Facility Location in the Sublinear Geometric Model

Abstract

In the sublinear geometric model , we are provided with an oracle access to a point set P of n points in a bounded discrete space [∆] 2 , where ∆ = n O (1) is a polynomially bounded number in n . That is, we do not have direct access to the points, but we can make certain types of queries and there is an oracle that responds to our queries. The type of queries that we assume we can make in this paper, are range counting queries where ranges are axis-aligned rectangles (that are basic primitives in database [36, 11, 17], computational geometry [1, 2, 6, 5], and machine learning [35, 31, 29, 28]). The oracle then answers these queries by returning the number of points that are in queried ranges. Let Alg be an algorithm that (exactly or approximately) solves a problem P in the sublinear geometric model. The query complexity of Alg is measured in terms of the number of queries that Alg makes to solve P . In this paper, we study the complexity of the (uniform) Euclidean facility location problem in the sublinear geometric model. We develop a randomized sublinear algorithm that with high probability, (1 + ϵ )-approximates the cost of the Euclidean facility location problem of the point set P in the sublinear geometric model using ˜ O ( √ n ) range counting queries. We complement this result by showing that approximating the cost of the Euclidean facility location problem within o (log( n ))-factor in the sublinear geometric model using the sampling strategy that we propose for our sublinear algorithm needs ˜Ω( n 1 / 4 ) RangeCount queries. We leave it as an open problem whether such a polynomial lower bound on the number of RangeCount queries exists for any randomized sublinear algorithm that approximates the cost of the facility location problem within a constant factor.