Output-sensitive algorithms for uniform partitions of points

Abstract

We consider the following one- and two-dimensional bucketing problems: Given a set S of n points in R¹ or R² and a positive integer b, distribute the points of S into b equal-size buckets so that the maximum number of points in a bucket is minimized. Suppose at most (n/b) + Δ points lies in each bucket in an optimal solution. We present algorithms whose time complexities depend on b and Δ. No prior knowledge of Δ is necessary for our algorithms. For the one-dimensional problem, we give a deterministic algorithm that achieves a running time of O(b⁴Δ² log n + n). For the two-dimensional problem, we present a Monte-Carlo algorithm that runs in sub-quadratic time for certain values of b and Δ. The previous algorithms, by Asano and Tokuyama [1], searched the entire parameterized space and required Ω(n²) time in the worst case even for constant values of b and Δ.