Combinatorial%20Designs

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Combinatorial Designs

• Dr. David R. Berman

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Sudoku puzzle
1 3 4
1
2 4 3
3 4 1
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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

1 2 3 4
4 1 2 3
3 4 1 2
2 3 4 1
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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
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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
5
4
6
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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)
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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

Table 1 Table 1 Table 1 Table 1 Table 1 Table 2 Table 2 Table 2 Table 2 Table 2
Round 1 8 0 v 4 5 1 3 v 2 6
Round 2 8 1 v 5 6 2 4 v 3 0
Round 3 8 2 v 6 0 3 5 v 4 1
Round 4 8 3 v 0 1 4 6 v 5 2
Round 5 8 4 v 1 2 5 0 v 6 3
Round 6 8 5 v 2 3 6 1 v 0 4
Round 7 8 6 v 3 4 0 2 v 1 5
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Research Strategies

• Use theoretical techniques to prove that a given
  design exists (or doesnt exist) for certain
  sizes.
• Use experimental techniques to prove that a given
  design exists (or doesnt 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 ? of Zp generates all non-zero
  elements, i.e., Zp 0 ?i 0 i p-2

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Whist with 13 players
out Table 1 Table 1 Table 1 Table 1 Table 1 Table 2 Table 2 Table 2 Table 2 Table 2 Table 3 Table 3 Table 3 Table 3 Table 3
R1 0 1 12 v 8 5 2 11 v 3 10 4 9 v 6 7
R2 1 2 0 v 9 6 3 12 v 4 11 5 10 v 7 8
...
R13 12 0 11 v 7 4 1 10 v 2 9 3 8 v 5 6
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Theorem

• If p is a prime of the form 4K1, 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?