Nissan Lev-Tov and David Peleg∗
Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
{nissanl,peleg}@wisdom.weizmann.ac.il
Abstract. This paper concerns geometric disk problems motivated by base station placement problems arising in wireless network design. We first study problems that involve maximizing the coverage under vari- ous interference-avoidance constraints. A representative problem for this type is themaximum weight independent setproblem on unit disk graphs, for which we present an exact solution whose complexity is exponential but with a sublinear exponent. Specifically, our algorithm has time com- plexity 2O(√mlogm), wheremis the number of disks. We then study the problem of covering all the clients by a collection of disks of variable radii while minimizing the sum of radii, and present a PTAS for this problem.
1 Introduction
1.1 Background
This paper deals with efficient algorithmic solutions for base station place- ment problems and related problems arising in wireless network design. The input for these problems consists of two sets of points in the Euclidean plane,
X ={x1, x2, . . . , xm} representing potential locations for placing base stations,
and Y ={y1, y2, . . . , yn} representing the clients. A base station located atxi
has a certain transmission range Ri, which could be either fixed or variable. A
client nodeyj is covered by a base station placed at xi if it is within its trans-
mission range, namely, if yj falls within the disk of radius Ri centered at xi.
However, coverage may not be enough; in certain models it is also necessary to avoid interferences between neighboring base stations whose transmission disks partially overlap. Hence our problems concern selecting a placement for the base stations (henceforth referred to in short asservers) that will guarantee adequate (interference-free) coverage for the clients while attempting to optimize certain cost functions.
Two design issues affect the nature of the optimization problem at hand. The first concerns the question whether interference avoidance must be enforced. In particular, when interferences are ignored, the problem to be solved is typically a disk-covering problem, namely, finding a minimum cost collection of servers covering all clients. In contrast, when interferences must be avoided, it might be Supported in part by a grant from the Israel Ministry of Industry and Commerce.
M. Penttonen and E. Meineche Schmidt (Eds.): SWAT 2002, LNCS 2368, pp. 90–99, 2002. c
impossible to satisfy all the clients simultaneously, hence it is of interest to study also variants of the problem aimed at maximizing the number of clients which are covered by exactly one server (henceforth referred to as supplied clients). A combined approach which is examined as well is to simultaneously take into account the number of supplied clients and the cost of the servers, by attempting to optimize totalprofit, defined as the gain from supplied clients minus the cost of the servers.
The second design issue is whether transmission radii are fixed or variable. When the radii are variable, we must decide on the transmission range of each server to be built, in addition to choosing its locations. The typical goal is to choose for each server xi a transmission radius Ri such that all the clients are
covered and the sum of radii is minimized. This target arises from the assumption that the cost of choosing a certain radius depends linearly on that radius. Such assumption is often made in various clustering and covering problems [1]. 1.2 The Problems
This paper considers problems of two main types. The first involves maximization problems on disks of fixed radius. Given the locations of the clients and servers and a fixed transmission rangeR, it is required to choose an active set of servers in an optimal way. Several optimization targets can be considered, all taking into account interferences between active servers.
We start with the problem of maximizing the number of supplied clients un- der the condition that theR-disks around the servers are disjoint. This require- ment is perhaps not the most natural in the context of base station placement, as overlaps over client-free regions should cause no problems. On the other hand, this problem can be handled as a special case of themaximum weight indepen- dent set (MWIS)problem on unit disk graphs, which is of independent interest. In the MWISproblem there are weights associated with the points inX, and it is required to choose a maximum weight subset of X such that theR-disks around them are all disjoint. To get our problem as a special case of theMWIS
problem, the weight of each vertexxi∈X is set to the number of clients covered
in theR-disk aroundxi.
We also consider a number of related problems, which capture the restrictions of the model more adequately. The first problem requires us to maximize the number of supplied clients under the constraint that no client is in the transmis- sion range of more than one chosen server. Again, this problem can be generalized into themaximum weight collision-free set (MWCS)problem, which is to find a maximum weight subset of servers such that theR-disks around the servers do not contain common clients. Another variant of this problem, namedmaximum supplied clients (MSC), which may be even more useful from practical point of view, is defined as follows. For any subset ˜X ofX, letSupp( ˜X) denote the set of clients supplied by ˜X. The problem is to choose a subset of servers ˜X maximiz- ing the number of supplied clients,|Supp( ˜X)|. We also consider a variant of the problem namedmaximum profit (MP), in which costs and benefits are treated in a combined manner. The goal is to choose a subset of servers maximizing the profitP( ˜X) =c1|Supp( ˜X)| −c2|X˜|, where c1, c2>0 are given constants.
The MWCS, MSCand MP problems turn out to be hard to manage for arbitrary inputs, and we consider them in a restricted setting referred to as the grid-based setting. In this setting, the set X of possible server locations is restricted to grid points of a given spacing, fixed for concreteness to be 1, and moreover, the transmission range R is assumed to be bounded by a constant. Without loss of generality we assume that the underlying unit gridG1is aligned
so that (0,0) occurs as a grid point. Throughout, the grid-based version of a problemPROBis denotedPROBg.
The second type of problems considered in this paper is disk-covering prob- lems, where the goal is to achieve minimum sum of radii. Given a set of clients
Y and a set of serversX in the plane, we aim to choose the transmission range
Ri of each server xi such that all the clients are covered and the sum of the
transmission ranges chosen,ϕ=iRi, is minimized. Although any radius Ri
can be chosen for a given server, every solution is dominated by a solution in which for each chosen radius Ri =0 the corresponding Ri-disk has a client on
its border. So the problem is equivalent to choosing from alln·mdisks centered at a server and with a client on their border. This problem is referred to as minimum sum of radii cover (MSRC).
We also study the variant of this problem where interferences are involved, meaning that the transmission ranges must be chosen such that no two disks intersect on a client. We call this problem minimum sum of radii cover with interferences (MSRCI). Its representation is more general as we are given a set of disks of various radii and we have to choose a subset of the given set of disks such that the chosen disks are all disjoint and minimum sum of radii is achieved. 1.3 Previous Work
The disk-covering problem on fixed radius disks is studied in [5], in a model where the server locations are not restricted to a given set of possible locations but rather may be chosen at any point on the plane. A PTAS is given for this problem using a grid-shifting strategy.
Fixed radius covering problems where only potential server locations are considered, with or without interferences, are studied in [4]. They consider op- timization problems for cellular telephone networks that arise in a traffic load model which also addresses the positioning of servers on given possible locations with the aim of maximizing the number of supplied clients and minimizing the number of servers to be built.
A technique calledslab dividing is proposed in [7]. It is used there to give a sub-exponential exact solution of time complexityO(nO(√P)) for the Euclidean
P-center problem. This approach is essentially based on a version of the √n- planar separator theorem of [8], suitably adapted to the Euclidean case.
TheMWISproblem on unit-disk graphs is shown to be NP-hard in [2], and is given a PTAS in [6]. The MWIS problem on general (arbitrary radii) disk graphs is considerably harder, and was only recently shown to have a PTAS using a sophisticated hierarchical grid-shifting technique [3].
1.4 Our Results
The paper presents exact and approximate solutions for the above problems. We begin in Section 2 by developing a variant of the slab technique of [7] which is suitable for handling maximum independent set and maximum covering set problems. We then apply our method for deriving an 2O(√mlogm) time exact
solution for theMWISproblem.
Using variations of our method it is possible to obtain similar results for a number of grid-based problems. In particular, the grid-based MWISg problem
can be given a slightly better 2O(√m)time solution, and a similar solution exists
for the grid-basedMWCSg problem. The grid-based problemsMSCgandMPg
enjoy 2O(√m+logn) time exact solutions. The details of these results are also
deferred to the full paper. In the full paper we also provide a PTAS for the grid-based MPg problem, using a grid-shifting strategy similar to that of [5].
We then turn to the variable radii model. In section 3 we present a PTAS for theMSRCproblem (with no interference constraints), based on a modified variant of the hierarchical grid-shifting technique of [3].
Note that all our results can be extended for the case where each client has a certain weight (say, representing the fee paid by this client or the significance of providing it with service), and the optimization targets refer to the sum of weights of the supplied clients instead of merely their number.
While our focus is on the natural 2-dimensional variants of the above prob- lems, we also studied their 1-dimensional variants. Our polynomial time solutions for the 1-dimensionalMWIS,MWCS,MSCandMPproblems and theMSRC
problems with and without interference constraints are deferred to the full paper.