Sarah Spence Adams Professor of Mathematics and Electrical & Computer Engineering Discrete Mathematics, Fall 2014 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 distribute cryptographic keys so that any two sensors share a common key? Camtepe and Yener, IEEE Trans. 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 2 10 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) a ij = 1 if point j is on line i, 0 otherwise

Incidence Matrix of a Design Rows labeled by lines Columns labeled by points a ij = 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 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

Applications of Designs To minimize energy within a wireless sensor network, points represent sensors and blocks 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 teams (or 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

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 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

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 K 5 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 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

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

Represent Z 7 Example with Fano Plane

Why Does Z 7 Example Work? Based on fact that the six differences among the elements of {0, 1, 3} are exactly all of the nonzero elements of Z 7 “Difference sets”

Your Turn! Find a 2-(13, 4, 1) using Z 13 Find a 2-(15, 3, 1) using the edges of K 6 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 v is a 2-(v, 3, 1) design 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

When do designs exist?

More general existence? In 1853, Steiner asked about t-(v, k, ) designs The case of t=2 received particular interest in the 1940’s 2-(v, k, ) designs were resolved by Wilson in the 1970’s 160ish years later, there are only a few known examples of designs when t>2

Conjecture Steiner’s 1853 conjecture: For every k, t and λ, if v is sufficiently large (and the parameters satisfy divisibility conditions), then there exists a t-(v, k, ) design Peter Keevash proved the conjecture (and then some!) last January 2014!! Keevash unveiled the results in a 25 minute talk at Oberwolfach  Uses hypergraphs, matchings, probabilistic methods, random greedy algorithms….

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

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