0 7
6 1
5 0
1+ 6
0 6 5 1+
7 1 0 6
8 7 2 1
9 8 7 3
1 9 8 7
3 2 9 8
5 1+ 3 9
2 3 1+ 5
1+ 5 6 0
6 0 1 2
9
5
0
2
1+
7
8
6
1
3
8 9
7 8
2 7
1 3
8 7 1
9 8 2
6 9 3
1 0 1+
3 2 5
5 1+ 6
7 6 0
0 1 7
2 3 8
1+ 5 9
1
9
8
7
2
3
1+
5
6
0
1
8
9
7
3 5 2 1+
2 1+ 3 5
9 3 1+ 6
8 9 5 0
8 6 1
3
7 0 2
1+
6 1 3
5
0 7 8
6
1 9 7
0
2 8 9
1
2
9
7
8
6
0
1
2
3
1+
5
9
8
7
COM BIN A TOR I A L
MAT HEM A TIC S
YEA R
Feb'PUCCl'y 1969 - June 1970
LECTURES ON THE THEORY OF SEARCH
I
by
2
ALFRED RENYI
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 600.7
MAY 1969
I. Research partially sponsored by the National Science Foundation under
Grant No. -GU-2059.
2University of North Carolina and Hungarian Academy of Sciences
LECTURES ON THE THEORY OF SEARCH 1
by
Alfred Renyi 2
Hungarian Academy of Sciences
CHAPTER 1.
Introduction
§l.l.
The aims of these lectures .
. Problems of search occur in practically every field of human activity.
Some typical examples of search are:
1)
search for a failure in a system, e.g. a computer, a car, etc.
2)
search for a lost or hidden object
3)
medical diagnosis
4)
chemical analysis
5)
search for a mistake in a series of computations, or in a
program for a computer, or in a proposed proof of a
theorem, etc.
6)
search for some items of information in a library
7)
search for a mathematical object with prescribed properties
(the root of an equation, the maximum of a function, the
construction of some combinatorial design, etc.)
8)
decoding a received message
9)
search for the parameter of a distribution by taking a
sample
10)
guessing games, like the game of 20 questions
Many particular problems of search have been investigated by setting
up a mathematical model of the situation.
However, up to now there does
not exist a general mathematical theory of search.
The aim of these
lectures is to deal with the theory of search in a more systematic way
Research nartially s~onsored by the ~~ational ~; cience Foundation under
Cra!1t !';o. GU-2059.
2. This "wrk vIas done ,"hile the author ,<las at the Dept. of Statistics at
the University of North Carolina.
L
-1.2as has been done previously.
We shall try to point out the outlines of
a general theory, hoping that this may contribute to the development of
this field of research, which, besides its obvious importance from a
practical point of view, presents many challenging unsolved mathematical
problems;
one of the aims of these lectures is to call attention to
these problems.
§1.2.
The relation of search theory to other branches of mathematics.
Search theory--as interpreted in these lectures--is a chapter of information theory.
If we are searching for something, even if this is a material
object, our endeavour can be described by saying that we want to get certain
information, about the thing searched.
The process of search can be char-
acterized as the process of collecting bits of information until the information obtained is sufficient for our purposes:
after this, the information
obtained has to be processed, the information needed has to be extracted
from the data, i.e. the information we need separated from the information
which is unnecessary for our purposes.
This being the case it is clear
that a close connection exists between search theory and other branches
of information theory, as coding theory, the theory of pattern recognition, etc.
Problems of search are also closely connected with mathematical
statistics, because the collection of information on the object searched
is a special sort of sampling.
To put it otherwise:
searching means
to make indirect observations and to extract from these the information
needed.
Search theory being closely connected with both information
-1.3theory and statistics, it is evident, that is is especially closely connected
with the information-theoretical point of view in statistics.
Looking at
problems of search as statistical problems, one usually can take the
Bayesian point of view.
Search theory leads to many highly interesting combinatorial problems;
conversely, results of combinatorial mathematics obtained originally for
other purposes can be often successfully applied in solving problems of
search.
Thus combinatorial mathematics is one of those branches of
mathematics which are most intimately connected with search theory.
§1.3.
The general mathematical model of a search process.
In what follows we shall deal with the following ,model of a search process:
There is given a nonempty set
search.
We shall call
S:
the set of all possible objects of
S , for the sake of brevity, the basic set.
We
suppose that the unknown thing we are searching for is an element of the
set
find
S:
we shall denote it by
x*
in
S.
x* = x*(w),
in
S:
(w
E
x*.
Thus the aim of the search is to
~),
(~,
defined on a probability space
x* ,
defined by
~(A) = P(x*
finite or denumberable, we suppose that H?(A)
S ,'
is a random variable,
A, P),
in other words we suppose that there is given on
distribution of
of
x*
In general we shall suppose that
in the general case
subsets of
S).
defined on
S
1P (A)
E
A)
•
S
with values
a prior
(In case
S
is
is defined on all subsets
is defined only on some
We suppose further that a family
F
a-algebra
of functions
of
f
is given, and that the search process consists in selecting
one after the other certain functions
f , f , ..•
2
l
from the family
F
-1.4and observing their values at the (unknown) point
x*
to be identified on the basis of the observed values
Each function
f
E
F is called a test-function,
family of admissible test-functions.
f c F
the functions
space
(~,
f(x, w)
on
A,
P).
f
has
fl(x*), f (x*),
2
F itself the
and
x ,.
it is called search in the
presence of noise (or simply noisy search) if each
is a random variable i.e.
x*
The search is called noiseless if
depend only on
XES
thus
=
f(x, w)
f
E
F
for each fixed
defined on the probability
[Of course the distribution of the random variable
has to depend on
x , otherwise it does not give any information
x.]
F one after the other the
A strategy is a rule for selecting from
test-functions
f , f , .•.
2
l
to be observed at the point
is sequential if the choice of the function
on the previous observations
f
k
,
f1(x*), fZ(x*), ""
x*.
(k=2,3, ... )
the strategy in advance,
f , f , ...
l
2
F
A strategy (independently whether it is
f
k
f
k
,
uniquely, or it may prescribe, at each stage, only
a probability distribution on the subsets of
the function
Thus in the case
is specified by
sequential or not) may specify the choice of the function
in
otherwise
(Sequential strategies may be also called
strategies with feedback.)
(k=1,2, ... ) ,
depends
fk_l(x*) ;
the strategy is called non-sequential (or predetermined).
of a non-sequential strategy the sequence
A strategy
has to be chosen from
F
F,
according to which
In the second case we
call the search procedure a random search.
The above described general model will be more specified in what
follows.
In the folling
§
we discuss the classification of search
problems according to several points of view.
-1.5-
F contains at least one function which takes on different
Of course if
values for different points in
point
x*
this single observation determines
search is finished.
of every function
lsi
x*
uniquely, and the
However in most problems of search no such function
F
is contained in
number
then observing such a function at the
S
As a matter of fact the number of possible values
f
F
in
is usually very small compared with the
of elements of the basic set
that the values of the functions
f
E
F
S
Usually we shall suppose
are integers, and the family
F
is finite or denumerable.
Let us introduce already here one more definition:
We call a
search problem simple, if the following conditions are satisfied:
S
is a finite set, and
(depending only on
only:
F consists only of such function
f(x)
x) which take on a finite number of different values
Without the restriction of generality we may suppose that these
values are
0, 1, " ' , q-l
for some
is one in which the unknown
x*
q,~
2.
Thus a simple search problem
may take on a finite number of values
only, the search is noiseless, and each admissible test function is
capable to take on the values
§1.4.
In this
§
0, 1, " ' , q-l
only.
Classification of search problems.
we shall classify search problems according to ten
different points of view, giving some comments on the different types of
problems.
A~
Search problems can be classified according to the cardinal
number of the basic set
S.
If
S
is finite or denumberable, we shall
-1.6call the search problem discrete.
discrete search problems.
In what follows we shall mainly consider
Evidently, if the search problem is not
discrete, and the set of possible values of each admissible function
f
E
F is finite, then the search procedure can in general be not terminated
after a finite number of steps, provided we want to specify
(see however
B)
x*
exactly
C) ).
The basic set
S
mayor may not possess some structure, which
is relevant for the search process.
usually no structure in
S
In case of discrete search problems,
is supposed (of course it may be made a
completely ordered set by labelling its elements); if
then usually there is some structure in
S
S ,is nondenumerable,
given, e.g.
S
is a metric
space or a Euclidean space with given dimension, etc.
C)
An important distinction is whether we want to find
or we are contented with localizing
x*
[this may be expressed also as follows:
certain function
g(x*)
of
x*].
if the problem is not discrete:
localize
x*
x*
to a "small" subset of
exactly,
S
We want to find the value of a
This point of view is usually taken
In this case it may be possible to
to a sufficiently small subset in a finite number of
steps, even if each admissible test function takes on a finite number of
values only.
D)
One can classify search problems as Bayesian or non-Bayesian,
according to whether a prior distribution of
However in the case when
S
x*
on
S
is given or not.
is finite, this distinction is not really
fundamental, because if a prior distribution of
we may suppose nevertheless that
x*
x*
is not specified,
is a random variable with uniform
-1,7distribution over
S , i.e. that
with the same prior probability
E)
x*
is equal to any element
x
of
S
lsi-I.
The important distinction between search under the presence of
noise and noiseless search
has been introduced already in
§1.3.
We want to add here only two remarks.
a)
It is not a restriction to suppose that in the case of
search in the presence of noise the random variables
f(x, w)
are defined on the same probability space as
x*(w) ;
if these random variables were defined on different
probability spaces, we could pass to the product space.
b)
In the case of search in presence of noise, we shall usually
suppose that for each fixed value of
iables
class
f(x, w)
F)
(where
f
are independent.
the observations
XES
the random var-
ranges over all functions in the
Of course this does not mean that
fl(x*(w), w), f (x*(w), w), ...
2
are independent.
The following classifications concern the strategy employed.
f)
We distinguish between systematic and random strategies.
A
systematic strategy is one that specifies at each stage exactly the element
f
E
F
to be chosen as the next test function which has to be observed
at the point
x*.
A random strategy is one that does not specify
exactly which function has to be chosen, it specifies only a probability
distribution on
chosen.
If
F
according to which the next function
f
has to be
A search process with a random strategy is called a random search.
F is finite or denumberable, and its elements are labeled by the
integers, i.e.
F = {f , f , ... , f n , ... } a random strategy may be
l
2
-1.8described as a strategy consisting in observing one after the other the
functions
f
k
~
f
1
k
~
random variables.
generality ~ that
(~~ A~ P)
space
•..
k2~'"
k ,
1
where
2
are positive integer-valued
Again it may be supposed without restricting the
k
l
~k2 ~
...
on which
also be called pure
are random variables on the same probability
x*(w)
strategies~
is defined.
Systematic strategies may
and random strategies mixed
strategies~
in analogy with the terminology of game theory.
Strategies can be c1assified--as already mentioned--as sequential
G)
or non-sequential (predetermined).
fl~
the selection of functions
f2~
... fN(x*)
in case we want to determine
means that we stop at the least value of
N
fl(x*)~ f2(x*)~
uniquely.
... fN(x*)
determine
random
variable~
then
N
duration of the search.
x*
x* .
usually depends on the value of
On the other
fi.xed~
hand~
uniquely~
this
such that the values
thus i f
~
give us the
x*
x*
In this case
N
is considered as a
is a random variable too.
We call
N
the
in the case of a non-
sequential strategy
N
fl(x*)~
are observed one after the other or simultaneously.
... ~ fN(x*)
However~
is
employed~
to be observed is of course con-
fl(x*)~ f2(x*)~
tinued until the observations
necessary information;
If a sequential strategy is
and it is irrelevant whether the values
even in the case of a nonsequential strategy it may occur that
a segment
(fl(x*)~ ... ~
fM(x*»
is sufficient to determine
depends on
x* .
~
x*
of the sequence
(fl(x*)~ ... ~
the case of a non-sequential
M~ N
uniquely where the value of
thus the duration of the search may
strategy~
on the value of
depend~
x*.
even in
A search
strategy with variable duration will be called a variable-length
and a strategy with fixed quration of search a
fixed-Ie~gth
fN(x*»
strategy~
strategy.
-1.9-
H)
A strategy may aim at determining
x*
with certainty (with
probability 1) or we may be contented with identifying
probability exceeding 1 -
E
,
(0
< E
same distinction as mentioned under
with a certain probability
p
>
I -
C)
€,
If we have identified
it may be anywhere in
be combined:
>
1 -
€
J)
S.
S, as with probability
Of course the two points of view may
e.g. we may aim only at confining
to a "small" subset of
x*
this does not mean that we have
localized it with certainty to a small subset of
l-p
with
"Note that this is not the
1),
<
x*
x*
with probability
S
Search problems can be classified according to requirements
on the strategy employed.
We may a1m for instance at minimizing the
duration of the search, or--if the durations is random--the expected
duration of the search.
More generally, different costs may be attached
to different test functions, and we may aim at minimizing the expectation
of the overall cost of the search process.
However we may have some other
objective, e.g. minimizing the maximal duration (or cost) of the search
process.
[Of course the word "cost" is used here in the same general
sense as in decision theory:
Thus for instance the cost of the use of
a test function may mean the duration of the observation and evaluation
of this particular test function,]
K)
A delicate distinction which is of importance especially in
connection with random strategies is the folloWing.
If we consider a
strategy, which is "good" in the sense that it leads to the identification
of the unknown
x*
by a relatively small expected number of steps,
it is not sure, that the same strategy works as well for all other values
of the unknown too.
Let us denote by
N(x, w)
the duration of the
-1.10-
search for
x
corresponding to a given strategy;
the expectation of
N(x*(w), w)
we may require that
should be small, or we may require more,
namely that the expectation of
Max N(x, w) should be small. In the
x
first case we may call the strategy "statistically good", and in the
second case "universally good".
These alternatives can be combined in various ways, which leads to
a great variety of types of search problems.
81.5.
Discussion of the scope of the model of
§1.3.
There are problems of search which at the first sight seem not
to be covered by the model described in
§1.3.
In this
§
we want to
show how some of these problems can be nevertheless treated by a model
of the type described in
§1.3.
In some problems of search we want to find instead of a single
element of a basic set
C, a certain subset
consd,der the well knpwn group of problems:
coin (or coins) in a set
C of coins.
c
of
C.
For instance
To find the counterfeit
It seems at the first sight, that
our model can be used only if it is known in advance that there is exactly
one counterfeit coin in the set
may take as our basic set
of all subsets of
S
C.
However if this is not the case, we
the power set of
C (i,e. the set
P(C)
C) and in this way the general problem of counter-
feit coins is covered by our model.
In problems of pursuit one aims at finding a moving object, i.e. the
location of the obj ect searched for changes during the search process.
Such problems can be described by a model as described in
§1.3.
by
-1.11-
taking as the basic set the set of all admissible trajectories of the
unknown.
In such a problem we may aim only at hitting the moving target
after a certain number of steps, and its trajectory may be irrelevant.
This means that'denotingby x*
of specifying
x*
say
x*
a trajectory, we may be interested, instead
exactly, only in finding the value of a function of
g(x*) .
This again fits into the model easily (see C».
There are problems of search in which different restrictions are made
concerning the choice of the
1st, 2nd, 3rd, ...
test function.
Such
restrictions may be taken into account, without changing the basic model,
as additional conditions on the choice of the strategy.
91.6.
Some examples
H. Steinhaus [1] has raised the following problem:
pairwise comparisons are needed to arrange
according to their order of magnitude?
solved;
different real numbers
The problem is not yet completely
L.R, Ford, Jr. and
the best results have been obtained by
S.M. Johnson (2),
£(n)
n
How many
Denoting the least number of comparisons needed by
the exact value of
£(n)
is known for
n
~
12
and is as follows: t
n
2
3
4
5
6
7
8
9
10
11
12
Q (n)
1
3
5
7
10
13
16
19
22
26
30
£(13)
is either 33 or 34, and in general upper and lower
bounds are known for
Q(n)
for arbitrary
n, the lower bound being
the least integer ~ log2n! , and the upper bound being larger only by
t For n=12 it has been shown by Wells [4 J that £( 12) =30 is the least
possible value. n=12 is the least number for which Q(n) is larger than
the lower bound {10g 2n!}. See also Cesari [5),
-1.12-
a term
~
cn
where
c
is a positive constant
(c=O,028 ... ) .
Let us
consider in detail the first nontrivial case (in which not all comparisons have to be made) that is the case
M(x, y)
=
max(x, y)
and
magnitude are
4
'
m(x, y)
min (x, y) .
and the same numbers arranged in increasing order of
x*
1
<
determining thereby
x
Let us put
If the 4 numbers in question are
A possible procedure is as follows:
Xl' x 2 ' x 3 ' x 4
n = 4.
x*
2
<
x* < x* ;
3
4
M(x , x )
l
2
compare first
m(x , x ) ;
2
l
and
M(x , x )
4
3
determining thereby
and
xl
with
x
now compare
m(x , x );
4
3
,
2
x
3
with
now compare
As we have clearly
x*
1
x*
=
x*
3
=
2
x*
4
by the mentioned 5 comparisons we obtain the sequence
xr,
x~, x~,
xZ •
Let us consider now how this problem of search can be described by
a model of the type considered in
the generality, that the
1, 2, ... n,
all
n!
n
§l.3.
We may suppose without restricting
numbers in question are the numbers
in some order, i.e. that the basic set
permutations of the first
n
S
is the set of
natural numbers and that
x* is an
-1.13-
element of
S
defined on
S
then
g . . (x)
~,J
F denote the family of all functions
Let
(1 ~ i
=
< j
~ n)
as follows:
x.
> X.
and
if
1
J
1
. (x)
~,J
... x )
If
n
. (x) = 0
g.
g.
The
if
~,J
admissible strategies in the Steinhaus problem are those which consist
in choosing a sequence of elements
the choice of
f
f
(k=l,Z, ... )
k
from
F such that
may depend on the previously chosen functions and on
k
their observed values.
Thus e.g. for
n
= 4 the strategy given above
may be described as follows:
=
a)
=
b)
if
fl(x*)
=
0
fZ(x*)
if
f (x*)
l
=
0
fZ(x*)
if
fl(x*)
=
1
fZ(x*)
if
fl(x*)
1
f (x*)
2
The choice of
f
4
and the choice of
=
=
c)
0
choose
f
1
choose
f
0
choose
f
1
choose
f
3
=
gZ4
3
=
gZ3
;:
3
3
=
depends similarly on the values
f
s
on the values
is defined as
follows
g14
g13
f (x*), fZ(x*), f (x*)
l
3
fl(x*), fZ(x*), f 3 (x*)
and
f (x*)
4
A search problem which is closely connected to, but nevertheless
different from that of Steinhaus has been considered by
R.J. Nelson [3].
R.C. Bose and
This problem consists in arranging a sequence of
n
given different real numbers in ascending order by a minimal number of
pairwise comparisons, so that after each comparison of two numbers if
that with the smaller index is larger the two numbers are interchanged,
while if that with the smaller index is smaller, the sequence is left
unchanged.
Denoting by
L(n)
the minimal number of such comparisons,
Bose and Nelson gave an algorithm, which can be performed by a computer,
.
-1.14-
and leads to the required permutation in
is a rather sharp upper bound for
we give the first 10 values of
n
2
I
3
4
1
I3
5
I
Notice that
9
I
= £(n)
Ll(n)
L(n) .
steps, where
Ll(n)
For the sake of comparison
Ll(n):
I
16
I
for
n
~
12
L (n)
1
8
I
9
10
11
19
I
27
32
38
4
and
Ll(n)
>
£(n)
5 ~ n ~ 11 •
for
The problem is of importance in digital computer programming as
it often occurs that a collection of data has to be arranged according
to their magnitude.
To make the comparison with the Steinhaus problem
easier, let us suppose, that the computer which carries out the algorithm
writes down a
a
each time when two elements have been interchanged and
1
if two elements have been compared, but no interchange was necessary.
0
Clearly the resulting sequence of zeros and ones determines the permutation
in question uniquely, and the permutations can be easily reconstructed
from this sequence, reading it backwards"
To describe this search
problem by a model of the type introduced in
§1.3,
let
Sand
have the same meaning as above, and to each function
correspond a transformation
•
x.
~
(1
and
~
i
x.
•
X
<
n
~
n) .
0
~,J
>
x
j
let
of the sequence
consisting in interchanging in
)
x.
if
J
< j
&
G.
in
F
and leaving
x
x
unaltered if
the elements
x.
~
<-
x.
J
With this notation the restrictions on the search
strategy in this problem are as follows:
a)
f
b)
If
i
has to be chosen from the set
f
l
= gl and Gl
F.
denotes the transformation corresponding
-1.15-
to
8
2
gl
E
then
gi
2
has to be of the form
f (x)
2
g2(G x)
1
where
F •
c)
form
f
f
Similarly if
3
f
= g3(G G x)
2 l
(i=1,2)
and
g3
= gl'
l
where
G
i
,F,
etc.
E
Notice that the functions
Moreover,
f
k
f
2
= g2
then
f
3
has to be of the
is the transformation corresponding to
f
k
do all belong to
can be any element of
Notice that for the special case
F,
for each value of
= 4,
n
F.
k.
the solution of the
Steinhaus problem is at the same time also a solution of the Bose-Nelson
problem, i.e. the sequence of comparisons and corresponding transpositions
(1, 2), (3, 4), (2, 4), (1, 3), (2, 3)
rearranges a se4uence of four
numbers always into an ascending sequence, i.e. we may put
The reason why for
n' 5
gl
= gl , 2
'
more comparisons are needed in the Bose-
Nelson case than in the Steinhaus case can be--rather heuristically-explained as follows:
In the Steinhaus problem at each stage all the
information obtained from the previous steps is available, while in
the Bose-Nelson case only a compound result of the previous steps is
available, Le. we have less information;
thus while at each step the
set of admissible functions is the same, in the Bose-Nelson case we have
less information concerning the most appropriate choice of the next
test function.
-1.16§1.7.
In this
the end of
Irreducible strategies.
we consider only simple sequential search, as defined at
§
§l.3.
Let us consider a strategy, which consists in observing
f , f , ...
2
l
one after the other the test functions
f
where the choice of
depends in general on the previously observed values
k
correct strategies, i.e. such that for each value of
a number
N(x*)
determine
x*
E
S
such that the values
uniquely.
such that
for which
XES
1
In what follows when speaking about strategies, we mean only
(i < k)
x*
f, (x*)
S (x*)
putting
o
k
~
f,(x)
J .
fk(x*)
for
x*
E
S
= 1, 2, ... N(x*)
k
For such a strategy and for each
N(x*)
let
= f,(x*)
J
Sk(x*)
for
j
there exists
k
and
denote the set of those
= 1, 2, ... k
Clearly we have,
= S for each x*
(1.7.1)
and
(1.7.2)
c
for
Sk_l (x*)
k
A strategy is called irreducible if for every
each
k
~
subset of
N(x*) ,
Sk(x*)
Sk_l(x*».
XES
x*
2, ... N(x*) .
E
k ( N(x*) ;
S
Sk_l(x*)
Notice that this implies that
than one element for every
of those
is different from
= 1,
as clearly
end for
(i.e. is a proper
Sk(x*)
Sk(x*)
has more
is the set
which are not yet excluded as possible values of
after having observed
f, (x*)
J
for
j
~
k ,
this means that if we
apply an irreducible strategy, we never continue the search once
has been identified.
x*
x*
More generally, the irreducibility of a strategy
can be interpreted as follows:
A strategy is irreducible if applying
this strategy no unnecessary observations of test functiqns are made.
-1.17References to Chapter 1.
[1]
H. Steinhaus, Mathematical snapshots,
[2]
Lester R. Ford, Jr. and Selmer M. Johnson, A tournament
problem, American Mathematical Monthly 66 (1959), p.387-389.
[3]
R.C. Bose and R.J. Nelson, A sorting problem, Case Institute
of Technology, Computing Center, Report No. 1043, 1961,
p.l-22.
[4]
M.B. Wells, Application of a Language for Computing Incombinatorics,
IFIP Congress, 1965.
[5 ]
Y.
,f
•
New York, 1950, p.37-40.
Questionnaire, Codage et Tris, Institut Blaise Pascal,
Paris, 1968.
Cesar~,
- 2 .1.CHAPTER 2.
Simple search and its description by a search
tree resp. by a code
§2.l.
Description of a simple search by a tree.
Let us consider a simple search problem, i.e. let us suppose that
there is given a finite basic set
the set
S
S, and a family
F of functions on
taking on only a finite number of values each.
We suppose
that the search is noiseless, and we consider only systematic sequential
strategies.
(A non-sequential strategy may be considered as a special
sequential strategy.)
Let us put
=n
lsi
~ 2
In this case
the search process consists in observing one after the other the values
fl(x*), f (x*), " ' , fN(x*)
2
where
N may depend on
x*.
Without
restricting the generality, we may suppose that each function
f
E
F takes on the values
0, 1, " ' , q-l
having observed the values of
r
~
N
= N(x*)
as follows:
,
the equations
=
fk(x)
x*
of those elements of
k
is for each
x*
fk(x*)
.)
S
k -< r
~
(1
k
~
r) hold.
Let us denote by
for which
~
2) .
After
f (x*)
where
r
obtained can be summarized
belongs to the set of those elements
set contains only
e
(q
fl(x*), f (x*), ... ,
2
the information on
x*
only
f (x)
k
(If
s(e , e ,
2
l
e
k
an element of the set
for
0, 1,
x
r
of
S
= N(x*)
... ,
1 -< k
... ,
for which
this
e )
r
the set
r
where
<
=
q-l
Let
us construct a graph, the vertices of which are all the sets
S(e , e , .•. , e )
r
2
l
and the set
set
S
S(e ) ,
l
S
with
itself.
e
k
E
{O, 1, ... , q-l}
which are not empty,
Let us connect the vertex corresponding to the
with all the vertices corresponding to nonempty sets of the form
and in general let us connect the vertex
nonempty set
S(e , ... e )
r
l
to nonempty sets of the form
(1
~
r)
cor~esponding
to a
with all the vertices corresponding
S(e , e , ... , e , er+l) .
r
2
l
-2.2In this way we get a rooted directed tree, the root of which is the
vertex corresponding to
S.
This tree has the following properties:
a)
The number of edges going out from every vertex is at most
b)
The number of endpoints of the tree is equal to
q.
n, and each
endpoint corresponds to a one-element set.
c)
The unique path leading from the rdot to the endpoint correspond-
ing to the set consisting of the single element
N(x*):
The
of those
x
k-th
E:
are satisfied.
d)
S
x*
S has the length
point along this path corresponds to the set
for which the conditions
(The sets
Sk(x*)
f. (x)
J
x
of
f, (x*)
J
(1,;. j ,;. k
have been introduced already in
The set corresponding to any vertex
all those elements
=
Sk(x*)
~
N(x*»
§1.7.)
V of the tree consists of
S which have· the property, that the unique
path from the root to the vertex corresponding to the one-element set
x
passes through
V.
Thus the tree which we have constructed reflects everything which is
of importance concerning the search process in question:
It is even un-
necessary to indicate the sets corresponding to the vertices, only the
one-to-one correspondence between the elements of
S
and the endpoints
of the tree has to be indicated.
If
q
= 2 , we call the simple search a binary search, and the
corresponding search tree a binary tree.
Fig. 2.1 shows the (binary) tree corresponding to the solution of
the Steinhaus problem given in
§1.6.
for
n
=4
.
It can be seen from Fig. 2.~that in 16 cases out of 24 the length
of the search is 5 steps, in the remaining 8 cases however, the length
is 4:
That is in the case of 8 permutations already four comparisons
(the first four) are sufficient.
-2.3-
,
123~
/321 31/2
2~/3 12B
I
/3~2 3~{1 2~31 2m 1m 73/2
LI2/3
217]
3/~2
1321
123/
-2.4Thus the average durat±on of the search process is 4,666 ... ,
provided that all 24 permutations are equiprobable.
It can be seen from Fig. 2.1 that the 8 permutations in which
already 4 comparisons are sufficient are those in which the greatest
and the smallest number occupy either the first two or the last two
places.
The reason why for these permutations the last comparison
is unnecessary is the following:
M(x , x ), M(x , x )
l
2
3
4
pair
(3, 4)
necessary.
( M(x , x )
2
l
is, except for the order, necessarily the
and thus the pair
ovder, the pair
In these cases the pair
(1, 2) ,
m(x , x ), m(x , x ), except for the
l
2
4
3
and thus the last comparison is really un-
Or, expressed otherwise, in these cases
M(x , x ) )
3
4
( m(x , x ) - m(x , x ))
l
2
3
4
is negative,
and therefore after the fourth comparison the permutation is uniquely
determined.
Let us consider now how the tree corresponding to a strategy of a
simple search reflects the fact, that the strategy is irreducible.
This clearly means that from each point which is not an endpoint, at least
two edges are going out:
Thus every point which is not an endpoint
has at least valency 3, except the root, the valency of which is at least 2.
In what follows we shall call a tree, which describes an irreducible
strategy of a simple search process, a search tree.
Thus a search tree
is a rooted tree, such that its endpoints are labeled, and the valency
of each point which is different from the root and not an endpoint is at
least 3, while the valency of the root is at least 2.
Conversely, every
such tree is the tree describing an irreducible strategy of a simple
search.
To show this, it is sufficient to remark, that given a search
-2.5tree as defined above the function
N(x*)
(1~ k~N(x*»
thereby uniquely defined
and the sets
Sk_1(x*)
for all x*
= S for all x*
S (x*)
o
Sand
E
S.
E
k
~
fk(x*)
fk(x)
=
Sk(x*)
N(x*) ,
is a proper subset
where by definition
Labeling the edges going out from any
0, 1, ... , v
point of a search tree by the number
define the function
(v
for
j
and defining
,
k
>
N(x*);
~
1) ,
let us
as follows:
j
k
if
~
N(x*)
and the
of the unique path leading to the endpoint labeled by
by
are
and satisfy the conditions
characterizing irreducible strategies, that
of
Sk(x*)
k-th
x*
arbitrarily (e.g. putting
fk(x*)
edge
is labeled
f
k
(x*)
= 0)
in this way we get an irreducible strategy, to which the
given tree corresponds.
Notice, that the irreducible strategy we have
constructed has the property, that the choice of the function
f
k
does
not depend on the previously observed values; nevertheless the strategy
is sequential in the sense, that whether we stop after
tinue the search by observing
fk+l(x*) ,
depends on
k
steps, or con-
x*
A search tree is called a regular (or homogeneous) search tree of
degree
q
(q
~
2)
if its rpot has the valency
which are not endpoints have the valency
search tree is always regular.
q
+ 1.
q
and all other points
Clearly a binary
A regular search tree describes such an
irreducible strategy of a simple search in which after having already
observed
k-1
test functions
(k
~
1)
if the search has to be con-
tinued at all, then the next test function chosen according to the
strategy takes on effectively exactly
of those possible values of
the previous observations.
x*
q
different values on the set
which have not yet been excluded by
-2.6As well known any tree with
N vertices has
N-1
edges, further
in every graph the sum of the valencies of all vertices is equal to the
double of the total number of edges.
follows immediately that if
and having
n
T
Using these well known facts it
is a regular search tree of degree
endpoints, then the total number
N
(2.1.1.)
n
+
N of vertices of
q
T
is
n - 1
q - 1
An even simpler way to verify (2.1.1) is the following:
Each eadge is at the same time an outgoing and an incoming
edge, therefore the total number of outgoing edges i.e.
(N-n)q has to be equal to the total number of incoming
edges, i.e. to N-l, and thus (2.1.1.) follows.
Thus such a graph can exist only if
(2.1.2.)
where
n
k
n-l
=
is a positive integer:
is divisible by
k(q-l) + 1
Conversely, if (2.1.2.) is fulfilled,
one can always construct a regular search-tree of degree
n
endpoints.
This is most simply proved by induction:
we simply take a root and connect it with
search tree of degree
q
q
and
k(q-l) + 1
q
endpoints.
q
For
and having
k
1,
Given a regular
endpoints, by connecting
additional points with one of the endpoints of the given tree, we get
a regular search tree of degree
Of course if
on
q-l, i.e. if
n:
q
= 2,
for every
endpoints.
n
satisfying (2.1.2).
with
(k+l)(q-l) + 1
endpoints.
the equation (2,1.1,) does not mean any restriction
~
In the next
search trees of degree
q
2
there exist regular binary trees with
§
n
we shall deal with the number of regular
q, with a given number
n
of endpoints,
-2.7A search tree can also be interpreted as describing a questionary
(see Picard [1]) .
Every point of the tree, which is not an endpoint, cor-
responds to a question, and the edges going out from the point as the possible answers to the question.
Thus a regular search tree of degree
describes a questionary in which every question admits
q
possible answers:
A binary search tree describes a questionary in which only questions,
which can be answered by yes or no, can be asked.
q
-2.8-
§2.2.
Enumeration of regular search trees.
We prove now the following:
THEOREM 2.2.1.
differen~ regular
endpoints
(q
>
Let us denote by T(q, k)
search trees of degree
2,
k
>
1) .
Proof.
and having
=
= k(q-l) + 1
(kq)!
k! (q!)k
We shall prove Theorem 2.2.1 by using Prlifer's Method [2].
method consists in coding labeled trees.
vertices, labeled by the numbers
Let
T
1, 2, .•. , n.
be a tree with
of which are elements of the set
different):
{l, 2, " ' , n}
n
n-2,
the elements
(not necessarily
This correspondence is one-to-one between the set T
all labeled trees with
n
This
To such a tree by
Prlifer's method there corresponds a sequence of length
n-2
n
Then we have
T(q, k)
(2.2.1.)
q
the total number of
of
n
vertices and all possible sequences of length
from elements of the set
1, 2, ... , n
which, incidentally, furnishes a proof,
(one of the most simple known) of the result, due to A. Cayley [3],
that there are
n
n-2
different labeled trees with
Prlifer code is obtained as follows:
n
The
vertices.
a) among the endpoints of
labeled with the smallest number is removed from the tree
T,
T
that
and the
number by which the unique point to which the removed endpoint was connected is labeled , is written down;
* As
b) with the remaining tree the same
pointed out in the previous §, in a search tree the endpoints (and, only
these) are labeled: Thus two regular search trees which are isomorphic,
but their endpoints are labeled differently, are considered as different.
(Of course if the tree has n endpoints, these have to be labeled by the
numbers 1,2, ... ,n) .
-2.9procedure is repeated, and this process is continued until there remains
only a single edge.
It is easy to show that a tree can be uniquely repro-
duced from its Prlifer-code.
Let us note the following fact about the
Prufer code, which follows immediately from its definition:
labeled tree with
n
vertices and
v
and labeled by the number
T
exactly
(v-I)
times:
k
P
then
k
root, having valency
q
T.
1, 2, ... , n
are not occurring
Now let us consider all regular
and having
n
= k(q-l) +
1
endpoints.
such trees have besides the
§
q
occurs in the Prlifer code of
Thus the numbers by which the endpoints are
at all in the Prlifer code of
shown in the previous
T is a
one of its vertices having valency
labeled, and only these among the numbers
search-trees of degree
If
and
k-l
n
points having valency
As
endpoints a
q + 1;
let
us label all the points which are not endpoints of all these trees by the
numbers
1, 2, ... , k,
in every possible manner.
The total number of
such labeled trees is, acoording to the mentioned property of the Prlifer
code, equal to the total number of all sequences of length
from the integers
occurs exactly
n + k-2
1,2, ... , k
q-l
n + k-2
in which one of the numbers
times, and all the others
q
formed
1,2, ... , k
times (notice that
= q-l + (k-l)q = kq - 1 ) . The total number U(q, k)
of such
sequences is clearly
(kq-l): k
(2.2.2.)
To get the number
U(q, k)
T(q, k)
=
(q-l): q:k-l-
we have to divide
=
(kg):
,k
q.
U(q, k)
by
k: ,
because
the labeling of the vertices other than the endpoints is irrelevant
according to the definition of a
sea~ch
tree.
Thus from (2.2.2) we obtain (2.2.1).
-2.10For
q
=
2 ,
we get from (2.2.1) that the total number of binary
search trees with
n
endpoints is
T(2, n)
(2.2.3.)
1.3.5 •.. (2n-3)
=
This result is due to A. Cay.ley [4] ,
following question:
(2n-3)::
=
who obtained it by solving the
Given a non-associative operation
interpretations of the ambiguous symbol
Al
A
2
0
The answer to this question is, that the number
AoB
how many
o A
0
are possible?
n
of possible inter-
C
n
pretations is
(2.2.4.)
T(2, n) 2n- 1
n:
C
n
and thus
(2.2.5.)
=
C
n
(2n-l)
To prove (2.2.4) it is sufficient to point out, that every possible
interpretations of the ambiguous symbol
bracketing this expression:
0
A2
For instance for
n
bracketings of the "product"
«A
(A
0
0
B)
(B
0
0
C)
C»
0
D ,
A
0
(B
0
D
A
0
«B
A
0
0
0
B
0
(C
0
D»,
C)
0
corresponds a tree in an obvious way.
B
CoD
D) .
0
A3
=4
0
•••
0
An
means
the possible
are the following:
(A
0
B)
0
(C
0
D) ,
To each bracketing there
For instance the five bracketings
A
on Fig. 2.2.1.
It can be seen from this figure, that the four endpoints
0
are always labeled with
binary trees with
n
listed above
co~respond
of the product
0
CoD
Al
to the trees shown
A, B, C, D from left to right.
As in enumerating
endpoints, any labeling of the endpoints is admissible,
-2.11-
D
A
B
c
/\
c
A
D
B
ABC
D
D
B
C
-2.12to obtain
C
n
from
T(2, n)
we have first to divide by
.
n'
,. however
as left and right make no sense with respect to a tree as a graph*, and
there are
n-l
branching points, we have to multiply by
all bracketings of a nonassociative product of
n
n 1
2 -
to obtain
factors; this explains
(2.2.4).
Let us add, that the bracketing of nonassociative products can be
directly interpreted in terms of search theory.
search tree to each vertex
basic set:
P
there corresponds uniquely a subset of the
The set of all those elements
path from the root the endpoint labeled by
bracketing
correspondi~g
Let us recall that in a
x
of
x
S
for which the unique
leads through
P.
The
to a binary search tree exhibits all these
subsets in a rather compact way:
This can be seen if for instance for
the trees shown by Fig. 2.2.1 one writes besides each vertex the corresponding subset of the set
and compares this with the bracketing
A, B, C, D
corresponding to the tree.
On Fig. 2.2.2 this is shown for the first
and the last of the trees in Fig. 2.2.1.
It is easy to see that if in a bracketing of a nonassociative
product qf
n
factors only the beginnings
retained and the endings
uniquely:
[") "]
Thus the sequence of
["("]
of a bracket are
are lent out, these can be reconstructed
n
letters and
n-1
"("
signs
determines the bracketing in question, and thus also the corresponding
tree as drawn on the plane.
If now instead of the siEn
a zero and instead of each of the
consisting of
n
ones and
n-l
n
"("
we write
letters a one, we get a sequence
zeros; this sequence still determines
* It has a meaning only with respect to the figure of the tree drawn
on the plane.
-2.13-
D
A
])
c
B
C
-2.14the tree as drawn on the plane, except for the labeling of its endpoints.
Thus the nU).1lber of all such sequences is
Cayley-sequences.
n
Clearly not all (only one out of
sequences of length
2n-l
consisting of
n
1,
tinue this as long as
n-l
of
ones obtained.
n
2n-l
can
We start from the sequence consisting of a
and we replace it by the group of three symbols
zeros and
zeros
)
= (2 n- 1) Care
n
after this we replace one of the ones, by the group
n-l
n-l
The family of Cayley sequences of length
be described as follows:
single
2n-l) possible
ones and
( 2n-l)
n
(the total number of such sequences being
Cayley sequences.
We call these sequences
C
011
011;
and we con-
replacements have been made, i.e. a sequence
Thus the set of all Cayley
sequences forms a formal language (see e.g. Bar-Hillel - Schlitzenberger
[5]).
For
n
= 4 the Cayley sequences of length 7 are the following:
0001111
0010111
0011011
0100111
0101011
Let us notice, that forming all cyclic permutations of these 5 Cayley
sequences we get all the 35 sequences of length
4 ones and 3 zeros.
This is true for every
Cayley sequences of length
2n-l
n-l
n :
consisting of
taking the
C
n
and taking all their cyclic permu-
tations, we get all sequences of length
and
7
2n-l
consisting on
nones
zeros, each once.
Another way to characterize Cayley sequences is the following:
A sequence consisting of
n
ones and
n-l
zeros is a Cayley sequence
-2.15if an only if among its
k
last terms the number of ones exceeds the
number of zeros for each value of
k
(1
<
k
~
2n-l)
This character-
ization implies among others that each Cayley sequence starts 'vith a
zero and ends with at least two ones.
The correspondence between a
binary tree and its Cayley code can be described as follows:
We consi-
der a binary tree drawn on the plane (i.e. in which it makes sense to
speak about the left-hand side edge or branch and the right-hand side
edge or branch going out from any point which is not an endpoint) having
n
endpoints and
to
n-l
2n-1
branching points.
as follows:
We first label all its vertices
from
1
We start at the root, and label it with
one:
We continue the labeling by proceeding along the tree so that we
proceed along the left-hand edge whenever this is possible (i.e. until
we get to an endpoint.
If this can not be done any more we go back
until we find an outgoing edge leading to a still unlabeled point, go
along this edge to this point, if possible proceed along the left-hand
side edge, etc.
After all points have been labeled in this way, the Cayley
codeword of the tree is obtained simply as follows:
The
codeword is one if the point labeled by the number
k
o otherwise.
k-th digit of the
is an endpoint and
Fig. 2.2.3 shows a binary tree, labeled according to the
rule given above, and the corresponding Cayley codeword.
The characterization of Cayley sequences given above can be formulated also as follows:
Each Cayley sequence, read from right to left
corresponds to a sequence of
n
"yes" votes and
n-l
"no" votes
such that at each stage of the voting the number of yes votes exceeds the
number of no votes.
The problem of counting the number of such sequences,
i.e. to determine the probability of such a sequence of votes, is called in
probability theory
the ballot problem:
It was solved by Desiree Andre.
_2.1.6-
13
8
9
00 0 1100
'r
; (1
I
0
11
2, 2- ' 3 '
I I
-2.17There are several other ways to prove Theorem 2.2.1, but we
have chosen to deduce it by Priifer's method, because we shall need this
method for the solution of other enumeration problems on trees, and
wanted to introduce the method anyway.
formula for the sequence
q,
T(q, k)
We shall prove now a recursion
(k=1,2, .•. )
for any fixed value of
which can also be used as the starting point of a quite different
proof of Theorem 2.2.1.
THEOREM 2.2.2.
Denoting by
search trees of degree
q
recursion formula holds
(2.2.6.) T(q, k)
=
and
T(q, k)
n = k(q-1) + 1
(and putting
(k(q-l)+l)
E
the number of different regular
edges, the following
T(q, 0) = 1,
k
~
1):
k-1
~
~tl~«j (q-l)+l)!) a j .a j !
II
a.
(T(q,j»
J
j=O
where the summation has to be extended for all sequences
aI' a 2 ,
~-1
of nonnegative integers which satisfy the conditions
k-l
k-1
E ja.
j=O J
Remark:
For
q = 2,
k-l
2: a.
j=O J
q •
the recursion formula (2.2.6) can be written
in the following simple form:
(2.2.6.)
T(2, k)
~
!
2 h=l
Proof of Theorem 2.2.2.
(k+1)T(2, h)T(2, k-h+l) .
h
To prove (2.2.6) it is sufficient to remark
that in a regular search tree of degree
endpoints each of the
q
~
j
~
and having
n = k(q-1) + 1
points connected with the root is the root of
a regular search tree of order
(0
q
q
and with
j(q-1) + 1
endpoints
k-1) provided that we consider a single point also as a regular
-2.18search tree of order
q
convention
1.
T(q, 0)
and having one endpoint.
We call these trees the subtrees of the given
If there are among these subtrees
tree.
(This explains the
a.
J
having
j(q-l) + 1
k-l
points (j=0,1, ... ,k-1) then the equations j~O a
q
j
k-l
.L a,(J'(q-l) + 1) = k(q-l) + 1 have to hold, i.e. the
J= O J
k-l
k-l
satisfy the equations .L a, = q and .E ja. = k-1.
O
J=
J
J= O J
how many ways the number
1, 2, " ' , n
end-
and
a,
J
have to
Counting in
can be distributed among the
subtrees, and in how many ways the subtrees can be chosen and labeled,
we get (2.2,6).
From (2.2.6) it is easy to deduce a functional equation of the
generating function of the sequence
THEOREM 2.2.3.
T(q, k)
Putting
00
F
(2.2.7.)
(k=O,l, ... ) .
q
(x)
T(q, k) xk(q-l)+l
(k(q-l) + l)!
L
k=O
we have the functional equation
(x»q
(F
(2.2.8.)
q
-
q!
o.
F (x) + x
q
Starting from Theorem 2.2.3, an alternative proof of (2.2.1) can
be obtained, by developing the inverse function of the function
x
=Y
_
L!.
q!
into a power series in
x
according to the Blirmann-Lagrange
formula (see e.g. G. P61ya and G. Szegg, [6]).
for arbitrary
q,
only for
= 2,
q
We do not give the details
for which case only an equation of
the second degree has to be solved, as in this case we get from (2.2.8)
(2.2.10.)
F (x)
2
=
1 - 11-2x
-2.19and thus, by the binomial expansion
00
(2.2.11.)
F (x)
2
T(2,.n) n
,x
n.
L:
n=l
(2n-3) !! n
,
x
n.
=
Thus we obtained a second proof of the formula (2.2.3).
As regards the general case, comparing Theorems 2.2.1 and 2.2.3,
we get a combinatorial proof of the expansion
(2.2.12.)
~
(kq):
x k (q-1)+1
k=O k!(q!)k (k(q-1)+1)!
y
y = y(x)
of the function
defined as that solution of the equation
q
~ - y + x = 0
which vanishes for
x = O.
One way of looking at this
result is, that we have obtained the power series expansion of a root
of a trinomial equation of the form
yq + Ay + B = 0,
by a purely
combinatorial argument.
From (2.2.1) we get, using Stirling's formula, the asymptotic formula
k k(q-1) ( q}K
T(q, k) ~ (-)
~ Jq
(2.2.13.)
e
q.
for
k+ +00
q
and
fixed,
which shows, that the power series (2.2.12) is convergent for
Ixl
(2.2.14.)
R
(Note that
3
is increasing with
that
R
1
q++oo q
e
lim -S = -
q
<
't(
q. q- l)q-j - 1
q-1
qq
R
q
= 212
and that
3
and tends to
+00
for
q
-+
+00,
R
q
in such a way
-z.zo§Z.3.
Description of "a simple search by a code.
Let us consider a simple search problem such that all admissible
test functions take on only the values
number of elements of the basic set
0, 1, ... q-l,
S be denoted by
and let the
n.
Let us con-
sider a strategy, consistingdin observing the values of the functions
f
k
(k=l,Z, ... )
be denoted by
one after the other, and let the duration of the search
N(x*)
This means that the element
determined by the sequence
x*
is uniquely
a sequence each element of which is one of the numbers
0, 1,
Thus there exists a one-to-one mapping of the elements of
set of
n
sequence
to x*,
this being
fl(x*), fZ(x*), ... fN(x*)(x*) ,
sequences formed from the numbers
(f l (x*), fZ(x*), ... ', fN(x*) ('.'K*»
" "
S
.,
q-l .
into a
0, 1, ... , q-l.
We call the
the codeword corresponding
and the set of all such sequences the code corresponding to the
search process.
The code corresponding to a search process has evidently
the following property:
No codeword is a continuation of another codeword (or expressed otherwise:
No codeword is a segment of another codeword).
Codes having
this property are called in informatT-on,,·the:oryprefix codes.
Prefix codes
have the property that if several codewords are written one after the other,
the resulting sequence can be uniquely decomposed into codewords:
Thus
in writing codewords one after the other it is unnecessary to leave
gaps between the codewords or indicate in any other way where a codeword
ends and the next begins.
As
N(x*) "in general depends on
x*,
the
prefix code corresponding to a search process is in" general a variable
length prefix code.
If in particular
fixed length prefix code.
N(x*)
is constant, the code is a
-2.21If the maximal number of values taken on by any oLthe testfunctions figuring in a strategy is
q
we may without restricting the
~
generality suppose that these values are the numbers
O~
...
l~
q-1 .
~
Moreover we may suppose that if a test function figuring in the search
process takes on only
O~
1~
...
~
r-1.
to a simple
<
va1ues~
q
then these values are
In what follows speaking about the code corresponding
search~
is satisfied:
r
As
we shall always
suppose~
that the mentioned condition
regards the code this"me-ansthat if among the code
words of the code starting with a given se-gment of length
the next
s~mbol
O~
the numbers
may take on
1~
... r-1
r
(r
different
~ l~
k
~
values~
a
1
~
a
2
~
...
then these values are
1) .
The code corresponding to a binary search is clearly a binary
code~
i.e. such that its codewords are sequences of zeros and ones.
Let us now consider what characterizes the codes corresponding to
irreducible strategies.
For the sake of simplicity we deal with this
question only in the binary case.
THEOREM 2.3.1.
We shall prove the following:
The binary code corresponding to a simple binary
search with an irreducible strategy has-the property that if from one
(or more) of the codewords one (or more)di-gits" are
omifted~
the
resulting set of codewords is no more a prefix code.
Remark:
The statement of Theorem 2.3.1 can be expressed simply
by saying that the code corresponding to an irreducible strategy in a
simple binary search can not be shortened without losing the prefix
property.
~
Such a code is called an irreducible prefix code.
Thus the
a k 1
-2.22statement of Theorem 2.3.1 can also be expressed as follows:
To an
irreducible strategy of a simple binary search there corresponds an
irreducible prefix binary code.
Proof of Theorem 2.3.1.
If a code does not have the prefix property,
then by leaving out a digit from any of its codewords, the code thus obtained does not have the prefix property either.
Therefore in proving
Theorem 2.3.1 it is sufficient to show that if we omit one digit from one
of the codewords of the code corresponding to an irreducible strategy,
we get a code which does not have the prefix property, that is it contains
at least two codewords, such that one is a segment of the other.
(In parti-
cular this may mean that the two codewords are identical, i.e. the set of
zero-one sequences is not a code at all).
ing simple lemmas.
To show this we need the follow-
For the sake of brev1ty in course of this proof we
call a code corresponding to an irreducible strategy of a simple binary
search a binary I-code.
LEMMA 2.3.1.
If a binary I-code
(e , e , e , ... en)
2
k
1
starting with
Proof:
(n
~
k)
C
contains a codeword
it contains also at least one codeword
(e , e , ... e _ , l-e ) .
2
l
k l
k
If this were not true, the strategy would not be irreducible,
as after having observed the first
the values
e , e , ... , e - ,
2
1
k l
function would be unnecessary.
k-l
test functions and obtained
the observation of the
k-th
test
-2.23Remark:
Notice that for the special case
Lemma 2.3.1 is as follows:
then the codeword
(e , ""
l
LEMMA 2.3.2.
sequence
s
s
If the codeword
e - , l-e )
n
n l
=n
k
(e , e ,
l
2
is also in
Given a binary I-code
the statement of
... ,
is in
e )
n
C
C .
C and an aribitrary finite
of zeros and ones, then either
s
is contained in
C, or
C or finally there is in
is a proper segment of a codeword in
one and only one codeword which is a segment of
s,
C
and these three
cases exclude each other.
Proof:
The fact that the three cases exclude each other follows from
the prefix property of the code
Thus to prove the lemma it is
C.
sufficient to show that if for a given sequence
s
the first two cases
do not take place, then the third case takes place.
the sequence
s
codeword of
does not belong to
Let
C.
c
Suppose namely that
C and is not a segment of any
denote that codeword in
C which has the
longest initial segment coinciding with the corresponding segment of
s;
such a
c
always exists, becuase
0
starting with a
the length
n
and also codewords starting with
if
n
~
k
= k,
+
s
has_length
k
because otherwise
1
then
want to prove.
c
is a segment of
Suppose therefore that
(k
+ l)-st digit of
c
is
l-e
1.
If
c
has
and its longest initial segment coinciding with the
initial segment of
length
C necessarily contains codewords
then
S'
s
,would be a segment of
s,
n
C
c
and that is exactly what we
>
k.
s , then by supposition the
but then by Lemma 2.3.1
necessarily has
Let
k
e
+
denote the
1 -st digit of
contains at least one codeword
-2.24c'
such that the first
digits of
digit of
of
c
c'
s
k
digits of
c'
coincide with the corresponding
and thus also with those of
is
e,
s,
and the (k
i.e. it coincides with the
+
l)-st
(k + l)-st digit
But this is in contradiction with our supposition that
c
has
the maximal initial segment coinciding with the corresponding segment
of
s.
This contradiction proves our lemma.
The statement of the
following lemma is obvious:
LEMMA 2.3.3.
codewords of
Let
C
be a binary I-code and let us consider those
k
C in which the initial segment of length
with a given sequence
s
of length
k
coincides
of zeros and ones.
from these codewords the initial segment
s.
Let us omit
The codewords thus obtained
(if their set is not empty) form also a binary I-code.
In view of Lemma 2.3.3 we can suppose, without restricting the
generality that we omit the first digit of a codeword
C.
I-code
S
Let
(e ' e , " ' , en)
Z
3
O.
Let
and let us denote it by
C denote the set of codewords obtained by replacing
c
by
C
in
Let us apply now Lemma 2.3.2 to
If
s
is contained in
in
C
then clearly
s
and the binary
C
C'
is not a prefix code.
Thus it is sufficient
the initial segment of which coincides with, s .
cases are possible:
C.
I-code
C or contains as initial segment a codeword
to consider the third case, in which there is a codeword
in
of a binary
We may suppose that the omitted first digit is a
the remaining codeword be
S
c
Either
c*
c*
Now two
may be chosen different from
c,
in
-2.25this case
is contained in
c*
C'
too,
code, or the only possible choice of
that
c
C'
and thus
c*
is
=
c*
c
is not a prefix
.
But this means
has the property that i f its first digit (which is a
omitted, the remaining sequence coincides with the first
of
c
.
But this is possible only i f
c
consists of
in this case by the remark to Lemma 2.3.1,
starting with
from
c,
n-l
zeros and ending with
n
) is
digits
zeros.
However
C contains also the sequence
1
and its initial segment of length
in contradiction with our supposition.
n-l
a
this sequence is different
n-l
coincides with
s ,
This contradiction proves
Theorem 2.3.1.
The situation in the general case is not so simple:
the code consisting of the codewords
00, 01,
02,
For instance
10, 11, 12
is a
prefix code, but is is not irreducible, because omitting the first digit
of the third codeword we get again a prefix code, namely
10, 11, 12.
00, 01,
2,
Nevertheless Theorem 2.3.1 can be generalized for regular
codes of degree
§2.4.
q, which will be introduced in the next
§.
Connection between the code and the tree describing
a'simple search.
As we have seen, a simple search can be described both by a tree and
by a code.
Let us now have a look at the question how these two descriptions
are related to each other.
If the tree corresponding to a simple search is given, we can obtain
from it several codes describing the search, by labeling all the edges
going out from the root by the numbers
0, 1, ... , v-I
if
v
is the
-2.26valency of the root, then labeling similarly all the edges going out
(i.e. leading nearer to an endpoint) from any other point which is not
an endpoint similarly.
Of course this labeling can be done in several
ways.
points which are not endpoints, the root has
If there are
valency
va
and the other
... , v k l
(v.-l)! .
1
q
k
k-l
vertices the valencies
then the number of possible labelings is
Especially if the tree is a regular search tree of
having
n
endpoints, where
number of possible labelings is
(q~)~
n
= k(q-l) +
x*
of
S
,then the
thus for a binary tree with
endpoints the number of possible labelings is
labeling to each element
I
n-l
2
.
n
For any particular
there corresponds an endpoint of the
tree, and a unique path leading to this endpoint.
The corresponding
codeword is defined as the sequence of numbers by which the edges of
this path have been labeled, starting from the root.
can be of course used also the other way around:
This correspondence
To every code we can
construct a corresponding tree, the edges of which are labeled, as
described above.
Using this correspondence between search trees and codes, we can
get from results concerning the enumeration of all search trees of
a given type the number of all possible codes of the corresponding
type (and conversely).
In counting codes we shall always suppose that
a code is an ordered set of codewords, i.e. we consider two codes
consisting of the same codewords, but in different order, as different
codes.
(This is of course a matter of convention.)
A code corresponding to a regular search tree of degree
be called a regular code of degree
code is of course always regular.
q.
q
will
A binary irreducible prefix
-2.27From Theorem 2.2.1 we get the following:
THEOREM 2.4.1.
degree
q
The total number C(q, k)
containing
(z.Lt, I)
where
codewords is
(kq) !
k!
n
is the number of regular search trees of degree
endpoints.
(2.4.2)
where
+ 1
C(q, k)
T(q, k)
having
n = k(q-l)
of regular search codes of
n
C .n!
n
is the sequence of Cayley defined by (2.2.4).
Remark:
Clearly
C(q,
=
(2.4.3.)
n-l
0)
(n::l mod(q-l»
n!
is the total number of unordered regular search codes of degree
consisting of
n
codewords.
Cayley's sequence.
C(2) = C
We have
Let us denote by
n
where
n
G (x)
00
(2.4.4.)
G (x)
q
=
L: c(q) xn
n=l n
n:::l mod (q-l)
Then in view of Theorem 2.4.1 we have
n-J.
00
G (x)
q
,-~
2: (q.)
,T(q,
n=l
n!
n=l mod (q-l)
n-l
0)
C
n
q
is
the generating function of
q
Le. let us put
the sequence
(2.4.5.)
and
In particular the total number of binary irreducible
C(2, n-l)
C
q
1
xn
q!
1
q -l
-2.28and thus
G (x)
satisfies the functional equation
q
(2.4.6.)
= x + [Gq (x)]q
G (x)
q
The equation (2.4.6) can be verified also directly, by the recursion
formula
c(q)
n
(2.4.7.)
=
L:
c(q)c(q) ... c(q)
n
n
n
1
2
q
n +n + ... n
l 2
q
=n
Enumeration of all search trees having
§2.5.
a given number of endpoints.
Let
n
V(n)
(labeled)
Let
having
denote the total number of different search trees with
endpoints:
V(n, k)
n
The aim of this
§
is to determine
V(n) .
denote the total number of different search trees
endpoints and
k
other points (including the root).
Then
we have
n-l
(2.5.1.)
V(n)
=
L:
V(n, k)
k=l
as
k
is clearly maximal if and only if the tree is binary.
We shall determine
V(n, k)
by using Prlifer's method.
According
to the definition, in a search tree only the endpoints are labeled.
V(n, k)
denotes the number of those trees which have
these there are
n
endpoints, labeled by the numbers
n + k
1, 2,
points, among
•
•
It
,
n ,.
among the remaining points there is one which is distinguished as the
root and has valency
valency
~
3
~
2,
while the remaining
If we label the root by the number
k-l
1
Thus
points have
and the
-2.29remaining
way, we get
k-l
points by the numbers
D(n, k)
(k-l)! yen, k)
coding it is evident that
of length
the numbers
n + k - 2
2, 3,
D(n, k)
in every possible
different graphs.
Now by Ptlifer's
is equal to the number of sequences
1
in which the number
2, 3, " ' , k
k,
occurs at least once ,and
occur at least twice.
let us solve first the following question:
number of those sequences of length
To determine
S (N, M)
Let
r
denote the
N formed from the numbers
1, 2, " ' , M in which each of these numbers occurs at least
(N ~ Mr)
D(n, k)
r
Then we have evidently the recursion formula
(2.5.2.)
S (N, M)
r
(~)S (N-j, M-l)
E
j=r
J
r
from which it follows that, putting
S (N
M)
N
-r-'
......:;.r_-..:- - x
N=Mr
N
(2.5.3.)
=
.6
r
(M, x)
we have
00
(2.5.4.)
.6
r
=
(1, x)
x
L
N=r
N
NT =
e
r-l . k
x
E~
k=O k!
and
( 2.5.5.)
.6
r
(M, x)
r
(M-l, x)
r
(1, x) .
Thus it follows
(2.5.6.)
.6 (M, x)
r
e
=
(
X
-
r-l xk)M
E-
k=O k!
(M=l, 2, •.. ) .
times
-2.30Thus especially for
r = 2 we get
S2(N, M) N
00
(2.5.6b.)
L
x
,
N=2M
N.
=
x
(e -I-x)
M
•
Sl (N, M)
Evident:y
In general
set of
's
are the Stirling numbers of the second kind.
M~
r
(N, M)
is equal to the number of partitions of a
M:
N elements into
M parts each containing at least
We shall need the numbers
S2(N, M)
=
elements.
because evidently
n-k
U(n, k)
(2.5.7.)
r
L
(
n-k+2
,
J' ) S2 (n+k-2-J, k-1) .
j=l
Thus, putting
00
L
U(n,k) xn+k-2
n=k+1 (n+k-2):
u(k, x)
(2.5.8.)
we have, in view of (2.5.6b),
u(k, x)
(2.5.9.)
=
x
x
k-l
(e -1) (e -l-x)
and therefore
00
L
(2.5.10.)
V(n,k~ x n+k-2
n=k+l" (n+k-2) .
x
=
Using (2.5.10) one can determine the numbers
x
(e -l)(e -I-x)
(k-l):
V(n, k)
k-1
and thus also
V(n) •
For instance we get from (2.5.10) the following values:
n
2
3
4
5
V(n. 1)
1
1
1
1
N(n, 2)
0
3
10
25
V(n. 3)
V(n. 4)
0
0
15
105
0
0
0
105
...
V(n)
1
4
26
236
-2.31Another way for computing the numbers
V(n)
considering all possible ways in which the
n
is the following:
By
endpoints can be distributed
between the main branches of the tree starting from the root, one obtains
the recursive formula:
~(~rj
n
L
V(n)
n:
(2.5.11.)
h=2
.
l:ja.=n
l:a~=h
a ..
J
J
J
where by definition
V(l) = 1.
From (2.5.11) we get for the power series
00
(2.5.12)
W(x)
l:
n=l
V(n)x
n
n:
the functional equation
(2.5.13.)
W(x)
= l2 (x + eW(x) - 1)
It can be shown that the power series (2.5.12) is convergent for
Ixl
<
4
In.
e
By taking successively the derivatives of both sides of
(2.5.13) one obtains successively the values
§2.6.
V(n) = W(n) (0)
(n=1,2, ... ) .
Enumeration of all codes corresponding to a search
process, and having a given number of codewords.
In the case of regular search trees of degree
where
n = k(q-l) + 1,
q
and
n
endpoints,
it was easy to get from the number of trees the
number of codes, by multiplying it by the factor
q
k
In the case of all
search trees, one can not pass so easily to the number of all corresponding
codes, because the factor by which one has to multiply depends not only
on the number of endpoints, but on the structure of the tree too:
The
-2.32k-l
vO:j~l
factor is namely equal to the product
the valency of the root, and
(v
vI' v 2 ' •.. , vk - l
other points which are not endpoints.
Let
- 1):
j
D(n)
denote the total
n
endpoints, and
the number of those of these codes, which correspond to search
trees containing besides
can not directly obtain
n
endpoints
D(n, k)
P2(N, M)
of the numbers
k
from
method which led us to determine
Let
is
the valencies of the
number of codes corresponding to search trees with
D(n, k)
where
further points.
V(n, k) ,
V(n, k)
While one
nevertheless the same
can be used to determine
D(n, k) .
denote the sum of the product of the occurrences
1, 2, ... M in a sequence of length
N formed from
these numbers, where the summation is extended over all such sequences
in which each of the numbers
1, 2, ... M occurs at least twice.
We
get, in analogy to (2.5.2)
=
(2.6.1.)
and thus, putting
00
(2.6.2.)
L
N=2M
J?2 (N,
M)
---::;--=::--_
it:
xN
=
we obtain
=
(2.6.3.)
Let
E(n, k)
numbers
denote the sum of the product of the occurrences of the
1, 2, ... , k
of the numbers
in a sequence of
1, 2, ..• , k,
at least twice.
elements, consisting
the summation being extended over all
such sequences in which the number
1,2, ... , k
(n + k - 2)
1
occurs at least once and all the numbers
Then we have, in analogy to (2.5.7)
-2.33-
~~k1 (j+1)! (n+~-2)p2(n+k-2-j, k-1) .
E(n, k)
(2.6.4.)
J=
J
.
Thus putting
00
=
e(k, x)
(2.6.5.)
E(n,k)
n=k+1 (iJ.+k-2)!
L:
x
n+k-2
we get
e(k,
(2.6.6.)
As however
D(n, k)
E(n, k)
(k-1)!
x)
we obtain finally for
2k
cc
(2.6.7.)
d(k,
x)
=
D(n,k) x n+k-2
n=k+l (n+k-2)!
L:
k=1,2, ••.
l (1
(k-1)!
x
+
(1_x)k+1
1)
(l_x)k
From (2.6.7) we get
(n-2)!(n + k - 1)!
k!(k-1)!(n - k - 1)!
D(n, k)
(2.6.8.)
Thus we obtained
surp~izing1y
an explicit formula for
D(n, k)
this
formula can also be written in the form
D(n, k)
(2.6.9.)
The first four values of
=
D(n, k) (k=1,2,3,4)
for
n = 2, 3, 4, 5
are
given by the following table:
n
D(n, 1)
D(n, 2)
D(n, 3)
D(n, 4)
D(n)
2
3
4
5
2
6
24
120
0
12
120
1080
0
0
120
2520
0
0
0
1680
2
18
264
5400
In the table we have indicated also
D(n)
and
D(n)
~
D(n)
~
1
3
11
45
I
-2.34There is another way to prove the formula (2.6.8).
show that for
D(n, k)
(2.6.10.)
D(n, .,k).
n!
One can easily
the following recursion formula holds:
D(n , k )
D(n Z' k 2 )
l
, l
,
n2 ·
nl •
e=2
n +n + .• '+n =n
1 2
e
k +k + •. '+k =k-l
1 2
e
n
D(n , k )
- e
, e'
n
e
.
L:
Let us put
co
00
k) n k
L D(n,
n!
x y
n=l k=O
(2.6.11.)
L:
= d(x,
y)
Then it follows from (2.6.10) that
(2.6.12.)
Thus we get for
2
x + yd (x, y)
l-d(x, y)
d(x, y)
d(x, y)
the explicit formula
/
2
1 + x - l(l+x) - 4x(y+1)
2 (y+1)
d(x, y)
(2.6.13.)
and developing the function
d(x, y)
into a double power series we
obtain (2.6.10).
Now we shall prove a simple explicit formula for the generating
function of the sequence
Recalling the meaning
(2.6.14.)
D(n)
---n:
D(n)/n~
of
.
D(n)
we get for
ben)
the recursion formula
n
(n ~ 2)
L:
h=2
from which, putting
co
d(x)
=
D(~) xn
n=l n.
2:
-2.35we get (taking into account that
D(l)
= 1) that
d(x)
(2.6.16.)
and thus we arrive to the following explicit formula:
d(x)
(2.6.17.)
=
1 + x -
11 - 6x + x 2
4
It is easy to see that the power series (2.6.14.) is convergent
for
Ixl ~ 3 - 21:2
function
d(x)
(x
=3
- 21:2
being the singular point of the
which is nearest to the origin);
one can prove by a simple
function-theoretic argument that the following asymptotic formula holds:
D(n)
---;;:
Of course
d(x~
1)
= d(x)
~
~
thus formula (2.6.171 could have been
obtained also from (2.6.13) by substituting into it
to show that if we are interested only in
we can get
(2.6.1~
D(n)
y
=1
and not in
~
but we wanted
D(n~
k)
directly.
Of course from (2.6.17) one can also get an explicit formula for
D(n) but it is not very
<2.6.19,)
useful~
D(n)
---n!
as it is a sum with alternating terms:
=
We get e.g. from (2.6.13) that
D(6)
GT
197 .
We shall solve now some more related problems of enumeration.
D (n) denote the total number of search codes having
r
n (ordered)
Let
codewords~
-2.36-
each of length
~
r,
(r=1,2, ... )
and let us put
D (n)
-r- xn
co
(20'6.20. )
=
d (x)
r
~
n!
n=l
By the same argument, that led us to (2.6.9) we get the recursion formula
D (n)
r
(2.6.21. )
=
and from this we deduce
2
d (x)
r
(2 •.6 ..22., )
=
dr_lex)
x + l-d _ (x)
r l
Thus we get
d (x)
l
=
x
I-x
d (x)
2
=
x +
x
(I-x) (1-2x)
d (x)
3
=
x+
2
2 2
x (1-2x+2x )
(I-x) (1-2x) (1-4x+4x 2-2x 3)
2
etc.
Of course we have
lim d (x)
( 2.6.23.•)
The number of codes consisting of an
length
~
r
B(r~'q,~rr)
=
d(x) .
r
X+Toc
a~bitrary
number of codewords of
can be determined for search codes of degree
q
too.
denote the number of{brdered) search codes of degree
consisting of codewords of length
formula, in analogy to (2.4.7)
~
r.
Let
q
We obtain easily the recursion
-2.37(2.6.,24. )
B(r,q,n)
n!
B( r-l , q ,U )
I
.. -_ ... ..,..
.
-
=
n +n + .. '+n =n
1
2
nl ·
B(r-l,q,n )
2 .•
----:--..::=-.
B(r-l,q,n)
q
n '
2'
'
n
'
q'
q
from which, putting
00
b (r, q, x)
=
E B(r-l, q, n) n
n!
x
n=l
n=l mod(q-l)
we obtain
x + (b(r-l, q, x»q
b(r, q, x)
(2.6.26. )
for
r ~ I
where
b(O, q, x)
x •
Clearly
~.6.
27. )
E
lim b(r,q,x)
r-H-oo
n-l
n-l)
,q-l
T(q, q-l q.
n
,
n.
n=l mod(q-l)
x
I
I
= ~-F
q!
Thus from (2.6.2'6) by passing to the limit
r
-+ +00
q -l q
(2.6.28.)
q
= 2,
i.e. for binary search codes we get
x + b(r-l, 2, x)
b (r, 2, x)
and thus we obtain successfully
bel, 2, x)
etc.
x
b (2, 2, x)
=
b (3, 2, x)
=
) = G (x) •
we obtain a new proof
of the formulae (2.2.8) resp. (2.4.7) .
Especially for
(xq!
q-l
+ x2
2 2
x + (x + x )
2
q
-2.38-
The total number of binary searchloodes consisting of codewords of length
~
r
irrespectively of the number of codewords is of course finite too:
Denoting this number by
B(r)
we have
B(r)
Thus we get for
B(r)
b(r, 2, 1) •
the recursion formula
(2.6.23.)
from which we can compute the values of
-B-(-:-)-
( B(O)
B(r)
successively:
We have
-2-':- -~~..:...;-I·· 45~330
--~---11---~-+---~-
= 1 means that the empty code is considered here as the unique
code consisting of codewords of length
0.)
-3.1CHAPTER III.
Uniform flows in cascade graphs.
§3.1.
Cascade ographs.
We shall call a directed graph
infinite~umber
G,
having a finite or denumerably
of points, a cascade graph, if it has the following
properties:
a)
There is in
any other point
b)
c)
k
q point
~--cal1ed
of
G
there is in
For any point
a
of
the same length
a
G
r(~)--cal1ed
G
the source--such that for
G each directed path from
the rank of
The number of points of
of all points of
G having rank
G having rank
has clearly the rank
Thus
V
o
0
k
by
a
has
is finite for every
V
k
As the supposition
the point
b
point in
V
k
r(~)
then
-
r(~)
b)
has to hold
and by
the outdegree of
D(~)
a
The source
G
G
for
only, i.e.
~ = ~
containing
also,
~
•
an edge from the point
V+
k l
(k=O,l, ... ) .
G ending at the point
the number of edges of
Clearly both
~
a
to
i.e. every edge starting from a
= 1
leads to a point in
the number of edges of
~,
and the set
(k=O,1,2, ... ) .
is a one-element set containing the element
{~}
V
and it is the only point with this property:
It follows further that if there is in
of
k
G by
it follows that there is no directed cycle in
d(~)
to
a.
a.
We shall denote the set of all points of
o
~
to
1 .
>
V
~
a directed path from
d(~)
and
G
Let us denote by
~,
i.e. the indegree
starting from
D(~)
~,
i.e.
are finite for every
-3.2a
E
We call a point
V .
a
V an endpoint of
E
shall denote the set of endpoints of
of rank
of
k
by
G
A.
i.e.
We put
number of those points of rank
denote further by
of
k
R = +oc
then
N
m
= 0
r<.~)
and
for
V is infinite.
Clearly
for
R
m> k
and
.
We
denote the
~= IVk - Ekl
k
, and ~ the
which are not endpoints.
the maximum of
R
if
k
IAI
let
denotes the total number of points of rank
N
k
and put
N = Ivkl
k
0
and the set of endpoints
For any finite set fA
E .
k
number of elements of
E
by
G
D<,~) =
if
G
a
V if
E
~= 0
if
We shall
V is finite
for some value
is finite, and conversely.
Let us consider now some examples of cascade graphs.
Example 1.
If
G is a roOted directed tree in w1:l'ich all edges are
directed away from the root, and
G,
then
D(~)
is finite for every point
a
of
G is a cascade graph, its source being the root of the tree.
Conversely if in a cascade graph
different from
~
G,
d(a)
(for which of course
=
I
d(~)
for all points
= 0)
then
a
G is a rooted
directed tree in which all edges are directed away from the root.
Example 2.
Let
S
be a finite set.
be all subsets of
S
(directed from
to
a
and connect
b )
Example 3.
Let
a
c
S
a c S with
i f and only i f
omitting one of its elemenns.
graph and for every
Let the points of the graph
The graph
one has
b
b
G
r(a)
:=
S
by an edge
is obtained from
a
by
thus obtained is a cascade
IS I - i a I •
S be a finite set and let the points of the graph
be all non-negative integral valued functions defined on
g
G
are two such functions, draw an edge from
f
to
g
S.
If
G
f
if and only if
and
-3.3g(x) ~ f(x)
for all
XES
x~s(g(x)-f(x»
and
get a cascade graph and the rank of a function
1.
f
is
In this way we
ref)
= XELsf (x)
•
Let us call a cascade graph simple, if it does not contain any infinite
directed path.
In a simple cascade graph to every point
a nonempty set
T(~)
a
E
V to
~
by a directed path,
of the point
b
E
V then
A subset
a.
T(£)
We call
T(~)
the
If there is a directed path from
~ T(~)
.
A of the vertices of a cascade graph
antichain, if for any two points
exist in
there corresponds
consisting of those endpoints of the graph which
can be reached from
target-set
a
a
G a directed path from
E
~
A and
to
b
E
A
G is called an
there does not
An antichain is called
b.
saturated, if it is not a proper subset of another antichain.
An antichain
is called a blocking antichain, if any directed path from the source to an
endpointand every infinite directed path stq~ting at the source, passes through
a (unique) point of the antichain.
Clearly every blocking antichain is
saturated but a saturated antichain is not necessarily blocking.
instance in Example 2, let
n
~
3,
S
and let the antichain
f2, 3, ... n l .
Then
containing the set
is not blocking.
{I}
be the set
S
= {I, 2, ... n} where
A consist of the two sets
is a subset of the set
Example 4.
G is infinite.
S
and
not
{2, 3, ... , n} but it
G is a rooted tree (see Example 1)
then a saturated antichain is always blocking if
necessarily if
il}
A is saturated, as every subset of
If the cascade graph
For
G is finite, but not
(See the following Example.)
Let the points of the graph
each term of which is one of the numbers
G be all finite sequences,
0, 1, "', q-l
where
q
>
=
2 .,
-3.4the empty sequence is also a point of
the point
sequepce
a
~
to the point
b
G.
Let there be in
if the sequence
b
is obtained from the
by adding to the end of the sequence
one of the numbers
0,1, ... , q-l
a
one more digit, i.e.
In this way we obtain a cascade
graph, which is a tree, and has no endpoints.
and let the antichain
G an edge from
Let us
A consist of the sequences
take
q
=
2
0, 10, 110, 1110,
A is clearly a saturated antichain, but it is not blocking as it does
not block the infinite path from the source (this being the empty sequence)
leading through the points
1, 11, Ill, •...
In a simple cascade graph the set of all endpoints is a blocking
antichain.
In any cascade graph the set
there are no endpoints of rank
k,
b
a
if there is a directed path from
ordered set*
a lattice.
is an antichain and if
then
Let us write, for any two points
~ <
V
k
V
k
and
a
is a blocking antichain.
b
to
of a cascade graph,
b
G is a partially
with respect to the order relation
but not necessarily
The following example shows that cascade graphs which are
lattices have remarkable properties.
Example 5.
Let
G.e. Rota [1]).
of
G be a finite combinatorial geometry (see H. Crapo and
Let
G be the graph, the points of which are the flats
G and there is an edge in
and only if
b
covers
a.
G from the flat
Then
lattice, and the rank of any flat
Let us remove the cascade graph
a
to the flat
b
if
G is a cascade graph, which is a
a
in
G is the same as in
G •
G its unique element with maximal rank:
* What we call a cascade graph is considered as a partially ordered
set a graded partially ordered sets: See Birkhoff [2] and Klarner [3],
where the graded partially ordered set with a given maximal ran~ and
given number of points are counted.
-3.5We obtain again a cascade graph, the endpoints of which are the copoints
of
G.
This cascade graph has among. others the following remarkable
property:
The target sets corresponding to different points are different,
and the target sets of points having the same rank form a Sperner-system
(i.e. none of them contains any other as a subset.)
We call a simple cascade graph in which the target sets corresponding
to different points are different, a search cascade.
Example 1 is a search cascade if and only if
a
E
~
D(a)
A rooted tree of
1
for all
V , i.e. if it is a search tree, describing a simple search without
noise.
A search cascade which is not a tree describes a search process
with noise.
One can construct a search process (with noise) which is described
by a given cascade graph
as follows:
G
Let
•.. , a
points to which there is an edge from the source.
sets be
(i=1,2, ... ,q) •
T(a.)
1
to the source, i.e.
graph
Then
G
elements of
flex)
X €
S.
j
be the
Let the corresponding
denote the set corresponding
is the set of all endpoints of the cascade
is the basic set of the search problem and if the
let the first test function
be defined as follows:
for which
is equal to
T
S
are
S
those values
sets
S
S
Let
q
x
S
flex)
€
takes on with equal probabilities
T(a,)
Notice that the union of the
J
thus
flex)
is defined for each
The subsequent test functions are defined similarly.
For
example, the search cascade shown on Fig. 3.3.1 can be interpreted
as describing the following search process:
0, 1, 2, 3
variables
and let
f..
1J
Let
S
F consist of the family of the
defined as follows
(0
~
i
< j
<
3) :
be the set
6
random
-3.6f
ij
i
(i)
=
or
j
.
1/2
f
, f ij (j)
0
then
1 ,
f .. ex)
while i f
is not equal to either
x
takes on the values
J.J
0
and
Let us now consider the following strategy:
1
with probability
We first observe
(x*)
=0
we select
(x*)
=
we select as the second test function
(x*) .
If
f
function.
If
f
f
If we obtained as the result of the first two observations
03
I3
(x*) .
03
03
1
f
02
we select as the third test function
00,
first two observations were
'
function
f
were
we select
11
12
01
or
10
(x*)
f
Ol
;
if the results of the
we select as the third test
while if the results of the first two observations
f
23
as the third test function.
between the observations and the true value of
x*
as the second test
x*
is equal to the sum of the three observations.
The correspondence
is as follows:
We may interpret
a search process with noise like the above example as a questionary,
with multiple choice questions such that in certain cases more than one
of the possible answers are correct, but only one of them must be
answered in the affirmative.
f
03
For instance observing the test function
is equivalent to getting an answer "yes" to one and only one of
the following two questions:
1) Is
x*
~
2 ?
2) Is
x* > 1 ?
If
x* = 0
then we shall get a positive answer to the first question;
if
x* = 3
we shall get a posttive answer to the second question, while
if
x* = 2
or
x*
=3
we may get any of the two possible answers.
Thus i f we get a positive answer to the first question, we know that
is not
3
i.e. is in the set
(0, 1, 2)
answer to the second question we know that
x*
while i f we get a positive
x*
is not
0
Le. it is
-3.7-
o
1
-3.8in the set
(1, 2, 3) .
Of course if we interpret such a questionary
so that if more than one answer is available, one of them is selected
at random, then this interpretation is not essentially different
from the interpretation as a search with noise.
However there is no need
to suppose definite probabilities for these choices, provided that the
strategy is such that it leads in a finite number of steps to finding
with certainty the object of the search.
If
a
is any point :of the cascade graph
set of all endpoints of edges starting at
points of
A we denote by
fA
G we denote by
a.
If
fA
= aEUAfa
If
the set of those points which are the
is any point of
b
the set of those points
§3.2.
a
for which
ab
G a non-negative number
(3.2.1.)
A,
G we denote by
i.e. we
f-lb
b E fa
Random walks on a cascade graph.
Let us assign to each edge
graph
the
A is any set of
endpoints of at least one edge starting at a point in
put
fa
(from
w(~,
L
£)
w(~,
a
to
b)
of a cascade
such that
£) =
l,
bEra
for all vertices
w(a, b)
a
of
G which are not endpoints.
defines a (Markovian) random walk on the edges of
The random walk starts always from the source
point
a
the point
point
Such a function
b,
of rank
~,
etc.
1
with probability
and proceeds to a
~
w(~,~)
G as follows:
;
the walk continues with probability
after arriving to
w(~,
£)
to a
Thus the random walk proceeds always along a directed
-3.9path
of
G
~
until it reaches an
the walk continues indefinitely.
a probability measure
endpoint~
while if the path is
Such a random walk defines uniquely
on the power set of the set of all paths starting
P
from the source (this set being finite or denumerable).
B denote
Let
a
the event that the random walk arrives eventually to the point
B
(In other words let
point
a.)
a
a.
denote the set of all paths containing the
Let us put
PCB )
w(a)
(3.2.2.)
A be any
Let
infinite~
antichain~
a
then by definition the events
B
a
~
(3.2.3.)
w(a)
A)
A
Thus we have for every antichain
are mutuallY exclusive.
(~ E
1.
<
aEA
If
A is a blocking
antichain~
then the events
complete set of events (i.e. the sets
their union is the set of all
paths)~
~
(3.2.4.)
B
a
B
a
(~ E
A)
form a
of paths are disjoint and
and thus we have
1 .
w(a)
aEA
§3.3.
Normal cascade graphs.
We shall call a cascade graph
probabilities
w(~~
only on the rank
(3.3.1.)
where
f(x)
£)
r(~)
G normal
if the transition-
can be chosen in such a way that
of
~ ~
w(~)
depends
i.e.
w(~)
=
f(r(a))
is a function defined on the set of non-negative integers.
-3.10Let
B
k
denote the event that the random walk does not stop before
arriving to a point of rank
k,
Ck
and let
denote the event that the
raijdom walk does not stop at an endpoint of rank
k.
Then we have
evidently*
(3.3.2.)
Now let
G be a normal cascade graph, and suppose that the transition
probabilities have been chosen so that (3.3.1) holds:
call the random walk a uniform flow.
G,
In this case we
In case we have a uniform flow on
clearly
(3.3.3.)
=
and
(3.3.4.)
It
follows
f(k+l)
(3.3.5.)
=
f(k)N~
k+l
and thus, as
f(O)
= 1, we get
f(k)
(3.3.6.)
for
where an empty product is by definition
eq~al
to
1.
k
~
1
Especially if
the cascade graph is finite and it has no endpoints of less than maximal
rank. (i.e. for every endpoint
*
e
one has
r(e)
= R) , or if the
denotes the conditional probability of
C
k
under condition
B .
k
-3.11cascade graph is infinite and it does not contain any endpoints, then
we have
M. = N.
J
for
J
j
<
R and thus
=
f(k)
(3.3.7.)
The following theorem is an immediate consequence of (3.2.3)
and ( 3 . 3 . 6) :
THEOREM 3.3.1.
Let
G be a normal cascade graph.
the total number of points of rank
of
G and
Let
~
points of rank
k
G and let
denote the number of points of rank
~
which are not
k
endpoint~.
Let
N
k
denote
the number of those
A be an antichain in
k
in
A.
Then
the inequality
R~
(3.3.8.)
I:
-
k=O Nk
holds; if
_~
1
<
J
is a blocking antichain there is equality in (3.3.8).
COROLLARY:
rank of points of
~
If
G
= N
k
for
k
<
R,
where
then for every antichain
R ~
(3.3.9.)
L
-
k=O Nk
with equality standing in (3.3.9) if
Remark:
~
J'~k N.
<
R is the maximal
A we have
1
=
A is a blocking antichain.
Notice that (3.3.9) can be written also in the equivalent form
(3.3.10. )
L
1
N
aEA rea)
~
1 .
We shall refer to Theorem 1. as the unifovm flow theorem.
In the next
we shall give a necessary and sufficient condition for the normality of
a cascade graph.
§
-3.123.4.
A necessary and sufficient condition for the
normality of a cascade graph.
A cascade gra£h
THEOREM 3. 4 • 1.
satisfies the following condition:
A of the set
V - E
k
k
G is normal if and only if it
For every
of points of rank
k
k
~
1
and for every subset
which are not endpoints,
one has
(3.4.1.)
N+
k l
where
k + 1,
is the number of points of rank
of points of rank
k
and
~
the number
which are not endpoints.
Clearly for every random walk (i.e. probability
Proof* of Theorem 3.4.1.
flow) on
G and for every set of points
(3.4.2.)
<
A
~
V-E
one has
w(~)
L:
bErA
Thus if
A
V - E
k
k
c
and the flow is uniform then (3.4.1) holds, i.e.
the condition is necessary.
Let us prove now its sufficiency, i.e.
that i f (3.4.1) holds, one can choose the transition probabilities
w(~,~)
so that the flow should be uniform.
of such transition probabilities
on the rank
1
w(~,~)
k
of
1
= N'
a.
1 .
all points
step by step, i.e. by induction
Clearly if we put for every point
a
of rank
<
k
w(~)
of rank
= 1N
so that (3.3.1) holds for all
We have to show that one can choose the values of
*
b
for every point b of
l
Let us suppose that we have already determined w(~,~) for
then we have
1
rank
w(~,~)
We shall prove the existence
The proof given here is due to Dr. G. Katona.
w(~,~)
a
E
V
k
for all
-3.13a
V
k
E
so that (3.3.1) holds with
k + 1
instead of
k
too.
In other words, we have to choose the transition probabilities
w(~,
£)
for 'a
E
Vk - Ek
in such a way that they should be non-negative
and should satisfy the following two sets of equations:
(3.4.3.)
w(a, b)
L:
for all
1
a
E
bEra
V - E
k
k
and
L:
( 3.4.4.)
ad
-1
w(a, b)
Let
and
V - E
k
k
of the. sets
G*
graph
by
a.
~,j
G*
Thus
V+
k l
and
(1 ,;, i ,;, ~
.
a
.
i oJ
G*
G*
in
has
1 ,;, j b N + )
k l
1 ,;, v ,;,
.
~)
B
= {b
U,v
G*
Let
2~Nk+l points which we denote
and
b
U,v
We connect the points
G
and edge from
}
a.
~,j
a.
~
b
U,v
j
A*
for any subset
Let
to
and
b
b
u,v
u
of the set
A I*A*
which are connected by at least one
A denote the set of those
is for at least one
contained in
(3.4.5.)
and
(3.4.6.)
We consider the auxiliary
is a bipartite graph with the two classes of points
denote the subset of those
~,J
respectively.
i f and oly if there is in
and
a.
denote the elements
defined as follows:
(1 ,;,U ,;, N + ;
k l
-in
for all
b
Ir*A*1
A*.
a
i
E
V - E
k
k
Then we have
for which
-3.14Thus (4.1) implres
Ir*A*1 =
(3.4.7.)
>
IA*\ .
But (3.4.7) means that the conditions of the marriage-theorem
(see e.g. Harper, L. and Rota, G.C. [4])
A
one-to-one matching between the sets
is matched with such a
G*,
b
for the existence of a
and
B
so that each
with which it is connected in
U,v
are fulfilled, and thus such a matching exists.
such a matching and let
s(i, u)
Let us take
denote the number of
which are matched to a
b
U,v
ai,j
(1 ~ v ~~)
•
Then we have evidently
(3.4.8.)
=
for
s (.1., u )
=
and
~L:
(3.4.9.)
i=l
further
s(i, u)
= 0 if a.1. and b U are not connected in G.
w(a., b )
(3.4.10. )
1.
U
=
S (1, u)
bE ra.)
u
Nk +1
the equations (3.4.3) and (3.4.4) hold.
Thus putting
1.
Thus Theorem 2 is proved.
From Theorem 3.4.1 one can easily deduce the following:
COROLLARY:
~,
G the out degree
which is not an endpoint. depends only on the rank
and the indegree
r{b)
If in a cascade graph
of
b
then
d(b)
of any point
G is normal.
b
D(~)
r (2;.)
of a point
of
depends only on the rank
2;..
-3.15Proof:
Let us denote the outdegree of a point of rank
an endpoint by
As
D
k
,
k
which is not
and the indegree of a point of rank
k
the total number of edges starting from some point of rank
equal to the total number of edges leading to a point of rank
by
d.
k
is
J
.
k + 1
we have
=
(3.4.11.)
Now let
A be any subset of
V - E
k
k
out from one of the points in
edges arriving to a point in
As the number of edges going
A can not be larger than the number of
fA,
we have
(3.4.12)
Multiplying both sides of
(3.4.12) by
we get that (3.4.1) holds, i.e. that
Remark.
N+ ,
k 1
and using (3.4.11)
G is normal.
Instead of deducing the above corollary from Theorem 2
one can prove its statement by constructing effectively the uniform flow
on the cascade graph.
not an endpoint we put
on the cascade graph
As a matter of fact, if for every
w(a, b)
=
Iraj-1,
a
which is
then we get a uniform flow
G satisfying the conditions of the corollary.
The cascade graphs satisfying the conditions of the above
of Theorem 3.4.1 are called semiregu1ar cascade graphs.
Corollary
As by the
Corollary every semiregu1ar cascade graph is normal, it follows that
the statement of Theorem 3.3.1 holds for every antichain of a semiregu1ar
cascade graph; this special case of Theorem 3.3.1 is due to Kirby A. Baker [5],
-3.16who has formulated this result in a slightly different terminology,
but the result expressed in our terminology is just the statement of
Theorem 3.3.1 for semiregular cascade graphs.
§3.5.
Kraft's inequality and Sperner's theorem as special
cases of the uniform flow theorem.
In spite of the extreme simplicity of its proof, Theorem 3.1.1
is a common source of several known results, thus for example Kraft's (see
e.g,.) Feinstein, [6]) inequality which we will need in the next chapter,
and Sperner's theorem (see e.g. Lubell [7]).
In this
we shall deduce
§
these theorems and some of their generalizations as special instances
of the uniform flow theorem.*
Let us deal first with the Kraft inequality.
the cascade graph described in Example 4:
Let us consider
Let the points of
finite sequences which can be formed from the digits
(q
~
2)
sequence
0, 1,
and let us draw a directed edge from the sequence
b
to the end of
if and only if
a.
b
be all
G
... ,
a
q-l
to the
is obtained by adding one more digit
Let the source be the empty sequence.
It is easy
to see that the cascade graph thus obtained is a tree, moreover a
regular search tree
in which there are
point and no endpoints.
Let
q
edges going out from every
A be an antichain in
G ,.
thus
A
is
a finite or denumerable family of finite sequences formed from the
digits
0,1, ... , q-l
such that no sequence in
segment (prefix) of another sequence in
A.
A is an initial
Such families of sequences
* I want to mention that I obtained the uniform flow theorem by analysing
Lubell's elegant proof of Sperner's theorem [7].
-3.17are called in information theory
codewords.
Evidently
q-ary prefix codes, and its elements
G is a normal cascade graph; to show this we do
not need Theorem 2, because it is easy to see that putting
for every edge
of
a b
Applying Theorem
1
= q -r(a)
-
w(~)
G we get
,
w(~,
-1
q
£)
i.e. a uniform flow.
we obtain the following result, called Kraft's
inequality:
A is a
If
(q ~ 2)
q-ary prefix code
of codewords of length
k
denotes the number
and
in the code, then the inequality
00
n
k
Ek"
(3.5.1.)
<
1
k=O q
holds; if
A has the property that every infinite sequence consisting
of the digits
0,1, " ' , q-1
contains one of the codewords from
A
as an initial segment, then there is equality in (3.501).
Clearly (3.5.1) can be formulated also in another form, speaking
about trees instead of codes.
It is natural to ask in general which rooted
trees, considered as cascade graphs, are normal.
The answer is very simple:
A rooted tree is a normal cascade graph if and only if it is semiregu1ar,
i.e. the outdegree of any point which is not an endpoint depends only on
the rank of the point.
N+
k 1
= Mk
. qk
If
D(a)
=
qk
for all
a
E
V - E
k
k
then
and thus we get from the uniform flow theorem the following
result, which reduces to the Kraft inequality in the special case when
q
k
=q
for all
If
D(a) = qk
k
.
A is an antichain in a semiregu1ar rooted tree such that
if
a
of the antichain
E
V - E
k
k
and i f
A having rank
nk
r
denotes the number of elements
then the inequality
-3.18-
~
k=O qlq2' ··qk
00
(3.5.3.)
L:
1
<
holds.
Of course Kraftts inequality (as well as its generalization)
can be proved quite easily
directly, but it is instructive to consider
this inequality as a particular instance of the uniform flow theorem.
Now let us consider how S-perner's theorem is obtained from
Theorem 3.3.1.
Let
Let the points of
G
be all
G
let there be an edge in
i f and only i f
b
be the cascade graph of Example 2 of
G
subsets - of an
from the set
is obtained from
To ~how that the cascade graph
G
a
n-element set
S
to the set
b
S
~
3.1.
§
and
S
c
~
by omitting one of its elements.
a
is normal it is
suffi~ient
to point
out that if we put
1
n-r(a)
(3.5.3.)
(notice that
rea) = n
(unique) endpoint of
(n-k)~
i.e.
only if
a
for every
~
<
n
G), then we get, taking into account that there are
having
k
1
i.e. there exists a uniform flow in
corollary of Theorem'3.4.l.)
G,
a
r(~)
for which
= n - k
elements, it follows that
=
A of the points of
r(~)
with
is the empty set and this is the
paths from the source to a point
to set
a
G.
for all points
(This follows also from the
It is easy to see further that a subset
i.e. a family of subsets of
if for any two different sets
a,
a
~
Sand
b
~
S
S
is an antichain
be~onging
to
A,
a
-3.19is not a subset of
b
i.e. if
A is a Sperner-system of subsets
S.
Thus the uniform flow theorem yields the following result:
If
and
A is a Sperner system of subsets of an
A contains
n
k
sets having
k
elements
n-e1ement set,
(k=O,l, ... )
then the
inequality
,;;;,1
(3.5.5.)
holds.
As
(3.5.6.)
=
we obtain from (3.5.6) the usual (though slightly weaker) form of
Sperner's theorem:
(3.5.7.)
Before going further let us add a remark.
Comparing the two
special cases just discussed, it turns out that the property of a
system of sets being a Sperner-system plays the same role in Sperner's
theorem as the prefix property of a code in Kraft's inequality.
As a
matter of fact, there is a real connection between these two concepts,
not only a superficial analogy.
As mentioned earlier the prefix property of a code implies that if
the codewords of such a code are written one after the other, without
indicating where one codeword ends and the next begins, and if the code
has the prefix property, the sequence of symbols can be uniquely decoded,
-3.20i.e. the codewords can be unambiguously separated from another.
Now
let us impose on the code the auxiliary restriction that a sequence of
codewords should be uniquely decodab1e even in the case when the letters
rearrranged~
within each codeword are arbitrarily
are unordered sets of
1etters~
i.e. if the codewords
and not ordered sets as usual.
If we
require further that the same letter should not occur more than once in
any codeword, then such a code is uniquely decodab1e if the codewords
(considered as unordered sets of letters) form a Sperner system.
Expressed in the language of search
theory~
a Sperner system corresponds
to such a strategy of search in which the values of the test functions are
obtained simultaneously and one does not know which value comes from
which test function.
For example let us consider the 19 numbers
0,1~2,3,4,5,6,7,8,12,13,14,16,17,18,l9~23,24,29
These numbers can be uniquely characterized by their residues mod 2, mod
mod 5, even if these three residues are given in a random order.
instance if we are told that the three residues are
easy to see that among the 19 numbers
residues, namely it is congruent to
e~listed
0 mod 2
For
0, 0, and 2
it is
above only 20 has these
and
mod 5
and to
2 mod 3 .
As a further application of the uniform flow theorem, we prove
the following generalization of Sperner's theorem:
THEOREM 3.5.1.
A
Let
of disjoint subsets of an
(AI' A , " ' , A )
2
r
and
be a family of ordered
n-element set
(B ,
1
B2 , " ' , Br )
S,
r-tuples
(r
>
such that if
both belong to the family
1)
3~
-3.21-
A then the relations
A
(j=1,2, ... ,r)
B.
J
c
j -
A
Then the number of elements of the family
IAI
(3.5.8.)
<
can not hold simultaneously.
satisfies the inequality
r
which is best possible, i.e. equality is possible in (5.8) by
A.
a suitable choice of the class
Proof of Theorem 3.5.1.
. t s 0f
Th e p01n
G
are a 11 pOSS1'b1 e
(AI' A2 , "', Ar )
the
Let us construct a cascade graph
r
= J='
.L: !A.1
1 J
rea)
-
to
following conditions are satisfied:
= r<.~) + 1
r(~)
J
to
for all but one value of
J
i --and
for
A. .
1
c
J -
B
j
in other words there is in
A. = B.
if
A.
B.
1
is obtained from
A.
1
j
as follows:
r-t.up1es
The source of
r-tuple each element of which is the empty set.
from
S
ordered
S
of disjoint subsets of
is
b
(r + l)n
G
The rank of an
and there is an edge
... ,
B )
for
j=1,2, ... ,r
G
(l ~
if and only if the
r
an edge from
j
~
G.
and
~
to
r)--say except
by adding one more element of
Clearly the conditions of Theorem 3.5.1. mean that
should be an antichain in
is
G
Thus if we show that
A
G is normal, we
can apply Theorem 1
We shall prove the normality of
G by verifying that the conditions
of the corollary to Theorem 3.3.1 are fUlfilled.
be any
~oint
in
r -
r(~)
a
= (A 1 , A2 , ... , Ar )
G which is not an endpoint; then we have
because the endpoints of
are
Let
elements of
G are the points with
S
r(~)
=
n.
r(~)
<
n
Thus there
which can be added to one of the sets
-3.22Al
to get a point
b
to which there leads an edge from
of possible choices is thus
r'
(n-r(~»
a E V - E.
only on the rank of
Corollary of Theorem 33 1
I
,
Let now
d(~)
a similar argument we get that
•
=
(~)
k
and
Mk
= Nk
b
r(~)
if
k
<
n,
=
r
The number
(n-r(~»
be any point of
•
G.
depends
By
Thus all conditions of the
G
is normal.
Thus
A and we get, in view of
that
{3.5.9.)
1,
<
denotes the number of elements of
where
D(a.)
are satis fied, and therefore
we can apply Theorem 3,3,1 ito tha antichain
Nk = r
Thus
a:
A
having rank
k.
As however
(3.5.10.)
rk(~)
Max
ok
, and
= IAI
n
it follows that(3.5.8) holds.
Clearly for
theorem.
A
of
r = 1 Theorem.3.5.l. is nothing else than Sperner's
To show that the inequality (3.5,8) is best possible take for
the family of all
S
such that
r-tuples
(AI' A , ... , A )
r
2
.~ IA. I = [(n+l)r]
J=l
[~~:i~r} of G);
Theorem 3.5.1 and
J
r+l
of disjoint subsets
(i.e. all points of rank
A clearly satisfies the requirements of the
IAI
is equal to the right hand side of (3.5.8).
Other generalizations of Sperner's theorem can also be obtained
from Theorem 3.3.1.
-3.23References to Chapter III.
[1]
Henry H. Crapo and Gian-Carlo Rota t On the foundation of
Combinatorial theory: Combinatorial Geometries t M.I.T. 1968.
[2]
G. Birkhoff t Lattice TheorYt
[3]
David A. Klarner t The number of graded partially ordered sets t
Journal of Combinatorial Theory 6 (1969) p.12-l9.
[4]
L. Harper and
M.I.T.
[5]
Kirby A. Baker t A generalization of Sperner's Lemma t
Journal of Comninatorial Theory 6 (1969) p.224-225.
[6]
A. Feinstein t Foundations of information theorYt
1958
[7]
D. Lubell, A short proof of Sperner's Lemma t
Combinatorial Theory 1 (1966) p.299.
G.C. Rota t
Amer. Math. Soc. ColI. Publ.
Matching theory; an introduction.
Me Graw - Hill t
Journal of