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

3

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

5

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?