Combinatorial Designs and Related Discrete Combinatorial Structures

Sarah Spence AdamsFall 2008

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”

“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?

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

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!)

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)

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”

Assigning Fertilizers to Farms

Rows represent farms Columns represent fertilizers Can see 1’s are “well mixed”

0 1 0 0 0 1 1

0 0 1 1 0 1 0

0 0 0 1 1 0 1

1 0 0 0 1 1 0

1 1 0 1 0 0 0

1 0 1 0 0 0 1

0 1 1 0 1 0 0

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

Combinatorial Designs

Incidence Structure

Set P of “points”

Set B of “blocks” or “lines”

Incidence relation tells you which points are on which blocks

t-Designs

v points

k points in each block

For any set T of t points, there are exactly blocks incident with all points in T

Also called t-(v, k, designs

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

Revisit Fano Plane

This is a 2-(7, 3, 1) design

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 so that any collection of t points is together on blocks

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

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 so that any collection of t points is together on blocks

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

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

Represent Z7 Example with Fano Plane

001 2

5

6 4 3

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”

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 three edges that are either the edges of a perfect matching or the edges of a triangle

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

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

Rich Combinatorial Structure

Theorem: The number of blocks b in a t-(v, k, designis b = 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

Incidence Matrix of a Design

Rows labeled by lines Columns labeled by points aij = 1 if point j is on line i, 0 otherwise

001

5

6 4 3

2

0 1 0 0 0 1 1

0 0 1 1 0 1 0

0 0 0 1 1 0 1

1 0 0 0 1 1 0

1 1 0 1 0 0 0

1 0 1 0 0 0 1

0 1 1 0 1 0 0

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

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 Fast Originally designed for long-distance telephony Now used in main memory of computers

Discrete Combinatorial Structures

CodesCodesGroups Graphs

Designs

Latin Squares

DifferenceSets

ProjectivePlanes

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