The Antimedian Function on Paths

An antimedian of a sequence ( ) 1 2 = , , , π k x x x  of elements of a finite metric space ( ) , X d is an element x for which ( ) , ∑ 1 =1 k i d x x is a maximum. The function with domain the set of all finite sequences on X , and defined by ( ) π AM = { x : x is an antimedian of π } is called the antime-dian function on X . In this note, the antimedian function on finite paths is axiomatically characterized.


Introduction
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 [1], Minieka [2], Ting [3], and Zelinka [4].In these investigations two solutions to the problem are given from an algorithmic perspective.The most appealing solution is called the antimedian, the points that maximizes the total distance from the facility to the users.Another solution is the anticenter, the points that maximizes the total distance from facilities to users.For more information about obnoxious facilities the reader is remitted to [5]- [7].In the case of tree graphs, Ting [3] published a linear algorithm to find the antimedian of a tree, and Zelinka [4] proved that the set of leaves of a tree contains an antimedian.This problem can be approached through the axiomatization of location functions.The input of a location function consists of some information with respect to the users of the facilities, and the output is related to the consensus reached based on the given information.The rationality of this process is supported by the fact that location functions must satisfy a number of consensus axioms.The mean function on tree graphs was the first location function studied from the axiomatic point of view by Holzman [8] in the continuous case (in the continuous case a tree contains an infinity number of elements, the edges of the tree are considered to be rectifiable curves, and a profile π and its members are allowed to be located anywhere on edges).After that Vohra [9] also characterized the median function in the continuous case; in addition the reader can see [10].In the discrete case, the center, the median, and the mean function have been characterized axiomatically on trees (see for example [11]- [20]).Not much research has been done with respect to the axiomatic characterization of obnoxious location functions, but recently Balakrishnan et al. [21] published a characterization of the antimedian function on paths.In what follows, we present a different axiomatic characterization, also on paths, of the antimedian function.Ortega and Wang have recently sent for publication an axiomatic characterization of the antimean function on paths.For more information about location theory and axiomatization we refer the reader to the following references [22]- [27].

Preliminaries
Let ( ) , X d be a finite metric space and set is the cartesian product of X .The elements of X * are called profiles and usually denoted by , , , k x x x π =


. Location theory and consensus theory are related to solve the following problem: Given a collection of k users (voters, customers, clients, etc.) with each user having a preferred location point in X , one attempts to find a set of elements of X that satisfy the preferences of the users with respect to some well-defined criteria.Modeling this situation requires the use of a location function on X , which is a , . We are interested in finite metric spaces defined in terms of connected graphs.Let ( ) connected graph, and let d be the usual distance on V , where ( ) , d x y is the length of a shortest path between x and y .It is well known that ( ) , V d is a metric space, and observe that a profile in a graph G is simply a sequence of vertices where repetitions are allowed.We will investigate some properties of the antimedian function on finite metric spaces defined in terms of a very special type of connected graphs, namely paths.

The Antimedian Function on Paths
In this section P or ( ) will denote a path of length p .We will label the vertices of P as 0,1, 2, , p  and assume that the order that the vertices have in the path is given by the order of the numbers 0,1, 2, , p  .Hence, P will be represented as Notice that the set of vertices is and also that vertex 0 is adjacent to vertex 1 , vertex 1 is adjacent to vertex 2 and so on.In the case p has an even number of vertices, we will write 2 1 p k = + .In the case p has an odd number of vertices, we will write 2 p k = .Let ( ) , , , n x x x π =  be a profile on P ; for any x V ∈ we define the status of x with respect to π to be the number The antimedian of π is the set In order to study the antimedian function on P , we will divide the paths in two classes.The set of paths that have an odd number of vertices will be called odd paths, and the set of paths with an even number of vertices will be called even paths.Let ( ) , , , n x x x π =  be a profile on P , the notation w π ∈ will indicate that there is = .We also use [ ] π to denote the set of all the different vertices included in π , and the number of vertices in the profile π counting repetitions is denoted by π .For example consider the profile ( ) , , , m y y y x x x y y y The profile α is called the concatenation of π and β .The following result related to the antimedian function has been proved in [21].
, , , m y y y The definition of the antimedian function implies the following characteristic of this function.
, , , n x x x π =  be a profile on P , and let σ be any permutation of { } , , , .
 The median function on finite tree graphs satisfies the following property that was proved in [13], and will be important in the proof of several results.

s S m S y S y S y
The property of the median function described by Lemma 3 will be called the increasing status property.Lemma 4 Let ( ) Proof.Notice first that a path is also a tree; consequently, we can apply to P the increasing status property.
We first obtain the set ( ) for some 0 t p ≤ ≤ , then we define the paths ( ) On the other hand, assume


We say that a profile π on P is of the form ( ) is a profile on a finite tree T , the median of π consists of all the vertices in the path .
Since a path P is also a tree, and if ( )  Lemma 6 Let P be a path of length p , and let ( ) , , , n x x x π =  be a profile that is not of the form Proof.Since P is a tree we can apply the increasing status property.We start by obtaining the set ( ) for some 0 t p ≤ ≤ , then we define the paths ( )  From Lemmas 4, 5, and 6 we obtain the following important result that characterizes the output of the antimedian function on paths of length p .

Lemma 7 If
( )  is a profile on a path P of length p , then
will play an important role in the following sections.

The Antimedian Function on Odd Paths
In this section 0 1 2 , P p = → → → →  represents a path such that 2 p k = , and note that in this case , , , n x x x π =  be a profile on P ; we will use π to define a new profile that will be denoted π * .This profile contains the vertex k repeated π times.In other words we are assuming that i x k = for all n i ≤ ≤ 0 .So, π * is the profile and Using ( 3), ( 4), and the definitions of ( ) In terms of π ∆ , defined by (2), and ( ) we deduce the following relation for The next result is corollary to the definition of the number π ∆ .

S p d p x d p x d p k d k x d p k d k x S p d k x d k x S p S k S k
The definition of π * and the fact that The following three lemmas establish an important relationship between the numbers ( ) , and π ∆ .These results will be used to characterize the antimedian of profiles π on P .

Lemma 8 If
( ) , and notice ( ) ( ) This implies 0 π ∆ =.Because of ( 5) and the fact that 0 π ∆ =, we obtain Replacing the equal sign with < and > in the proof of Lemma 8, we obtain the next two results.

Lemma 9 If
( ) , , , n x x x π =  is a profile on P , then ( ) ( ) We end this section with an important result that characterizes the antimedian of a profile π on odd paths that is not of the form ( ) , , , n x x x π =  be a profile on an odd path P .If π is not of the form ( ) Assuming 0 π ∆ = and because π is not of the form ( ) 

The Antimedian Function on Even Paths
In this section 0 1 2 , P p = → → → →  represents a path where 2 1 p k = + ; so, we have 1 2 , and in this case Let π be a profile on P .Using similar ideas as in the last section, we can obtain a relationship between the numbers ( ) , π , and π ∆ .Since the profile k i≤ π contains all the vertices of π whose index is less or equal to k , then ( ) ( ) ( ) Using the profile From ( 6) and ( 7) and the definition of ( ) In terms of π ∆ , we have the relation Observe that .
The next three results show some properties of the numbers ( )  By replacing the equal sign with < and > in the proof of Lemma 12, we obtain the following two results.

The Axioms and the Main Result
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 L be a location function on a path P of length p with 2 p k = . Let π ∆ be defined as in (2); if π is not a profile of the form ( )  Some of these axioms are not independent.For example it is clear that (Ex) is a particular case of (G-Ex) when 1 s = .Next we prove that if a location function satisfies (C) and (Ex), it also satisfies (G-Ex).Lemma 16 If L is a location function on P that satisfies axioms (C), (A), and (Ex), then L satisfies axiom (G-Ex).

where 2 X
denotes the set of all subsets of X .Three well known examples of location functions are: a) the center function, denoted by Cen, and defined as median function, denoted by Med, and defined as mean function, denoted by Mean, and defined as

1 sLemma 5 1 s
≥ , if π contains exactly s times the vertices 0 and p .For example the profile Let π be a profile the form ( ) ≥ on a path P of length p .Then and we can reorder the vertices of π to define the profile


The next result characterizes profiles π on a paths of length p that satisfy the condition that are not of the form ( ) partition of the profile π as follows: denote by k i≤ π the profile such that [ ] [ ] [ ]

.
Notice that π can be seen as the concatenation of profiles Using a similar argument as above, we obtain

(
Let L be a location function on a path P of length p with 2 (2); if π is not a profile of the form ( ) Let L be a location function on P .If Let L be a location function, and π be a profile on P .If Let L be a location function, and let π be a profile on P .If π is of the form ( ) The next lemma is an important result because it characterizes the antimedian of profiles π , on even paths, that are not of the form ( ) Let π be a profile on P .If π is of the form ( )