# Output-Sensitive Algorithms for Uniform Partitions of Points

## Abstract

We consider the following one- and two-dimensional bucke- ting problems: Given a set *S* of *n* points in ℝ^{1} or ℝ^{2} 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 pre- sent 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*(*b* ^{4}Δ^{2} 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 *Ή*(*n* ^{2}) time in the worst case even for constant values of *b* and Δ.

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