•
~~This
research was supported in part by an NSF Grant No. GP19568.
ON RANDOM SEARCH USING BINARY SYSTEMS
DERIVED FROM FINITE PROJECTIVE PLANES~'
May 1973
I.M. Chakravarti
Depar·tment of Statistics
UniiJeJ'sity oj' North CaroUna at ChapeZ H1:U
Institute of Statistics Mimeo Series No. 869
•
ON Rtt-JDOM SEARCH USING BINARY SYSTEtItS
DERIVED FRQ"v1 FINITE PROJECTIVE PLPNES
by
I. M. Chakravarti
Department of Statistics
University of North Carolina at Chapel Hill
SUMMARY
If the points of a finite projective plane with
are identified with the elements of the basic set
lines with the functions of a system
s+l
Sn
points on a line
of search and the
F of search, then the incidence
matrix of the plane defines a separating system of search on
F.
Bounds
on probabilities of termination of search processes when functions are
chosen by random sampling with replacement from the line set have been
worked out using combinatorial properties of the point-line incidence in
a finite projective plane, relevant to search.
on
PG(2,2) and
and Manglik [1].
PG(2,3)
This extends the results
search systems earlier obtained by Chakravarti
•
1.
INTRODUCTION
1.1
Renyi ModeZ for Search.
for search as follows.
Let
elements
In ([4], [5]),
S
be a finite set of
n
described a model
n
distinguishable
It is required to determine the identity of an
unknown element
x
be longing to
S
from the rest of the elements in
possible to observe
x
or equivalently, to separate it
n
Sn'
It is assumed that it is not
directly but one can choose a sequence of functions
f , f , •.. , f
from a given family or system
2
l
m
the values of these functions at
x.
R~nyi
We shall consider
F = {f l ,f , ... ,f M}
2
F
x
F
in order to determine the identity of
to be a finite family of
Sn = {a ,a 2 , ... ,an}'
l
and
of functions and observe
M functions,
If
F contains a function
which takes on different. values for different elements of S,
a single
n
observation of this function at
x
will reveal its identity.
F is much smaller
the number of different values taken on by a member of
than
n.
In practice,
Here, we shall consider the special case where each function
takes on only two values 0 and 1.
Such a family of functions will be
called a binary search system.
Problems of sorting, questionnaire, information retrieval, decoding
a received message, search for a lost or hidden object, search for a
mistake in a computer program, search for locating a failure in a complicated mechanism (a car, a computer, an airplane etc.) can be treated under
this model of search.
Separating
1. 2
S
L
If
A system
F
of functions defined on the set
is a separating system if to every pair of distinct elements
n
:t.
Sy:~ temc.
f
of
u.
J
F
S
/l
,
there exists in
F
a function
f
such that
a. ,
J
f(a.) -f f(a.).
J
1
is /lot a separuting system, there will exist at least two distinct
elements
u.
1
and
a.
J
in
S,
n
such that
f(a.) = f(a.)
1
J
for all
f
in
F.
-2-
•
the other hand, if
On
F is separating, then there exists at least one
sequence of functions, namely the one consisting in observing the values
of all the functions in
of
at
F
x,
which will identify all the elements
S .
n
A separating system
F can be characterized in an alternative manner.
Let
N
=
((f. (a.))), i
1
J
Mx n
denote the
=
=
1, 2, •.• , M, j
matrix whose
1, 2, ... , n
f. (a.).
(i,j)-entry is
J
1
Then
F
is a
N are
separating system if and only if all the columns of the matrix
distinct.
1.3 Notions of Homogeneity.
the function
1T
g
defined by
J
1T
J
(a. ,a. , ... ,a. )
J2
(2 ~ k ~ n)
k
in
of distinct elements from
F
such that
k-plet
as well,
F
n
Sn'
k-plet
the number
~
of
a. , ... , a.
J1
Jk
F is defined to be a strongly homogeneous system of order
A system
k, if for every k-p1et of distinct elements
a sequence of
the number
F,
= f(a. ), does not depend
Jk
f(a. )
J1
on the choice of the
in
S , is called weakly
if for every
Jk
f
f
{1,2, ... ,n}.
F of functions defined on the set
homogeneous of order
functions
belongs to
g(a.) = f(a (.))
is any permutation of the integers
A system
J1
of functions defined on
F
Sn' is called completely homogeneous, if for every
the set
where
A system
k
elements
1\ (y 1 ' ... , y JI,
f(a. )
J1
Yi , ... , Yi
1
)
(a. , ... ,a.)
Jl
Jk
of
S
n
and
(not necessarily all distinct),
k
of functions
f
in
F
for which
k
=
YR, , ••• , f(a. )
1
Jk
=
does not depend on the choice of the k-p1ct
yR,
k
(a. , a. , ... , a. ) ,
J l J2
Jk
but it
-3-
•
may depend on the sequence
(Yt , ... 'Yt )
1
k
The following properties hold for the different types of homogeneity
defined above.
(a)
If a system
F is weakly homogeneous of order 2 and
(which is always true except when
R
2
<
R
l
F consists of constant
functions only), then it is also a separating system.
(b)
A completely homogeneous system containing at least one constant
function is a separating system.
(c)
Strong homogeneity of order
orders
(d)
t::o;
implies weak homogeneity of all
k
imples strong homogeneity of
k.
Strong homogeneity of order
every order
k
t < k.
However, for a weakly homogeneous system,
the analogous property does not hold.
(e)
If a binary system
t
::0;
F is weakly homogeneous of all orders
2k, then it is also weakly homogeneous of order
In particular, a binary system
order
2k+l (k=1,2, ... ).
F which is weakly homogeneous of
2 (k=l) is also weakly homogeneous of order 3.
Many weakly and strongly homogeneous binary systems of search have
been constructed by Manglik [2], from finite geometrical structures.
2•
RANDO'-1 SEARCH
2. 1
Random Search Using a Separating Sys tern.
A method for the
successive choice of the functions
f , ... , f
from a given family F,
l
m
which leads in the end to the determination of the unknown x will be called
Let
a
F
strategy of search.
be a binary system of
system for the elements of
from
A strategy may be systematic or random.
Sn
Rl
functions, which is a separating
A sequence of functions is selected
F by random sampling with replacement.
Let
f l , f 2 ,· .. , f m be
-4-
•
the sequence which happens to be chosen.
m rows and
fj(a k ).
n
Consider the matrix
N having
columns, in which the k-th element of the j-th row is
Then a fixed unknown element
the elements of
x-th column of
Sn
N,
x
by the functions
(f1 (x) , ... , f (x))
m
is separated from the rest of
f , f , ... , f
if and only if the
2
m
l
T
is different from every other
column of N.
On the other hand, the functions
element of
Sn
from the rest if all the columns of N are distinct.
The probability that a sequence of
with replacement from the system
x
f , f , ... , f
separate each
m
l
2
m functions chosen by random sampling
F,
from therest of the elements of
will separate a given unknown element
Sn
The probability that a sequence of
is denoted by
m functions chosen by random
sampling with replacement from the system
elements of
Sn
Renyi [4]
F,
from one another is denoted by
has shown that if
PI (n,m,F,x).
will separate all the
P (n,m,F).
2
F is a weakly homogeneous system of
order 2,
(2.1 )
for all x
(2.2)
PI(n,m,F,x)
in
Sn
(2.3)
1 -
Cn-I)
[~r
and,
P (n,m,F) ;: : 1
2
Further, for a binary system
and hence
;:::
-(~] [:J .
F which is weakly homogeneous of order 2
of order 3,
PI (n,m,F,x)
<;
R2]m
1 - (n-l) ~
[
+
(n-lJ[R3]m
2
R~
-5-
•
3.
SEARQi SYSTEMS FRQ'v1 FINITE PROJECTIVE PUlNES
3.1
ProbabiLities of Termination of Search Processes.
points of a finite projective plane
TI(s)
are identified with the elements of
Sn
of
with
s+l
If the
points on a line,
and the lines with the functions
F then the incidence matrix of the plane defines a strongly homogen-
eous system of order 2 (and hence a weakly homogeneous system of order
3)
and thus a separating system on the line set
L(s)
of the plane.
We
note that for the above correspondence, we have
(3.1)
n = s
2
+ s+ 1, M = s
2
+ s+l = R
l
1
2
R = 1 + s (s-l) = s - s +1, R = 2(3R
2
2
3
- Rl ) = (s-l) 2 .
Hence using (2.1) , (2.2) and (2.3) we have
(3.2)
(3.3)
P 1 (s
2 + s+1,m,L(s)x)
P2 (s
2 + s+l,m,L(s))
The exact values of
for
s = 2,3
PI (s
~
~
2
1 - (s
2+ s) (:22 _ S+l]m
+ s+l
2
1_[5 ;5+1] [:~
+ s+l,m,L(s),x)
_ S+l]m
+ s+l
and
P (s
2
2
+ s+l,m,L(s))
obtained by Chakravarti and Manglik [1], are
(3.5)
P (5,m,L(2),x) = I - 6 (~J m
1
7
(3.6)
P2 (5,m,L(2))
= 1
+
..!.!..m •
3)m-1
7
+ 2 -
3 (-
7
(1)m-1
7
(3. 7)
(3.8)
P2(13,m,L(3))=(13
m-1
m
m
m
m
m
m-1
-6.7 +27.5 +20.4 -162.3 +216.2 -120)/13
-6-
•
3.2
Some CombinatoriaL Properties of Incidence in a Finite Projective
PLane~
ReLevant to the Termination of Search Processes.
In order to calculate the probabilities (3.5) through (3.8), it was
necessary to enumerate all incidence patterns of points and lines in
~(2)
and
TI(3), leading to the termination of the search processes.
enumeration of all such configurations in
TI(s)
where
s
>
3
The
is quite
complicated and hence, it is difficult to get the exact values of the
probabilities for termination of the search process for higher values
of
s.
However, the following combinatorial properties (relevant to the
termiantion of search) of finite projective planes can be used to improve
considerably the bounds given in (2.1), (2.2) and (2.3).
Let
L(x;s) denote the set of lines in
a given point
x,
x, L(x,y;s)
with
L(x,s)
= L(s)
- L(x;s)
TI(s) , which are incident with
the set of lines not incident
the set of lines incident with
Suppose that a set of
r
but not
x.
distinct lines have been selected in
steps of random sampling with replacement from
A.
y
F
m
= L(s).
For determining the identity of a given unknown element
x we
distinguish the following mutually exclusive cases.
(i)
At least two distinct lines from
In that case, the element
L(x;s)
have been selected.
x will be distinguished from the
rest of the elements.
(ii)
Exactly one line
t
is selected from the set
that case, the element
other points on
t
t.
are incident with at least one of the
is from
Hence.
L(x;s)
incident onc each wi til the
•
are needcd .
In
x will be recognized only if the
remaining lines selectcd.
which onc.
L(x;s).
s
distinct lines of
5+1
and
s
others from
points (other than
L(x;s)
x) on
t
-7-
(iii)
None of the lines selected is incident with
case. the lines will detect
other
s
2
+ s
There exists a set of
which covers all the
2
(s - s+ 1) lines in
Proof:
with
s
Let
y
and
x.
from
s (s+ 1)
points other than
L (x; s)
covers all the s(s+l)
x
R,)
The
lines incident with
s
the remaining
x.
and let
Further, any set of
points other than
x.
be the line incident
R,
y
and
(s-l)
lines (distinct
(s-l) points, other than
y
x,
(2s-l) lines, which consists of the
incident with
incident one each with the
R,)
on
(2s-l) lines, none incident with
Consider the set of
lines (other than
x only if each one of the
We prove the following.
be a point other than
y
In that
points be incident with at least one of
the selected lines.
Lemma 3.1
x.
2
will cover
s +1
(s-l) points are covered by the other
x
and
y,
points and
(s-l) lines
properly
chosen.
The second part of the lemma is proved by noticing that there are
exactly
(s2_ s ) lines in
with
where
y
Thus, in
y
is a given point other than
x nor
x.
from
L(s)
we consider
El that the only distinct line selected is from L(x;s);
that exactly one distinct line R, from L(x;s) and (s2_ s ) or less
E
2
lines from
b)
(other than
2
which are incident neither with
m random drawings with replacement
three events:
S
L(x;s)
-5
that
(3.9)
a)
L(x;s) where the latter lines do not cover all the
x)
on
and
c)
or less distinct lines from
E
3
that no line from
L(x;s)
are selected.
L(x;s)
s
points
and
Then we conclude
•
-8-
where
peEl)
= (s+l)(l/M)m
2
s -s
I
P(E 2) = (s+l)
t=l
2
s -s
I
where
t=l
is the probability that in
distinct units,exactly
B.
=
I (_l)V(~)(t_V)m/Mm
v=o
m random drawings with replacement from
t
given units appear.
Any set of
points of the plane
L(s).
2
TI(s).
x
one line incident with both
x
lines incident neither with
x nor
and
and
number of lines (functions) such that
Thus if in
We prove the following:
(s -s+2) distinct lines will separate all the
Given a pair of points
Hence any set of
(which
S
n
TI(s)) by lines selected by random sampling
with replacement from
Lemma 3.2
M
See Feller [3].
Consider the case of separating all the elements of
are the points of
Proof:
~tom/Mm
s 2 . s+2
y
y
in the plane, there is exactly
and there are exactly
with
y.
Hence,
R.(x) = R.(y)
R2 ,
2
(s -s)
that is the
s
is equal to
lines will separate all the points of
m random drawings with replacement from
L(s),
2
- s+1.
II (s) .
we have
(s2 _ s+2) distinct lines or more, the search will terminate with the
separation of all the points of the plane.
(3.10)
P (s
2
2
+
s+l),m,L(s))
R= s
•
Hence,
2
= s+2, M = s2 + s+1.
•
REFERENCES
[1]
1. M. Chakravarti and V. P. Mang1ik (1972). "On random search using
binary systems derived from the incidence matrices of PG(2,2)
and PG(2.3)." Journal of Cybernetics, 2, 2, 26-47.
[2]
V. P. Mang1ik (1972). "On some strongly and weakly homogeneous binary
search systems." Journal of Cybernetics, 2, 2, 61-77.
[3]
W. Feller (1968).
[4]
A.
•
An Introduction to Probability Theory and its
Applications. Vol. 1, John Wiley and Sons, New York.
R~nyi
(1965).
"On the theory of random search."
BuZZ. Amer.
Math .. Soc., 71, 809-828.
[5]
•
•
•
•
A.
R~nyi
(1969). "Lectures on the theory of search." Institute of
Statistics ~meo Series No. 600.7, University of North Carolina
at Chapel Hill, N.C. 27514.