combinatorial designs and related discrete combinatorial structures sarah spence adams fall 2008
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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
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