Sarah Spence AdamsProfessor o f Mathemat ics and

Electr ica l & Computer Engineer ing

COMBINATORIAL DESIGNS AND

RELATED DISCRETE AND ALGEBRAIC STRUCTURES

Wireless sensors: Conserving energy

Modern wireless sensors can be temporarily put into

an idle state to conserve energy. What is the optimal

on-off schedule such that any two sensors are both on

at some time?

Zheng, Hou, Sha, MobiHoc, 2003

Wireless sensors: Distributing cryptographic keys

Wireless sensors need to securely communicate with

one another. What is the best way to distributecryptographic keys so that any two sensors

share acommon key?

Camtepe and Yener, IEEE Transactions on Networking, 2007

More on 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

Team Formation

Can you arrange 15 schoolgirls (a class of Olin students) in parties (project teams) of three for seven days’ walks (projects) such that every two of them walk (work) together exactly once?

Kirkman, The Lady's and Gentleman's Diary, Query VI, 1850

Design of Statistical Experiments

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 which settings to test?

Bose and others, 1940s

Examples of Statistical Experiments

Combinations of drugs for patients with varying profiles

Combinations of chemicals at various temperatures

Combinations of fertilizers with various soils and watering patterns

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!)

Agriculture Example – Version 1

7 brands of fertilizer to test

7 different types of soil (7 different farms)

Insufficient resources to have managed plots to test every fertilizer in every condition on every farm

Facilitating Farming – Version 1

Test each pair of fertilizers on exactly one farm

Test each fertilizer 3 times

Requires 21 managed plots (reduced by an order of magnitude)

Conditions are “well mixed”

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 fertilizers are tested together on 1 farm Each fertilizer tested three times

Agriculture Example – Version 2

Pairs of crops are sometimes beneficial to one another

Suppose you have 7 crops you want to test

Want to test every pair, only have 7 plots, can plant three crops per plot

How to organize the crops?

Facilitating Farming – Version 2

Lines are plots

Points are crops

3 points on every line represent crops tested on that farm Each pair of crops is tested on one farm Each crop is tested on three farms

Conditions are “well mixed”

Combinatorial Designs

Incidence structure

Set P of “points”

Set B of “blocks” or “lines”

Incidence relation tells you which points are on which blocks

Incidence Matrix of a Design

Rows labeled by lines (farms/plots)Columns labeled by points

(fertilizers/crops)

aij = 1 if point j is on line i, 0 otherwise

01

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 linesColumns labeled by points

aij = 1 if point j is on line i, 0 otherwise

Design Matrix Code

The binary rowspace 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

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

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

Applications of Designs

To minimize energy within a wireless sensor network, points represent sensors and block represent sensors who are “on” at a given time step

For cryptographic applications, points represent sensors/people, and blocks represent sensors/people who share a particular cryptographic key

In team formation (and more general scheduling problems), points can be people and blocks can be time slots

In statistics, points can be the factors to compare, and blocks can be the directly compared factors

In general, points are what we're connecting/comparing, and blocks are how we're connecting/comparing them

Rich Combinatorial Structure

Theorem: The number of blocks b in a t-(v, k, ) l design is 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

Revisit Fano Plane

This is a 2-(7, 3, 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 l so that any collection of t points is together on l 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

Connections with Graph Theory

A graph is set of vertices and edges, with an incidence relation between the vertices and edges

Graphs also have incidence matrices and adjacency matrices

Complete graphs are used to model fully connected social or computer networks

All graphs are subgraphs of complete graphs

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 largest t and l so that any collection of t points is together on l 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

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

01 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 3 edges that you define so that the design works

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

Discrete Combinatorial Structures

CodesGroups Graphs

Designs

Latin Squares

DifferenceSets

ProjectivePlanes

Discrete Combinatorial Structures

Heaps of different discrete structures are in fact related

Often 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