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.