One-Dimensional r-Gathering Under Uncertainty

Abstract

Let C be a set of n customers and F be a set of m facilities. An r-gathering of C is an assignment of each customer \(c \in C\) to a facility \(f \in F\) such that each facility has zero or at least r customers. The r-gathering problem asks to find an r-gathering that minimizes the maximum distance between a customer and its facility. In this paper we study the r-gathering problem when the customers and the facilities are on a line, and each customer location is uncertain. We show that, the r-gathering problem can be solved in \(O(nk+mn\log n+(m+n\log k+n \log n+nr^\frac{n}{r})\log mn)\) and \(O(mn\log n +(n\log n +m) \log mn )\) time when the customers and the facilities are on a line, and the customer locations are given by piecewise uniform functions of at most \(k+1\) pieces and “well-separated” uniform distribution functions, respectively.