1 / 17

Combinatorial Designs

Combinatorial Designs. Dr. David R. Berman. Sudoku puzzle. Sudoku puzzle solution. Sudoku is Latin square with additional property. Latin square of order n : Each number {1, 2, 3, …, n} appears exactly once in each row and column. Order 4 Latin square, not a Sudoku:. The Fano plane.

penha
Download Presentation

Combinatorial Designs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Combinatorial Designs Dr. David R. Berman

  2. Sudoku puzzle

  3. Sudoku puzzle solution

  4. Sudoku is Latin square with additional property Latin square of order n: Each number {1, 2, 3, …, n} appears exactly once in each row and column. Order 4 Latin square, not a Sudoku:

  5. The Fano plane Seven points Three points on each line Every two points define a line Seven lines Three lines through each point Every two lines meet at a point

  6. The Fano plane as a set system 0 {0,1,4}, {0,2,5}, {0,3,6}, {1,2,6}, {4,2,3}, {4,5,6}, {1,3,5} 1 3 2 4 5 6

  7. Round robin tournament Directed edge between every pair of vertices X  Y means X beats Y {(1,2),(1,4),(2,4),(3,1),(3,2),(4,3)}

  8. Doubles tournament • Each game: a, b v c, d • Tournament has many games • Tournament usually has structure (e.g. everyone plays in the same number of games)

  9. Whist tournament every pair of players partner once and oppose twice. Tournament is played in rounds. Example: Whist with 8 players

  10. Research Strategies • Use theoretical techniques to prove that a given design exists (or doesn’t exist) for certain sizes. • Use experimental techniques to prove that a given design exists (or doesn’t exist) for certain sizes.

  11. Field • Operations + and * with properties: commutative, associative, identity, inverses, distributive • Examples: real numbers, complex numbers • Finite field: integers modulo a prime (Zp) • Primitive elementω of Zp generates all non-zero elements, i.e., Zp – {0} = {ωi: 0 ≤ i ≤ p-2}

  12. Whist with 13 players

  13. Theorem If p is a prime of the form 4K+1, then there exists a whist tournament with p players.

  14. Examples of experimental work • http://people.uncw.edu/bermand/Java.txt • http://people.uncw.edu/bermand/C.txt • http://people.uncw.edu/bermand/Mathematica.pdf

  15. Applications of combinatorial designs • Experimental designs (statistics) • Coding, cryptography • Software and hardware testing • Network design and reliability

  16. Resources • C.J. Colbourn, J.H. Dinitz, Handbook of Combinatorial Designs, second edition, 2007, http://www.emba.uvm.edu/~dinitz/hcd.html • C.J. Colbourn, P.C. van Oorschot, Applications of combinatorial designs in computer science, ACM Computing Surveys, 1989. (Available in ACM Digital Library at Randall Library web site.) • D.R. Berman, M. Greig, D.D. Smith, Brother Avoiding Round Robin Doubles Tournaments II, submitted to J. Comb. Des, http://people.uncw.edu/bermand/BARRDT.pdf

  17. Thank you Are there questions?

More Related