^{1}

^{*}

^{1}

^{*}

An antimedian of a sequence

The problem of finding one optimal location for schools, drug stores, police stations, and hospitals requires facilities to be placed near the users in order to minimize, for example, the distance traveled to reach them. Location theory deals with this type of optimization problem. Location functions such as the median, the center, and the mean have been used to solve these type of problems. On the other hand, there are circumstances where placing one or more facilities as far as possible from the users is the best solution. For instance, it is necessary to locate nuclear power plants far from cities or towns to minimize the risk of radiation problems. Similar problems include the determination of suitable locations for observatories, radio stations, airports, and chemical plants. The solution to the problem of finding an optimal location for these types of obnoxious facilities on networks has been studied by Church and Garfinkel [

Let

where

a) the center function, denoted by Cen, and defined as

where

b) the median function, denoted by Med, and defined as

where

c) the mean function, denoted by Mean, and defined as

where

We are interested in finite metric spaces defined in terms of connected graphs. Let

In this section

Notice that the set of vertices is

A vertex

The antimedian of

In order to study the antimedian function on

will indicate that there is

For example consider the profile

If

The profile

Lemma 1 Let

The definition of the antimedian function implies the following characteristic of this function.

Lemma 2 Let

where

The median function on finite tree graphs satisfies the following property that was proved in [

Lemma 3 Let

The property of the median function described by Lemma 3 will be called the increasing status property.

Lemma 4 Let

a)

b)

Proof. Notice first that a path is also a tree; consequently, we can apply to

and

By the increasing status property we have

and

Observe that

if

if

On the other hand, assume

and

By the increasing status property we have

and

If

if

We say that a profile

Lemma 5 Let

Proof. It is well known that if

from

Since a path

and the definition of the antimedian function implies that

By Lemmas 1 and 2 we obtain

The next result characterizes profiles

Lemma 6 Let

Proof. Since

and

By the increasing status property we obtain

and

Observe that:

if

and

By the increasing status property we have

and

Note that

From Lemmas 4, 5, and 6 we obtain the following important result that characterizes the output of the antimedian function on paths of length

Lemma 7 If

Assume

is a path of length

we define a partition of the profile

will play an important role in the following sections.

In this section

represents a path such that

We want to establish a relationship between

and

Using (3), (4), and the definitions of

In terms of

The next result is corollary to the definition of the number

Corollary 1 If

The definition of

These relations imply

The definition of

The following three lemmas establish an important relationship between the numbers

Lemma 8 If

Proof. Assume first

This implies

Because of (5) and the fact that

Replacing the equal sign with

Lemma 9 If

Lemma 10 If

We end this section with an important result that characterizes the antimedian of a profile

Lemma 11 Let

Proof. Assuming

If

If

In this section

represents a path where

profile on

Using the profile

From (6) and (7) and the definition of

In terms of

Observe that

Using a similar argument as above, we obtain

The definition of

This identity provides the following relation between

The next three results show some properties of the numbers

Lemma 12 If

Proof. Assume first

Conversely, if

By replacing the equal sign with

Lemma 13 If

Lemma 14 If

The next lemma is an important result because it characterizes the antimedian of profiles

Lemma 15 Let

Proof. Assuming

and Lemma 6 indicates

If

If

The next result is a corollary to Lemma 15.

Corollary 2 Let

Proof. Notice that in this case the profile

Finally, Lemma 15 implies

The axioms listed below are among the desirable properties that a general location function should satisfy, and it is not difficult to verify that the antimedian function satisfies these properties.

Oddness (O): Let

Evenness (E): Let

Consistency (C): Let

Extremeness (Ex): Let

Generalized Extremeness (G-Ex): Let

Anonymity (A): For any profile

have

Some of these axioms are not independent. For example it is clear that (Ex) is a particular case of (G-Ex) when

Lemma 16 If

Proof. Let

With the axioms listed above we will give two axiomatic characterizations for the antimedian function. The next theorem contains the first of these characterizations.

Theorem 1 Let

Proof. It is clear that if

Case 1. Assume first

Case 2. Assume

If

Case 3. Assume

profile on

strates

we get

If

We leave it to the reader to prove that the axioms used in the proof of Theorem 1 are independent. Notice that in the proof of Theorem 1 we needed to use three axioms to establish Case 1. We want to improve the demonstration of this result using fewer axioms. We achieve this objective using axiom (G-Ex) in the following theorem which is our main result.

Theorem 2 Let

Proof. It is clear that if

Case 1. Assume first

Case 2. Assume

Case 3. Assume

If

A similarly argument can be used to demonstrate that if

Notice that the definition of axioms (O), (E), and (G-Ex) indicate that they are independent. So it is not necessary to add a proof for the independence of these three axioms. More research is needed to find an axiomatic characterization of the antimedian function on tree graphs.