Information Processing Letters 109 (2009) 386–390 Contents lists available at ScienceDirect Information Processing Letters www.elsevier.com/locate/ipl Multi-dimensional dynamic facility location and fast computation at query points S. Abravaya a ,∗ , D. Berend b a Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel b Departments of Mathematics and Computer Science, Ben-Gurion University, Beer-Sheva 84105, Israel a r t i c l e i n f o a b s t r a c t Article history: Received 20 March 2007 Available online 24 December 2008 Communicated by F. Dehne Keywords: Computational geometry Minimax criterion Facility location We present O ( n log n ) time algorithms for the minimax rectilinear facility location problem in R 1 and R 2 . The algorithms enable,once they terminate,computing the cost of any given query point in O (log n ) time. Based on these algorithms, we develop a preprocessing procedure which enables solving the following two problems: Fast computation of the cost of any query point in R d , and fast solution for the dynamic location problem in R 2 (namely, in the presence of an additional facility). Finally,we show that the preprocessing always gives a bound on the optimalvalue,which allows us in many cases to find the optimum fast (for both the traditional and the dynamic location problems in R d for any d).  2008 Elsevier B.V. All rights reserved. 1. Introduction Numerous studies in operations research deal with the facility minimax location problem, as well as many vari- ants (cf. [2–4]). Megiddo [5] introduced linear time algo- rithms for linear programming in R d for any fixed d > 0. Ogryczak and Tamir [6] solved the problem of minimizing the sum of the k largest functions out of n functions, each defined on R d , over a polyhedral set Q ⊂ R d . By represent- ing any instance of the location problem in R d as a linear programming problem,it can be solved in linear time by any of these linear programming based algorithms (with k = 1 for the second algorithm). Clearly, these solutions neither support query points computation nor resolve effi- ciently the problem in the presence of an additional facility (namely,the dynamic version of the location problem). In Section 2 we introduce the model formally, introduce the problems,and state the main results.In Section 3, al- gorithms for the problems in R 1 are presented.Based on these solutions,we provide in Section 4 algorithms for the problems in R d for any fixed d. * Corresponding author. E-mail addresses: abravaya@cs.bgu.ac.il (S. Abravaya), berend@math.bgu.ac.il (D. Berend). 2. The main results In our model we have n facilities at points ( a i1 , . . . , a id ) ∈R d ,1 i n, and a new facility is to be located at a point ( x 1 , . . . , x d ) ∈ R d . The maximum cost incurred due to the distances between the new facility and the exist- ing facilities (where w i 0 and g i are constants for each 1 i n) is given by f ( x 1 , . . . , x d ) = max 1 i n w i d j =1 | x j − a i j | + g i . (1) Problem 1 (Single facility location). Find a point x∈R d min- imizing (1). Theorem 2.1. For d = 1, 2, Problem 1 can be solved by an algo- rithm requiring O ( n log n ) time. The algorithm is presented in Section 3. Note that there are some linear time algorithms for the problem, which are based on linear programming. The importance of our algo- rithm is that it enables efficientsolutions for Problem 2 and Problem 3 infra.Thus, it can be regarded as a prepro- cessing for the algorithms solving those problems. 0020-0190/$ – see front matter  2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2008.12.014