Continuous Optimization
A geometric characterisation of the quadratic min-power centre

https://doi.org/10.1016/j.ejor.2013.09.004Get rights and content

Highlights

  • Complete geometric description of the min-power centre of a finite set of nodes under quadratic cost.

  • Various results relating the min-power centre to the centroid and the 1-centre.

  • Linear-time algorithm for the construction of the quadratic min-power centre from the convex hull.

  • Upper bound for the performance of the centroid as an approximation to the quadratic min-power centre.

Abstract

For a given set of nodes in the plane the min-power centre is a point such that the cost of the star centred at this point and spanning all nodes is minimised. The cost of the star is defined as the sum of the costs of its nodes, where the cost of a node is an increasing function of the length of its longest incident edge. The min-power centre problem provides a model for optimally locating a cluster-head amongst a set of radio transmitters, however, the problem can also be formulated within a bicriteria location model involving the 1-centre and a generalised Fermat-Weber point, making it suitable for a variety of facility location problems. We use farthest point Voronoi diagrams and Delaunay triangulations to provide a complete geometric description of the min-power centre of a finite set of nodes in the Euclidean plane when cost is a quadratic function. This leads to a new linear-time algorithm for its construction when the convex hull of the nodes is given. We also provide an upper bound for the performance of the centroid as an approximation to the quadratic min-power centre. Finally, we briefly describe the relationship between solutions under quadratic cost and solutions under more general cost functions.

Introduction

One of the most important problems in the optimal design of wireless ad hoc radio networks is that of power minimisation. This is true during the physical design phase and when designing efficient routing protocols (Montemanni et al., 2008, Yuan and Haugland, 2012, Zhu et al., 2012). The most appropriate fundamental model in both cases is the power efficient range assignment problem, where a communication range ri is assigned to each transmitter xi such the resultant network is connected and that total power riα is minimised (see Althaus et al., 2006). The exponent α is called the path loss exponent and most commonly takes a value between 2 and 4, with α = 2 corresponding to transmission in free-space. In this paper we study a type of power efficient range assignment problem which allows for the introduction of a single additional transmitter in the plane. We show that this formulation is related to a continuous version of the cent-dian problem (Colebrook & Sicilia, 2007) from location analysis, and present a geometric method (based on farthest point Voronoi diagrams) of constructing the optimal solution when α = 2.

The classical range assignment problem, which does not include the option of introducing new transmitters, is a type of disk covering problem, where the centres of the disks are given nodes, the radii (ri) of the disks are transmission ranges, and the directed graph induced by the disks must satisfy a given connectivity constraint (for instance strong connectivity, biconnectivity, etc.) whilst minimising riα; see Fig. 1(a), where the graph drawn is not necessarily optimal. Observe that graphs that result from some assignment of ranges are similar to unit-disk graphs, except that the disks do not all have the same radius in our case. For any α > 1 the range assignment problem is NP-hard, even in the case when only 1-connectivity is required of the resultant network (Fuchs, 2008).

The idea of extending the classical range assignment problem by allowing the introduction of additional transmitters is justified because, during the design or maintenance of ad hoc radio networks, it is often pertinent to introduce relays or cluster-heads for the processing of aggregated data and for the improved routing efficiency that takes place in such hierarchical structures (see Dhanaraj and Murthy, 2007, Paul and Matin, 2011, Shi et al., 2009). Solving the range assignment problem whilst allowing for the introduction of a bounded number of additional nodes anywhere in the plane constitutes a very general and highly applicable geometric network problem, which has only been solved in certain restricted settings (see for instance Brazil et al., 2012, Brazil et al., 2010). Since the optimal locations of the cluster-heads must be found, as well as the optimal assignment of ranges on the complete set of nodes, this so called geometric range assignment problem is at least as difficult as the classical range assignment problem. Note that this problem is a type of continuous location problem, since the cluster-heads are free to be located anywhere in the plane.

This paper considers the problem of optimally locating a single cluster-head amongst a given set of transmitters, where each transmitter can send and receive data directly to and from the cluster-head. Not only is this an interesting and applicable model in it own right, but it is also a necessary first step in understanding the local structure of optimal networks with multiple cluster-heads. The graph induced by the assignment of ranges, in this case, contains an undirected star with the cluster head as its centre and the complete set of transmitters as its leaves; see for instance Fig. 1(b), where the white node represents a cluster-head.

In more formal terms we denote the given finite set of transmitters by XR2 and the cluster-head by sR2. The power of any x  X is Px = s  xα and the power of s is Ps = max{∥s  xα:x  X}, where ∥ · ∥ is the Euclidean norm. The total power of the system is denoted by P(s)=Ps+xXPx, and a min-power centre of X is a point s which minimises P(s). The min-power problem, which is the problem we address in this paper, seeks to locate a min-power centre of a given set X in the plane. Minimising only Ps is clearly equivalent to the 1-centre problem, i.e., the problem of finding the centre of a minimum spanning circle for X. Minimising only Px is a generalised Fermat-Weber problem (Brimberg & Love, 1999), which becomes the classical Fermat-Weber problem when α = 1 and the problem of constructing the centroid when α = 2.

A concept in facility location that is related to the min-power problem is the computation of the centre-median (or cent-dian) of a finite set of points (Colebrook & Sicilia, 2007); although, strictly speaking, the cent-dian is only defined for α = 1. The cent-dian problem requires one to find the Pareto-optimal solutions to the vector function Φ(s)=(Ps,Px), which is equivalent to finding the optimal value of λPs+(1-λ)Px for every λ   [0,1] (see Duin and Volgenant, 2012, Fernandez et al., 2001). The min-power problem may be viewed as a type of cent-dian problem, since it results by setting λ = 1/2 and by allowing other values of α besides α = 1. The cent-dian problem (with α = 1) has been considered in the rectilinear plane (McGinnis & White, 1978), and for α = 2 in the Euclidean plane (Ohsawa, 1999). Besides optimally locating a cluster-head amongst given transmitters, the cent-dian also has another application in wireless ad hoc networks, namely, finding a so called core node (Dvir & Segal, 2010). Bicriteria models such as the cent-dian have been described as seeking a balance between the antagonistic objectives of efficiency (i.e., the minisum component) and equity (i.e., the minimax component).

There are numerical methods, for instance the sub-gradient method, that optimally locate a min-power centre to within any finite precision. However, structural results for the min-power centre problem, of the type described in this paper, are necessary for optimally constructing more complex geometric range assignment networks (which is the overarching goal of our research). This fact is particularly manifest in the design of algorithmic pruning modules, where one develops strategies based on properties of locally optimal structures for eliminating suboptimal network topologies from the exponential set of possible topologies. The ultimate benefits of good pruning modules has been demonstrated a number of times for problems similar to the geometric range assignment problem (Brazil et al., 2012, Brazil et al., 2010, Warme et al., 2000).

In this paper we mostly focus on the quadratic case, α = 2, and develop a complete geometric description of the solution in terms of farthest point Voronoi diagrams and Delaunay triangulations. In terms of cluster-head placement the α = 2 assumption means that radio transmission takes place in free space, that is, in an ideal medium with zero resistance. Path loss exponents close to 2 frequently occur in real-world wireless radio network scenarios. This is true for transmission in mediums of low resistance, and in mediums of higher resistance when there is a degree of beam forming (constructive interference) (Karl & Willig, 2007). The quadratic case also applies to certain classical facility location problems, including the location of emergency facilities such as hospitals and fire stations (Fernandez et al., 2001, Ohsawa, 1999, Puerto et al., 2010). Furthermore, as demonstrated in the final section, it is anticipated that theoretical developments in the α = 2 case will lead to solutions and approximations for other α > 1.

Single-facility location problems under very general norms and convex cost functions have been studied by Durier (1995) and by Durier and Michelot, 1985, Durier and Michelot, 1994. Some of these formulations include the min-power problem as a special case, however, a constructive method of producing the min-power centre does not directly follow from their work. The principle contribution of the research presented by Durier and Michelot in these papers involves a description of the set of solutions to generalised Fermat-Weber problems, which is a feasible and interesting endeavour when the objective cost function is not strictly convex (as is the case with the min-power problem). Nickel, Puerto, and Rodríguez-Chía (2003) generalise this approach even further by allowing each given point (facility) to be replaced by a set of points, where an optimal centre is required to “serve” at least one point from each given set. A stochastic version of generalised single-facility location problems has also been studied (see Puerto & Rodríguez-Chía (2011)).

In Section 2 we provide definitions and set up a Karush–Kuhn–Tucker formulation of the min-power centre problem and its dual for any α > 1. When α = 2 the geometric construction of the min-power centre becomes tractable, allowing us to provide a characterisation of the solution in terms of the farthest point Voronoi diagram on X, and its dual, the farthest point Delaunay triangulation. This characterisation is described in Section 3, where we also present a new linear-time algorithm for the construction of the optimal quadratic min-power centre. For α = 2, Section 4 explores the significance of the centroid and the 1-centre of a set of points for the min-power centre problem, and provides a characterisation of point sets for which the min-power centre and the 1-centre coincide. Section 4 also provides a bound for the performance of the centroid as an approximation to the min-power centre. The final section briefly explores the general α > 1 case.

Section snippets

Definitions and analytical properties

Let X = {xi:i  J} be a given finite set of points in the Euclidean plane with index-set J, and let α > 1 be a given real number. For any point sR2 let G = G(s, X) be the undirected star with centre s and leaf-set X. The power of G with respect to s is denoted byP(s)=Pα(s,X)=iJs-xiα+maxiJ{s-xiα},where ∥ · ∥ is the Euclidean norm. The first terms of P(s) can be thought of as representing the total power required by the existing transmitters for communicating with the cluster-head, while the second

The geometry of α = 2

In order to study the geometry of the case where α = 2 we first rewrite the KKT conditions in terms of the 2-centroids of X.

Definition: Recall that the centroid (or centre of mass) of X, which we denote as M, is defined by M=1njJxj. We define the set of 2-centroids of X to be the set M={Mj:jJ} where each Mj=1n+1xj+iJxi.

Observe that M is the image of an affine transformation on X. A consequence of this observation is the following lemma. Let conv (·) denote the convex hull of a set of points.

The significance of the centroid and the 1-centre to the min-power centre problem

It is clear from the results so far that the centroid M plays a central role in the min-power centre problem when α = 2. In particular, it is intuitive that M is always relatively close to the min-power centre, s, since both lie in conv(M). This fact leads to a theorem at the end of this section which places an upper bound on the performance ratio of approximating the min-power centre by the centroid. The fact that M is a “special” point in the min-power centre problem was also observed in

A brief look at general α > 1

The situation when α  2 is complicated by the fact that the level curves of the Fj are no longer circular. In particular, this implies that there are no easily constructed and useful fixed points directly analogous to the Mj. Even though the farthest point Voronoi diagram still plays a crucial role for general α, there is no simple dual structure analogous to the Delaunay triangulation. We may consider argminss-xjα (the generalised Fermat-Weber point) as an analogue for M, however, a compass

Conclusion

In this paper we solve the quadratic min-power centre problem in the Euclidean plane. We provide a complete geometric description and method of construction for the optimal point by means of farthest point Voronoi diagrams. The solution leads to various structural results relating the min-power centre to the centroid and the 1-centre, and, in particular, allows us to construct an explicit bound on the performance of the centroid as an approximation point. We anticipate that the mathematical

References (28)

  • M. Brazil et al.

    Approximating minimum Steiner point trees in Minkowski Planes

    Networks

    (2010)
  • Dhanaraj, M., & Murthy, C. (2007). On achieving maximum network lifetime through optimal placement of cluster heads in...
  • A. Dvir et al.

    Placing and maintaining a core node in wireless ad hoc networks

    Wireless Communications and Mobile Computing

    (2010)
  • R. Durier

    The general one centre location problem

    Mathematics of Operations Research

    (1995)
  • Cited by (6)

    This research was supported by an Australian Research Council Discovery grant.

    View full text