteMathematics.html [Accessed 05 October 2010]. Kleinberg, J., 2003. The Mathematics of Algorithm Design. [Online] Available at: http://www.cs.cornell.edu/home/kleinber/pcm.pdf [Accessed 2 October 2010]. Knuth, D.E., 1980. ALGORITHMS IN MODERN MATHEMATICS AND COMPUTER SCIENCE. [Online] Available at: http://webcache.googleusercontent.com/search?q=cache:5zmlIGf7Wn0J:ftp://reports.stanford.edu/pub/cstr/reports/cs/tr/80/786/CS-TR-80-786.pdf+hard+work+of+logicians+of+mathematics+in+the+1930s,+anD+the+study+of+algorithms&hl=en&gl=pk [Accessed 30 September 2010]. Locke, S., 200. Graph Theory. [Online] Available at: http://math.fau.edu/locke/GRAPHTHE.HTM [Accessed 02 October 2010]. Peschl, M., 2010. Introduction to Mathematical Logic, Algorithmics: The Spirit of Computing, The Design and Analysis of Algorithms. [Online] Available at: http://psychology.jrank.org/pages/1914/algorithm.html [Accessed 02 October 2010]. Strom, I., 2010. The Importance of Algorithms. [Online] Available at: http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=importance_of_algorithms [Accessed 04 October 2010]. Wolfram Research, Inc. , 2010. Discrete Mathematics. [Online] Available at: http://mathworld.wolfram.com/DiscreteMathematics.html [Accessed 05 October 2010]. Read More