Combinatorial Designs and Related Discrete Combinatorial Structures

1. Combinatorial Designs and Related Discrete Combinatorial Structures Sarah Spence Adams Assoc. Professor of Mathematics and Electrical & Computer EngineerING

2. Wireless sensors need to securely communicate with one another. What is the best way to distribute cryptographic keys so that any two sensors share a common key? {Camtepe and Yener, IEEE Transactions on Networking, 2007

3. Cryptographic Key Distribution You and your associates are on a secure teleconference, and someone suddenly disconnects. The cryptographic information she owns can no longer be considered secret. How hard is to re-secure the network? Xu, Chen and Wang, Journal of Communications, 2008

4. 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?

5. 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?

6. 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

7. 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!)

8. 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)

9. 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”

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

11. 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

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

13. 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

14. 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

15. 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

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

17. 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

18. 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

19. 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

20. 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

21. 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

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

23. 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”

24. 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

25. 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

26. 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

27. 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

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

29. 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

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

31. 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