Discrete Mathematics

Transcription

1 Outline Applied Mathematics Division Department of Mathematical Sciences University of Stellenbosch, South Africa Hons Program Presentation October 10, 2011

2 Outline Outline

3 Outline

4 Some of the Topics Trees and Searching Algorithms Vertex Traversal and Hamiltonian Graphs Edge Traversl and Eulerian Graphs Planar Graphs Scheduling and Graph Colouring Tournaments

5 Outline

6 in Classical Problems The Travelling Salesman Problem The Chinese Postman Problem Selected Topics Genetic Algorithms Greedy Algorithms Simulated Annealing Tabu Search Genetic Algorithms

7 Outline

8 Encryption/Decryption How do I encrypt a message so that only the receiver is able to understand it? How do I decipher a message that I intercept? Application of Number Theory Content Block Ciphers Stream Ciphers Public Key Systems

9 Outline

10 Error Correction Designing of error-correcting codes to ensure the integrity of information An application of vector spaces over finite fields

11 Outline

12 Grobler The appearance of Fibonacci numbers in the arrangement of leaves and florets of plants The existence and designing of Mutually Orthogonal Latin Squares Benecke A survey and development of algorithmic methods for graph domination problems An analysis and solution to the Guess Who Problem

13 The Cartesian Product Definition The Cartesian product of two graphs G and H, V(G) = {v 1, v 2,..., v m }, V(H) = {w 1, w 2,..., w n } Denoted G H V(G H) = {(v i, w j ) : i = 1, 2,..., m, j = 1, 2,..., n} (v i, w j )(v k, w l ) E(G H) if and only if j = l and v i v k E(G), or i = k and w j w l E(H)

14 Example Consider G = P 3 and H = C 3 (v 1, w 1 ) (v 1, w 2 ) (v 1, w 3 ) Definition v 1 Vertex set V(G H) = {(v i, w j ) : i = 1, 2,..., m, j = 1, 2,..., n} (v i, w j )(v k, w l ) E(G H) iff j = l and v i v k E(G), or i = k and w j w l E(H) v 2 v 3G (v 2, w 1 ) (v 3, w 1 ) H (v 2, w 2 ) (v 3, w 2 ) w 1 w 2 (v 2, w 3 ) (v 3, w 3 ) w 3

15 Domination Definition A subset D V(G) is a dominating set if any vertex u D is adjacent to some vertex v D. The domination number γ(g) is the minimum cardinality over all dominating sets D of G.

16 Domination Algorithms Project Survey best algorithms to determine the domination number of the Cartesian product graph Study and implement the method by Livingston & Stout, as described by Benecke & Mynhardt Survey best algorithms for and investigate application to other graph products and/or other domination parameters Content Programming

17 The Guess Who Game Questions Generalize classic game to k 1 mystery people. What is the best strategy? What is the best question at any stage? How does one design a balanced game board? Under which conditions can the game always be resolved? Content Game Theory Probability Theory Programming

18 References A MENEZES, P VAN OORSCHOT AND S VANSTONE, Handbook of Applied Cryptography, CRC Press, JA BONDY AND USR MURTY, Graduate Texts in Mathematics -, Springer, Z MICHALEWICZ AND DB FOGEL, How to Solve it: Modern Heuristics, Springer, 2000.

19

20

21