Download Nanglik, V.P.; (1970)On the construction of systems and designs useful in the theory of random search."

1
The researeh in this report 7JJas supported in part by a National, Science
Foundation Eeseareh Grant and lJaB lJritten under the direction of Professor
I. M. Chakravarti.
fhti,tten
JU'{,y~ 19'10~
printed
October~
19'10.
N07JJ at the Department of MathematiCB~ EUsabeth Ci.ty State
Elizabeth City~ North CaroZina 2'1909
ON THE CONSTRUCTION OF SYSTEMS AND DESIGNS
USEFUL IN THE THEORY OF RANDOM SEARCH1
V. P. Mang1ik
*
Department of Statistics
University of North Carolina at Chapel, Hilt
Institute of Statistics Mimeo Series No. 717
October, 1970
2
lJniversity~
ON THE CONSTRI1crION OF SYSTEMS AND DESIGNS
USEFUL IN THE THEORY OF RANDOM SEARCH
by
V. P. Manglik
University of North CattoZina at Chapel Bill
ABsTRACT
This paper presents some methods of constructing systems and designs useful
in random search theory.
A brief introduction to basic concepts useful in simple
search and relation of search systems with important combinatorial configurations
is given in Section 1.
Section 2 gives some new pairwise balanced (PWB) designs.
Construction of the systems is discussed in Sections 3 and 4.
In Section 5, a
criteria for comparing search systems is given together with some methods for
constructing systems optimal in this sense.
The results are given without proof
because proofs will appear elsewhere [1].
1. I NTR>DLCTION
Problems of search occur in almost every field. such as medical diagnosis,
the parameter of a probability distribution, search for an unknown or hidden object, etc.
The general search problem is to determine an unknown object
x e S,
the basia set of all possible objects of search, by selecting one after the other
certain test funations
f ,f , •• from a family
l 2
F
defined on
S and observing
The research in this report tUas supported in part by a National Scienae
Foundation Research Grant and was wPitten undeP the direotion of ppofessop
I. M. ChakpavaPti.
2
their values at
x
till we have enough information
. finite consis ting of, say,
a
j
~
S
n
elements
a
( i · 1,2, ••• ,m) (j • 1,2, ••• ,n)
O,l,.,q-l
to find
a ,., an;
2
l
f
~
i
and taking only
associate an
mx n
matrix
at the point
E F
A system
a
~
j
S.
depending on
q
distinct values
M are all different.
of values of
b
jt
f
r
Lt=l b jt = n.
M
correspond to the flmctions and
S.
is called sepazaating on
S
A separating system
F is a separating system on
Y
be the finite set
different points of
Renyi [2] introduced
corresponding to the value of
The rows of
F of functions on
proper subset of
at
is
To a simple search problem, we can
M III «fi(a »)
j
the columns to the elements of
of
S
such that at each stage the choice of a function depends to some ex-
important notions of simple search theory.
i
When
F
tent on chance, the search is called simple ztemdom seazaoh.
f
x.
f
S,
j
III
F
S.
S
on
e: F,
j
III
S
f E F
such that
1,2, ••• ,m;
Associating with each element of
iff the columns
is called minimaZ if no
For every
{Yl'Y2' ••• 'Yr}
S
t·
let the set
fj(a). Yt
1,2, ••• ,r,
a probability of
P(fj(x) = Yt) = bjt/n III Pjt' entropy of f j i$
r
H(fj ) = Lt=l Pjt·tagU/Pjt). Then for any separating system
lin,
we
have
t;lIIlH(fj)
~ logn, and
an
F
on
S,
optimaZ separating system is one for which the equality
holds.
For a choice of
R (ai ,ai , ••• ,ai )
-It
12k
III
k distinct elements of S,
k
elements.
k
if
j
if,
R..
-It
(ai ,8t , ••• ,ai ) = R..,
12k
-It
Further, let us choose a sequence of
,Yt .••• ,yp
of Y also and let
1
2'K
l{f: f(aij)=YR ; j=1,2, ••• ,k; fEF}I.
k
let
12k
Yt
O'l"de'l"
8t,8t , ... ,ai;
12k
I{f: f(ai )=f(ai ) •••• =f(ai ), fEF}I. System F is
weakZy homogeneous of orde'l"
choice of
say,
~(Yt ,Yt ' ••• 'YI ; ai'
1
2
lc
1
k
k
elements
ai ) =
k
Then the system is strongZy homogeneoua of
~(Ytl,Yt2,••• ,Ytk; ail,ai2, ••• ,aik)
dependent of the choice of
independent of the
elements of
S.
III
~(Ytl'Ylz,••• ,Ytk)
in-
3
A system
noted by
F
F(m,n)
of
m functions defined on
and its
m
x
n
systems will thus be denoted by
MW
S
matrix by M.
SHS(m,n)
and
of
n
elements will be de-
Strongly and weakly homogeneous
WHS(m,n)
with matrices
M
S
and
respectively.
Following result is an easy consequence of above definitions and of partial-
ly balanced arrays ([ 3]) :
A partially balanced array of
t
N columns,
and parameters A(Y1'Y2"" 'Yt)
SHS(N,m)
of
order
t
in
s
m rows,
s symbols,
strength
is equivalent to transposed matrix of a
symbols with parameters
Rt (Y1'Y2""'Yt) =
A(Y1'Y2""'Y t )'
PWB designs ([ 4]) are found to be closely related to search systems.
posed matrix of
PWB design.
SHS(m,n)
of order
2
in 2 symbols gives incidence matrix of
However, converse is not ture, in fact there are examples when the
system is not even WHS.
In Section 3, a stronger result is given:
Incidence matrix of a PWB design is related to the matrix of a SHS
2
Trans-
in 2 symbols iff each treatment has equal number of replications.
of order
(These PWB
designs have been termed as SYmmetrical Unequal Block (SUB) arrangements by [5].)
2. Sat£ New PAB I£slGNS
2.1. PWB designs based on finite geometrical structures:
Using the results on finite geometrical structures ([6]) following PWB designs can be obtained by considering points not lying on the structure as treatmen ts and the lines as blocks.
(a)
PWB«q+1)(q+1-n);
k
{(n-1)q+n, n} - Me.
1
= q+1,
in
k
2
PG(2,q)
= q+1-n';
with
q
A. 1),
= O(modn),
from an
n'
= q/n.
4
(b)
PWB(v. q2(q2_q+l);
PWB(v
= q(q 2+1);
quadric in
k
2
k
k
2
• q,
k
= q-l;
3
and lines as blocks.
k
• q(q+l),
l
from
=
in
• q+l"
l
PG(3,q)
PWB(v • q(q +1);
• q(q-l); ~. 1)
2
PG(2"q2), q
pm.
• q2"
l
=0
x q+ l + yq+l + zq+l
(c)
k
k
= 1)
from elliptric
Also
2
2
~
~:II
• q;
q+l)
i, obtainable by
taking planes as blocks.
2.2. PWB designs based on finite Baer subplanes:
A sUbplane of a projective (affine) plane
and lines
L such that
(ii)
V
V
p'
E:
P
L'
E:
P
3
3
A proper sub-plane
is a subset
Q of points
p
Q is itself a projective (affine) plane, relative to
incidence relation given in
(i)
P
P.
A Baer sub-plane is a sub-plane such that
a unique line
L
of
is incident with
p EQ ~ L'
a unique point
Q(2"m)
Q 3 p'
E
P(2,s)"
is incident with
s . pn
L
p.
is a Baer sub-plane iff
2
s. m
([1]).
Considering points not in
as treatments and lines as blocks following
Q
PWB designs are obtained.
(a)
3
PWB(v· m(m -1);
k
• s-m,
l
k
= s;
2
A. 1)
from Baer projective sub-
plane.
(b)
2 2
PWB(v. m (m _l);
sub-plane.
k
l
.. s-m"
k
l
• s-l,
k
3
• s;
~.
1)
from Baer affine
5
3.
STRaG.Y
HoMxiENEous
Consider" system F(m,n)
SYsTEMS OF
ORJER 2 IN 2 SVM30LS
and its matrix M having entries, say,
0
and
For i-th and j-th columns of M let
Aij • Number of rows having 1 in both the columns
lPij .. Number of rows having
0
in both the columns
Yij .. Number of rows having
1
in i-th and
0
in j -th column
~ij
0
in i-th and
1
in j-th column.
We then have
A+
~
For a
~
+r +A
A•
(ii)
r .. A,
matrices.
mJ,
n
II:
SHS(m,n)
A and
y.
~
mx n
NN'
J - N
l
x
n
matrix with all entries
1.
6 • ~Jn
..
(m"'A-2~)J
n
•
of order 2 in 2 symbols is specified by only two parameters
..
N..
n
n
If we denote
C
is
n
r .. yJ ,
H'
..
yI
by
n
+
N,
then
AJ •
n
corresponds to
parameters
(vi)
J
of order 2 we shall have
nxn
(v)
where
Jl.Jn ,
Thus a
r .. A'.
are symmetric and
SHS(m,n)
(i)
(iv)
n x n
4
A and
Matrices
of rows having
.. Number
m - A - 2y
submatrix M
l
of
SHS(m,n)
and
M
mxn
of order 2 in 2 symbols with
y.
will correspond to
SHS(m,n)
of same
1.
6
order and with same
pa~ameters
for
leting columns from the matrix of
u
~
l
SHS
n.
This means that de-
does not affect the prop-
erty of strong homogeneity.
Above properties of
SHS(m,n)
of order 2 in 2 symbols lead us to following
theorem.
THEOf£M 3.1.
with entries
A
necessary and sufficient condition for an
and
0
to be the matrix of a
1
bols with parameters
A and
(A+Y)1 = r
(i)
1'M •
(i1)
C · H'M
III
x
n
matrix M
of order 2 in 2 sym-
V is that
1
VI + AJ
i.e. each column of H have same number of l's
i.e. diagonal entries of
off-diagonal entry is
CoRou.ARv 3.1.
SHS(m,n)
m
Care
r
and every .
A.
A necessary condition for
SBS(m,n)
of order 2 in 2 sym-
m ~ n.
bols is
In view of this theorem, we conclude that incidence matrix of a SUB arrangement is closely related to a
V = (r-A).
A and
with
k
i
=k
SHS(b,v)
Remembering that
of order 2 in 2 symbols with parameters
BIB designs are a special case of these
this includes all BIB designs.
Therefore, for constructing
SHS(m,n) of order 2 in 2 symbols, the whole force of BIB designs is available together with other SUB arrangements.
4. WeAKLy HOMOOENEOus
SYSTOO OF
ORIER 2 IN 2 SVf4nS
In the notations of Section 3, for a WHS(m,n)
of order 2 in 2 symbols, we
have
A+
~
+ r + /). = mJ,
A+
~
•
aJ
7
where
a ij
"ij+41ij • Number of rows having i-th and j-th columns same and for
III
WHS(m,n) of order 2
a ij • a
for all
i
and
j.
These properties give follow-
ing theorems.
THEOREM 4.1.
Let
A be a
AA'
then
A'· M represents a
Let
AA'
(m-a)I
III
n
THECREM 4.3.
AI
+ BJ,
n
•
1
2
= •••
such that
' 'J
~
"
i. .. .. rn
i..
of order 2 only if
r
l
n x m matrix with entries
M to represent
where
A be a
Let
or
l A •••
= A. r.2 •••• A
WHS(m,n)
A be a
a necessary condition for
that
0
• r
III
r ,
n
was is a SUS.
Furthermore, in this case
THEOREM 4.2.
n)( m matrix with entries
1
and
-1,
then
was (m,n) with parameter a is
a = 2a - m,
n)( m matrix with entires
1
and
-1
such
that
+ aJ
(i)
AA'
III
(ii)
-"31
:S
i!.m
:S
(ii1)
m and
a
are both even (or odd),
then
(m-B)!
1
A' .. M represents a
WHS(v,v)
WHS(m,n)
with parameter
a
III
(B+m)/2.
can be obtained from the 2 and 3-class association schemes.
In
this direction following results are obtained.
THEOREM
Bl
4.4.
A sufficient condition
for the
in a 2-class association scheme to represent a
is that
a
III
v x v
association matrix
WHS(v,v)
with parameter
a
8
Tt£OREM 4.5.
represent a
Association matrix
WHS(v,v)
with parameter
B
1
a
in a 3-c1ass association scheme will
if
3
r
=
a
pii + 2P23 i-I
THEOREM 4.6.
v x v
1
+
Consider a 2-c1ass association scheme in
association matrices
with entries
2
3
\' 3
L Pii
i=l
and
BO,Bl'B 2 -
Then
-1 will represent a
v x v
v
3
+ 2P23
objects with
2
N .. Ii=oCiB i
matrix
with parameter a rI v
WHS (v, v)
if
either
(i)
.
(1,1,-1) ,
t
2
P12 - P12
.£ =
(1,-1,1) ,
pt
.£
or
(ii)
It can be noted that
THECREM 4.7.
v x v
eter
(a)
(b)
(c)
(d)
Nand
a
~
v
- p
-N
2
12
1,
a
..
V - 2pi2
.
-1,
a
•
2
V - 2P12-
represent same
WHS_
Consider a 3-class association scheme in
association matrices
3
N • t i=oCiBi
12
..
with entries
B ,
i
1
i . 0,1,2,3_
and
Then the
-1 will represent a
v
v x v
objects with
matrix
WHS(v, v)
with param-
if one of the following holds.
1
1
2
2
3
3
P13 + P23 • P13 + P23 • 1 + P13 + P23'
1
1
a
v - 2(P13+P23)
2
2
3
3
1
1
e = (1,1,-1,1) , P12
+ P23 • 1 + P12 + P2 3 • P12 + P23'
1
1
a • v - 2(P 2+P23)
1
1
1
2
2
3
3
.£ = (1,-1,1,1) , 1 + P12 + PI3 • P12 + P13
P12 + P13'
2
2
a
V - 2(P +P12)
13
1
1
2
2
3
3
(1,1,-1,-1), P12
c
1 + P1 2 + PI3 • 1 + P12 + P1 3'
+ P1 3
1
1
a ... v - 2(P12+P13)
.£ =
..
(1,1,1,-1) ,
-
..
..
..
9
(e)
£
..
(1,-1,1,-1),
a
(f)
£
•
•
v -
1
1
1 + PIJ + P2J •
(1,-1,-1,1),
J
3
v - 2(P 1 3+P23)
(1,-1,-1,-1), a ..
£ •
..
2(pi2+P~J)
a •
(g)
1 + Pi2 + pi3
V -
2.
In view of the theorems 4.4 to 4.7, we have to look for the association
schemes which satisfy conditions of anyone theorem.
Among the 2-c1ass assa-
ciation schemes condition hold good for the following.
(a)
Triangular Association schemes with
(b)
L (n)
(c)
NLg(n)
(d)
SLB
(e)
Pseudo Geometric Association schemes with
schemes with
g
schemes with
schemes with
t .. k-r+l
For a
a = R
2
2
for
2
v. n ,
k· 21:'-1
or
5.
v .. n
n" 2g.
and k . 21+1.
PI(n,N,F,x)
R2/RI
fine a
tk
[(r-l)(k-l)+t],
for
RANooM SEARcH
of order 2 in 2 symbols
0
SYSTEMS
and
I
[:~t
'
with parameter
following result is given in [2].
using family
of
v ..
t.. k-r-l.
P1(n,N,p,x)
where
v .. 6.
n . 2g.
ca.PARISOi OF SIKJL.E
WHS(RI,n)
v .. 15,
WHS
F
1 - (n-l)
is probability of determining
of functions.
for fixed
~
n.
But
to be optimal WHS
this bound is attained.
x
~
S
uniquely in
N steps
Maximizing this probability means minimization
~/R1 ~
(n-2)/2(n-l),
for fixed n
if
R
2
(for proof see [2]).
and
R
I
De-
are such that
10
For a
y
we have
SHS(Rl'n)
R /R
2
III
l
of order 2 in symbols
R - 2y.
l
0
and
1,
Thus we look for these
with parameter
SUB
"
and
arrangements for
which
R1-2y
THEOREM
• 2n-2
(n-l)
5.1.
Only
4y(n-l)
,
or
PWB
or
n
b
4(r-") (v-I)
•
v
design with above parameters is the one for which
all blocks are of same size.
THEOREM 5.2.
k • t,
"III
t-l
BIB design
gives the optimal WHS (4t-2 ,2t) •
THEOREM 5.3.
of an optimal WHS
ments.
SHS
D with parameters
(in fact
of an optimal
4t-2,
SHS)
of order 2 on a set
S
of
4t
n = 2t
4t
III
2t,
r
III
2t-l,
was is SHS.
imply existence
S with
n
III
2t
ele-
elements, there exists a
Pl(n,N,F,x)
is maximised
exists.
Existence of Hadamard matrix of order
WHS(4t-l,4t)
v
In fact this
S such that the lowest bound for
if Hadamard ma trix of order
THEOREM 5.4.
III
Existence of a Hadamard matrix of order
In other words, for every set
of order 2 on
b
4t
imply the existence
of order 2 in 2 symbo,ls with parameter
This research was done mder the direction of
r.
M. Chakravarti.
a
III
2t-l.
11
·w
REfERENCES
[1]
Manglik, V.P. (1970), "Construction of Designs and Systems Useful in
the Theory of Random Search,"
Ph.D. Dissertation under preparation at University of North
Carolina at Chapel Hill.
[2)
B4nyi, A. (1965),
[31
Chakravarti, I.M. (1963), "Orthogonal and Partially Balanced Arrays and
their Application in Design of Experiments,"
Butt. Amer.
"On the Theory of Random Search,"
Math. Soc... 71, p. 808.
Mem'ka.. Heft 3, p.231.
[4]
Bose, R.C. (1968), ''Notes on the application of combinatorial theory to
design of experiments."
[5]
Kishen, K. (1940),
"Symmetrical Unequal Block Arrangements, rr
Sankhya, 5, p. 329.
[6)
Barlotti, A. (1965), "Some topics in finite geometrical structures,"
lust. of Stat. Mimeo Series No. 439, University of North Carolina
at Chapel Hill.
[7}
Dembowaski, P. (1968),
[8]
Blackwelder, W.C. (1966), "Construction of BIB designs from association
schemes," Inst. of Stat. Mimeo Series No. 481, University of
North Carolina at Chapel Hill.
[9]
Hall, M. (1967),
Finite Geometries.
CorrhinatoPia1. Theory..
Blaisdell Pub. Co.