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Maximizing Voronoi Regions of a Set of Points Enclosed in a Maximizing Voronoi Regions of a Set of Points Enclosed in a
Circle with Applications to Facility Location Circle with Applications to Facility Location
Bhaswar B. Bhattacharya
University of Pennsylvania
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Recommended Citation Recommended Citation
Bhattacharya, B. B. (2010). Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with
Applications to Facility Location.
Journal of Mathematical Modelling and Algorithms,
9
(4), 375-392.
http://dx.doi.org/10.1007/s10852-010-9142-0
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/statistics_papers/655
For more information, please contact repository@pobox.upenn.edu.
Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with
Applications to Facility Location Applications to Facility Location
Abstract Abstract
In this paper we introduce an optimization problem which involves maximization of the area of Voronoi
regions of a set of points placed inside a circle. Such optimization goals arise in facility location problems
consisting of both mobile and stationary facilities. Let
ψ
be a circular path through which mobile service
stations are plying, and
S
be a set of
n
stationary facilities (points) inside
ψ
. A demand point
p
is served
from a mobile facility plying along
ψ
if the distance of
p
from the boundary of ψ is less than that from any
member in
S
. On the other hand, the demand point
p
is served from a stationary facility
p
i
∈
S
if the
distance of
p
from
p
i
is less than or equal to the distance of
p
from all other members in
S
and also from
the boundary of
ψ
. The objective is to place the stationary facilities in
S
, inside
ψ
, such that the total area
served by them is maximized. We consider a restricted version of this problem where the members in
S
are placed equidistantly from the center
o
of
ψ
. It is shown that the maximum area is obtained when the
members in
S
lie on the vertices of a regular
n
-gon, with its circumcenter at
o
. The distance of the
members in
S
from o and the optimum area increases with
n
, and at the limit approaches the radius and
the area of the circle
ψ
, respectively. We also consider another variation of this problem where a set of
n
points is placed inside
ψ
, and the task is to locate a new point
q
inside
ψ
such that the area of the Voronoi
region of
q
is maximized. We give an exact solution of this problem when
n
= 1 and a (1 − ε)-
approximation algorithm for the general case.
Keywords Keywords
computational geometry, optimization, stationary and mobile facilities, Voronoi diagrams
Disciplines Disciplines
Applied Statistics | Business | Business Administration, Management, and Operations | Business Analytics
| Management Sciences and Quantitative Methods | Mathematics | Statistics and Probability
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Maximizing Voronoi Regions of a Set of Points Enclosed in a
Circle with Applications to Facility Location
Bhaswar B. Bhattacharya
Indian Statistical Institute, Kolkata - 700 108, India
bhaswar.bhattacharya@gmail.com
Abstract. In this paper we intro duce an optimization problem, which involves maximization
of the area of Voronoi regions of a set of points placed inside a circle. Such optimization goals
arise in facility location problems consisting of both mobile and stationary facilities. Let ψ be
a circular path through which mobile service stations are plying, and S be a set of n stationary
facilities (points) inside ψ. A demand point p is served from a mobile facility plying along the
circumference of ψ if the distance of p from the boundary of ψ is less than that from any mem-
ber in S. On the other hand, the demand point p is served from a particular member p
i
∈ S
if it is closer to p
i
than from all other members in S and also from the boundary of ψ. The
objective is to place the stationary facilities in S, inside ψ, such that the total area served by
them is maximized. We consider a restricted version of this problem where the members in S
are placed equidistantly from the center o of ψ. It is shown that the maximum area is obtained
when the members in S lie on the vertices of a regular n-gon, with its circumcenter at o. The
distance of the members in S from o and the optimum area increases with n, and at the limit
approaches the radius and the area of the circle ψ, respectively. We also consider another vari-
ation of this problem where a set of n points is placed inside ψ, and the task is to locate a new
point q inside ψ such that the area of the Voronoi region of q is maximized. We give an exact
solution of this problem when n = 1 and a (1−ε)-approximation algorithm for the general case.
Keywords: Computational geometry, Optimization, Stationary and mobile facilities, Voronoi
diagrams.
1 Introduction
The main objective in any facility location problem is to judiciously place a set of facilities,
serving a set of users (or demand points), such that certain optimality criteria are satisfied.
Facilities can be stationary, like shops, factory outlets, hospitals, or mobile, which supply
provisions to the users while on the move. The set of users, on the other hand, is either
discrete, consisting of finitely many points, or continuous, i.e., a region where every point
is considered to be a user. Provided all the facilities are equally equipped in all respects, a
user always avails the service from its nearest facility. Thus, each facility a has its service
zone Z(a), consisting of the set of users that are served by it. The service zone may be
a finite set of points or a continuous region. Many variations of facility location problems
in both the discrete and continuous category, under several optimality criteria, have been
studied [12]. Maximizing the cardinality of the service zone(s) of a (type of) facility is one
such criterion. In the discrete case, it generally denotes the number of users, and in the
continuous case it generally represents the area served by that particular (type of) facility.
In the discrete case, the problem of placing a new facility amidst existing ones, such that the
number of users served by it is maximized, has been addressed very recently by Cabello et
al. [8]. They proposed a general technique for solving the problem under different relevant
metrics. Recently, Bhattacharya and Nandy have addressed the problem of simultaneously
placing two new facilities amidst other existing facilities such that the total number of
users served by the two new facilities is maximized [6]. There remain several open problems
for continuous demand regions. Dehne et al. [11] addressed the problem of locating a new
facility p amidst a set of existing facilities, such that the area of the region served by the
new facility is maximized.
We study two variations of a facility placement problem consisting of both mobile and
stationary facilities. Here the set of facilities consists of (i) a circle ψ, and (ii) a set S of
stationary points inside ψ. The boundary of the circle ψ represents the path of a few mobile
facilities so that every point on ψ is assumed to be a facility point. Our objective is to place
the points in S inside ψ such that the total area served by the members in S is maximized.
We consider another variation of this problem, where the boundary of ψ and an existing
set S is serving the region inside ψ; the objective is to place a new facility q inside ψ such
that the area served by q is maximized.
Both these problems are of emerging interest in the context of designing a service network
for disaster management. Imagine a situation where a large locality/island is under certain
disaster, e.g. flood, and the population therein needs urgent help from outside. Most of the
affected region has been rendered inaccessible. In Figure 1(a), we demonstrate a situation,
where a few mobile service stations are plying along a path R (on roads or on rescue ships)
surrounding the affected region, supplying provisions to the distressed people. A few more
stationary locations (for example p
1
, p
2
as in Figure 1(a)) are marked at some less hazardous
locations, where facilities can be established, or provisions be supplied/air-dropped by the
rescue team. At any point of time, a person is likely to observe an accessibility condition,
i.e., he/she will only approach the nearest stationary service station, or the nearest point
on the surrounding path, whichever is nearer. Since service from a mobile station cannot be
obtained as soon as the user reaches its nearest point on the boundary of ψ, the motivation
of the proposed problems is to minimize the users’ dependency on the mobile stations for
getting the service.
Road R
(a)
Z(p
1
)
Z(p
2
)
p
1
p
2
ψ
p
1
(b)
p
2
β
1
β
2
o
Z(p
1
)
Z(p
2
)
Fig. 1. Formulation of the problem
For the second problem, we again imagine a similar situation. But now apart from the
circular road plying with the mobile facilities, we have some existing stationary facilities
placed in the disaster area. A new and better-equipped service station is to be established
amidst the existing ones and our obvious aim would be to maximize the area served by
it. This problem can also be considered as an extension of the competitive facility location
problems related to Voronoi games [1, 10].
These two problems can be mathematically modeled as follows. Let the surrounding
path R be approximated by the circumference of a circle ψ of radius r, with the center at o
(Figure 1(b)). It can be shown that the zone Z(p
1
) of a stationary service station p
1
inside
the circle ψ is an ellipse that includes the center o of ψ. For a pair of service stations p
1
and
p
2
, the service zones Z(p
1
) and Z(p
2
) are no longer complete ellipses, but form elliptical
sectors as shown in Figure 1(b). The first problem then reduces to maximizing the area
covered by Z(p
1
) ∪ Z(p
2
), whereas the goal of the second problem is to maximize the area
covered by Z(p
2
), given the position of the point p
1
.
These two problems can be easily formulated in terms of maximizing the area of a
Voronoi region of a set of points placed inside a circle. The Voronoi diagram of a set S of n
points in R
d
, denoted by V (S), is a partition of the space into |S| mutually non-overlapping
regions (excepting the boundaries) {V R(p, S)|p ∈ S}, where the region V R(p, S) = Z(p) is
the set of points in the space that are closer to the point p than to any other point q ∈ S.
This idea can be extended to the case when the members in S are general objects instead
of points [2, 15, 17]. In this paper, we consider the Voronoi diagram V (S ∪{ψ}) of the circle
ψ and a set S of points placed inside ψ, under the Euclidean metric. The Voronoi region of
a point p ∈ S is denoted by V R(p, S ∪ {ψ}) and that of ψ is denoted by V R(ψ, S ∪ {ψ}).
The optimization problems which we address can now be stated formally as follows:
P1: Given a circle ψ, place a set of n points S = {p
1
, p
2
, . . . , p
n
} inside ψ such that
Area{
S
n
i=1
V R(p
i
, S ∪ {ψ})} is maximized.
P2: Given a circle ψ and a set of n points S = {p
1
, p
2
, . . . , p
n
} placed inside ψ, locate a
new point q inside ψ such that Area{V R(q, S ∪ {ψ, q})} is maximized.
The properties, importance, and usefulness of Voronoi diagrams have been extensively
studied in the literature [4, 5, 7, 16] over the past few decades. The problem of maximizing
the area of Voronoi regions has been considered in the context of Hotelling game or Voronoi
game [1, 10, 15]. Maximizing the area of the Voronoi region of a particular point is addressed
in [9, 11], where the objective is to locate the position of a new point q amidst a set of n
existing points S such that the Voronoi region of q is maximized. Dehne et al. [11] showed
that the area function has only a single local maximum inside the region where the set
of Voronoi neighbors does not change, when the given points are in convex position. They
gave a numerical algorithm for locating the optimal point based on Newton’s approximation.
Cheong et al. [9] proposed a near-linear time algorithm for the same problem that locates
the new optimal point approximately, when the points in S are in general position.
Our framework considers similar area maximization problems in a different scenario. In
Section 2 we prove some basic results, which are used later in our analysis. In Section 3 we
solve a restricted version of problem P1, where the points in S are assumed to be placed
equidistantly from the center o of ψ. Under this assumption, Area{
S
n
i=1
V R(p
i
, S ∪ {ψ})}
is maximized when the points in S lie on the vertices of a regular n-gon with circumcenter
at o. The optimum distance of the members in S from o and the optimum area increases
with n, and at the limit approaches the radius and the area of the circle ψ, respectively.
In Section 4, we study the second problem (Problem P2). We give an exact solution of the
problem for n = 1. Moreover, for n ≥ 1, using the techniques of Cheong et al. [9], we give an
O(n/ε
4
+n log n) algorithm, which locates a point x
a
such that Area{V R(x
a
, S∪{ψ, x
a
})} ≥
(1 −ε)OP TArea, where OP T Area = sup
x
Area{V R(x, S ∪ {ψ, x})} and ε > 0. Finally, in
Section 5 we summarize our work and give some directions for future work.