Combinatorial Designs and related Discrete Combinatorial Structures

1. Combinatorial Designs and related Discrete Combinatorial Structures Discrete Mathematics Olin College Sarah Spence Adams Fall 2007

2. Kirkman Schoolgirl Problem (1847) • Can you arrange 15 schoolgirls in parties of three for seven days’ walks such that every two of them walk together exactly once? • Answered by looking at certain designs

3. Selection of Sites Problem • Industrial experiment needs to determine optimal settings of independent variables • May have 10 variables that can be switched to “high” or “low” • May not have resources to test all 210 combinations • How do you pick with settings to test?

4. Statistical Experiments • Combinations of fertilizers with types of soil or watering patterns • Combinations of drugs for patients with varying profiles • Combinations of chemicals for various temperatures

5. Designing Experiments • Observe each “treatment” the same number of times • Can only compare treatments when they are applied in same “location” • Want pairs of treatments to appear together in a location the same number of times (at least once!)

6. Farming Example • 7 brands of fertilizer to test • Want to test each fertilizer under 3 conditions (wet, dry, moderate) in 7 different farms • Insufficient resources to test every fertilizer in every condition on every farm (Would require 147 managed plots)

7. Facilitating Farming • Test each fertilizer 3 times, once dry, once wet, once moderate • Test each condition on each farm • Test each pair of fertilizers on exactly one farm • Requires 21 managed plots • Conditions are “well mixed”

8. Assigning Fertilizers to Farms • Rows represent farms • Columns represent fertilizers • Can see 1’s are “well mixed”

9. Fano Farming • 7 “lines” represent farms • 7 points represent fertilizers • 3 points on every line represent fertilizers tested on that farm • Each set of 2 points is together on 1 line

10. Combinatorial Designs • Incidence Structure • Set P of “points” • Set B of “blocks” or “lines” • Incidence relation tells you which points are on which blocks

11. t-Designs • v points • k points in each block • For any set T of t points, there are exactly l blocks incident with all points in T • Also called t-(v, k, l) designs

12. Consequences of Definition • All blocks have the same size • Every t-subset of points is contained in the same number of blocks • 2-designs are often used in the design of experiments for statistical analysis

13. Revisit Fano Plane • This is a 2-(7, 3, 1) design

14. Graph Theory Example • Define 10 points as the edges in K5 • Define blocks as 4-tuples of edges of the form • Type 1: Claw • Type 2: Length 3 circuit, disjoint edge • Type 3: Length 4 circuit • Find t and l so that any collection of t points is together on l blocks

15. Graph Theory Example Continued • Take any set of 4 edges – sometimes you get a block, sometimes you don’t • Take any set of 3 edges – they uniquely define a block • So, have a 3-(10, 4, 1) design

16. Vector Space Example • Define 15 points to be the nonzero length 4 binary vectors • Define the blocks to be the triples of vectors (x,y,z) with x+y+z=0 • Find t and l so that any collection of t points is together on l blocks

17. Vector Space Example Continued.. • Take any 3 distinct points – may or may not be on a block • Take any 2 distinct points, x, y. They uniquely determine a third distinct vector z, such that x+y+z=0 • So every 2 points are together on a unique block • So we have a 2-(15, 3, 1) design

18. Modular Arithmetic Example • Define points as the elements of Z7 • Define blocks as triples {x, x+1, x+3} for all x in Z7 • Forms a 2-(7, 3, 1) design

19. Represent Z7 Example with Fano Plane 5 1 2 0 6 3 4

20. Why Does Z7 Example Work? • Based on fact that the six differences among the elements of {0, 1, 3} are exactly all of the non0 elements of Z7 • “Difference sets”

21. Your Turn! • Find a 2-(13, 4, 1) using Z13 • Find a 2-(15, 3, 1) using the edges of K6 as points, where blocks are sets of 3 edges that you define so that the design works

22. Steiner Triple Systems (STS) • An STS of order n is a 2-(n, 3, 1) design • Kirkman showed these exist if and only if either n=0, n=1, or n is congruent to 1 or 3 modulo 6 • Fano plane is unique STS of order 7

23. Block Graph of STS • Take vertices as blocks of STS • Two vertices are adjacent if the blocks overlap • This graph is strongly regular • Each vertex has x neighbors • Every adjacent pair of vertices has y common neighbors • Every nonadjacent pair of vertices has z common neighbors

24. 5 1 2 0 3 6 4 Incidence Matrix of a Design • Rows labeled by lines • Columns labeled by points • aij = 1 if point j is on line i, 0 otherwise

25. Incidence Matrix of a Design • Rows labeled by lines • Columns labeled by points • aij = 1 if point j is on line i, 0 otherwise

26. Design  Code • The set of all combinations of the rows of the incidence matrix of the Fano plane is a (7, 16, 3)-Hamming code • Hamming code • Corrects 1 error in every block of 7 bits • Relatively fast • Originally designed for long-distance telephony • Used in main memory of computers

27. Discrete Combinatorial Structures Designs Groups Graphs Codes Latin Squares Difference Sets Projective Planes

28. Discrete Combinatorial Structures • Heaps of different discrete structures are in fact related • Often times a result in one area will imply a result in another area • Techniques might be similar or widely different • Applications (past, current, future) vary widely

29. Rich Combinatorial Structure • Theorem: The number of blocks b in a t-(v, k, l) designis b = l(v C t)/(k C t) • Proof: Rearrange equation and perform a combinatorial proof. Count in two ways the number of pairs (T,B) where T is a t-subset of P and B is a block incident with all points of T