Combinatorial Designs
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Combinatorial Designs Dr. David R. Berman
description
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. - PowerPoint PPT Presentation
Transcript of Combinatorial Designs
Combinatorial Designs
Dr. David R. Berman
Sudoku puzzle
1 3 4
1
2 4 3
3 4 1
2
Sudoku puzzle solution
1 2 3 4
4 3 2 1
2 1 4 3
3 4 1 2
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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:
4
1 2 3 44 1 2 33 4 1 22 3 4 1
The Fano plane
Seven pointsThree points on each lineEvery two points define a line
Seven linesThree lines through each pointEvery two lines meet at a point
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The Fano plane as a set system
{0,1,4}, {0,2,5}, {0,3,6}, {1,2,6}, {4,2,3}, {4,5,6}, {1,3,5}
0
54 6
3
2
1
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Round robin tournament
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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)}
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)
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Whist tournament
every pair of players partner once and oppose twice. Tournament is played in rounds.
Example: Whist with 8 players
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Table 1 Table 2
Round 1 ∞ 0 v 4 5 1 3 v 2 6
Round 2 ∞ 1 v 5 6 2 4 v 3 0
Round 3 ∞ 2 v 6 0 3 5 v 4 1
Round 4 ∞ 3 v 0 1 4 6 v 5 2
Round 5 ∞ 4 v 1 2 5 0 v 6 3
Round 6 ∞ 5 v 2 3 6 1 v 0 4
Round 7 ∞ 6 v 3 4 0 2 v 1 5
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.
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Field
• Operations + and * with properties: commutative, associative, identity, inverses, distributive
• Examples: real numbers, complex numbers• Finite field: integers modulo a prime (Zp)
• Primitive element ω of Zp generates all non-zero elements, i.e., Zp – {0} = {ωi: 0 ≤ i ≤ p-2}
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Whist with 13 players
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Theorem
If p is a prime of the form 4K+1, then there exists a whist tournament with p players.
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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
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Applications of combinatorial designs
• Experimental designs (statistics)• Coding, cryptography• Software and hardware testing• Network design and reliability
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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
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Thank you
Are there questions?
