COMBINATORIAL DESIGNS AND RELATED DISCRETE AND ALGEBRAIC STRUCTURES

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21 Νοε 2013 (πριν από 3 χρόνια και 4 μήνες)

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Sarah Spence Adams

Prof essor of Mat hemat i cs and

El ect ri cal & Comput er Engi neeri ng






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 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 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

0

1

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


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
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
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
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 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

0

1

2

5

6

4

3

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 non0
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 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

Codes

Groups

Graphs

Designs

Latin

Squares

Difference

Sets

Projective

Planes

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