sarah spence adams professor of mathematics and electrical & computer engineering combinatorial...
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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