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.
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Ahmed, S., Nakano, Si., Rahman, M.S. (2019). One-Dimensional r-Gathering Under Uncertainty. In: Du, DZ., Li, L., Sun, X., Zhang, J. (eds) Algorithmic Aspects in Information and Management. AAIM 2019. Lecture Notes in Computer Science(), vol 11640. Springer, Cham. https://doi.org/10.1007/978-3-030-27195-4_1
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