Application of discrete mathematics in IT
- 1. In IT / CS Discrete Mathematics Applications of
- 2. Learning Outcomes 1 2 3 4 Meaning of Discrete Mathematics About Different fields of Discrete Mathematics How Discrete Mathematics is used in almost every field of Computing Every day applications of Discrete Mathematics After this presentation, Students will be able to understand:
- 3. Welcome!
- 5. Professor Dr. Aqeel M.Sc. in Mathematics University of Engineering and Technology University of Lahore Discrete Mathematics Course Code: MATH2113 M.Phil. in Mathematics Ph.D. in Mathematics University of Engineering and Technology Lecturer at University of Education, Lahore
- 6. Lets Start!
- 7. Meaning & Definition "Discrete Math" is not the name of a branch of mathematics, like num ber theory, algebra, calculus, etc. Rather, it's a description of a set of branches of math that all have in common the feature that they are "discrete" rather than "continuous". Discrete mathematics is the study of mathematical structures that are fundame ntally discrete rather than continuous. In contrast to real numbers that have th e property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in thi s way, but have distinct, separated values.
- 9. Networks
- 10. INTRODUCTION 2 1 A network is, in its simplest form, a collection of points joined together in pairs by lines Networks are thus a general yet powerful means of representing patterns of connections or interactions between the parts of a system 3 In the jargon of the field the points are referred to as vertices or nodes and the lines are referred to as edges.
- 11. Continued… Scientists in a wide variety of fields have, over the years, developed an extensive set of tools – mathematical, computational, and statistical – for analyzing, modeling and understanding networks. In this article we introduce the basic theoretical tools used to describe and analyze networks, most of which come from graph theory or better to say from discrete mathematics the branch of mathematics that deals with networks. To begin at the beginning, a network – also called a graph in the mathematical literature – is, as we have said, a collection of vertices joined by edges. Vertices and edges are also called nodes and links in computer science,
- 12. Table Vertices and edges in networks Network Vertex Edge Internet Computer or router Cable or wireless data connection World Wide Web Web page Hyperlink Friendship network Person Friendship
- 13. Examples Throughout this article we will normally denote the number of vertices in a network by n and the number of edges by m, which is common notation in the mathematical literature. There are a number of different ways to represent a network mathematically. A us ual representation of a network for presen t purposes is the adj acency matrix. The a djacency matrix A of a simple graph is the matrix with elements Aij such that { } 1 if t here is an edge betw een vertices and ,, , 1,2, .
- 14. Continued… Two points to notice about the adjacency matrix are that, first, for a network with no self– edges such as this one the diagonal matrix elements are all zero, and second that it is symmetric, since if there is an edge between i and j then there is an edge b etween j and i. Many of the networks in science and theory have edges that form si mple on/off connections between vertices. Either they are there or they are not. In s ome situations, however, it is useful to represent edges as having a strength, weigh t, or value to them, usually a real number. Thus in the Internet edges might have we ights representing the amount of data flowing along them or their bandwidth. In a fo od web predator–prey interactions might have weights measuring total energy flow between prey and predator. In a social network connections might have weights rep resenting frequency of contact between actors. Such weighted or valued networks can be represented by giving the elements of the adjacency matrix values equal to the weights of the corresponding connections. Thus the adjacency matrix
- 15. Computers run software and store files. The software and files are both stored as huge strings of 1s and 0s. Binary math is discrete mathematics Computers Tablets Laptop Mobile
- 16. Compact Discs Compact discs store a lot of data, which is encoded using a modified Reed-Solomon code (a binary code, and thus discrete math) to automatically correct transmission errors.
- 17. Reed - Solomon Code 4 3 2 1 Storage devices (including tape, Compact Disk, DVD, barcodes, etc) Satellite communications Digital television / DVB Wireless or mobile communications (including cellular telephones, microwave links, etc.). 5 High-speed modems such as ADSL, xDSL, etc. Reed-Solomon codes are block-based error correcting codes with a wide range of applications in digital communications and stor age. Reed-Solomon codes are used to correct errors in many systems including:
- 18. 01 Computer graphics Computer graphics (such as in video games) use linear algebra in order to transform (move, scale, change perspective) objects. That's true for both applications like game development, and for operating systems. 02 Cell phone communications Making efficient use of the broadcast spectrum for mobile phones uses linear algebra and information theory. Assigning frequencies so that there is no interference with nearby phones can use graph theory or can use discrete optimization.
- 19. Applications in Different Fields uses discrete mathematics to merge images or apply filters. Digital image processing Hidden Markov models cybersecurity speeding up Facebook performance. Robot arms which are part of linear algebra, are used for large vocabulary continuous speech recognition. Graph theory and linear algebra can be used in speeding up Facebook performance. Graph theory is used in cybersecurity to identify hacked or criminal servers and generally for network security. are a type of linkage, the study of which is part of discrete geometry. 01 04 03 02 05
- 20. Google Maps
- 21. How Google Maps Calculates The Shortest Route Edsger W. Dijkstra’s Algorithm Is used to calculate shortest route Google Maps knows your position via the Global Positioning System. Co-ordinate geometry is used. Google Maps uses discrete mathematics to determine fastest driving routes and times.
- 22. Example Working of algorithm There is a simpler version that works with small maps and technicalities involved in adapting to large maps.
- 24. Encryption and decryption are part of cryptography, which is part of discrete mathematics. Cryptography Discrete mathematics to create ciphers Statistics to break them. Cryptography is the science of using mathematics to encrypt and decrypt data.
- 25. Mathematical Fields Most encryption is based heavily on number theory, most of it being abstract algebra. Calculus and trigonometry isn't heavily used. Additionally, other subjects should be understood well; specifically probability (including basic combinatorics), information theory, and asymptotic analysis of algorithms.
- 26. How it Works? Cryptography - Crypto -----> "Kryptos" --------> Hidden - Graphy -----> "Graphein" -------> To Write
- 27. Classification Cryptography Symmetric key cryptography Asymmetric key cry ptography (Public key cryptography) Modern cryptographyClassical cryptography Transposition Substitution Stream Block cipher cipher cipher cipher
- 28. Applications Defense services Secure data manipulation E –Commerce Business transactions Internet payment systems User identification systems Data security Access control
- 29. Different Fields 02 01 03 04 like deciding which nurses should work which shifts, or which airline pilots should be flying which routes, or scheduling rooms for an event, or deciding timeslots for committee meetings, or which chemicals can be stored in which parts of a warehouse---are solved either using graph coloring or using combinatorial optimization, both parts of discrete mathematics. One example is scheduling games for a professional sports league. Scheduling problems A food web describes the ways in which a set of species eat (and don't eat) each other. They can be studied using graph theory. Food webs uses discrete math: deciding how to expand train rail lines, train timetable scheduling, and scheduling crews and equipment for train trips use both graph theory and linear algebra. Railway planning Scheduling tasks to be completed by a single machine uses graph theory. Scheduling tasks to be completed by a set of machines is a bin-packing problem, which is part of discrete optimization.. Machine Job Scheduling Applications
- 31. Magic Behind Google Success 01 02 03 When Google went online in 1990’s, one thing that set it apart from other search engines was its search result listings which always delivered “good stuff”. Search Engines like Google have to do three basic things : Look the web and locate all web pages with public access. Indexing of searched data for more efficient search. Rate the importance of each page in the database, so when the user does a search, the more important pages are presented first.Big part of the MAGIC behind Google success is its PageRank Algorithm.
- 32. PageRank Algorithm PageRank Algorithm, developed by Google’s founders, Larry Page and Sergey Brin, when they were graduate students at Stanford University. 01 02 03 the number of links found in a page Outgoing Links the number of times other pages have cited this page Incoming Links - A value representing the page's relative importance in the network. Rank Three features for determining PageRank : PageRank is a link an alysis algorithm that r anks the relative impo rtance of all web page s within a network.
- 33. Represent Internet as Graph Represent Graph as Stochastic Matrix Find Dominant eigenvector of Google Matrix ⇒ PageRank Make stochastic matrix more convenient ⇒ Google Matrix Mathematical Model of Internet: Internet as a Graph Link from one web page to another web page. Web graph : Web pages = nodes, Links = edges PageRank – How it Works ?
- 34. Continued… Web graph as a Matrix 4 Links = nonzero elements in matrix S is a Sparse Matrix, as most of the entries are zero. Probability that surfer moves from page i to page j. 1 2 3 5 Every page ‘i’ has li≥1 outlinks. Sij = 1/li if page I has link to page j 0 otherwise 0 1/2 0 1/2 0 0 0 1/3 1/3 1/3 S = 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 4
- 35. Importance of Linear Algebra Using techniques of Linear Algebra, one can compute a unique solution for PageRank Problem. It gives importance of all webpages in terms of PageRank Eigenvector corresponding to each webpage. No other successful technique other than Linear Algebra is available to solve this problem.3 2 1
- 36. Applications 01 02 03 04 05 06 07 08 Analog Clock Apportionment Electronic health care records Voting systems Delivery Route Logistics Archaeology Power grids 09 10 11 12 Neuroscience DNA sequencing Kidney donor matching Measuring the evolutionary distance between genomes
- 37. Continued… 13 14 15 16 17 18 19 20 Design of radar and sonar systems Changing patterns in lizard skin Modeling traffic Producing rankings Balancing chemical equations Understanding molecular structure Determining how best to add streets to congested areas of cities Data Mining
- 38. And Many More…
- 39. Thank you
- 40. Any Question?