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 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 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 ⁴Δ² 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.

Work by the first author was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by NSF grants EIA-9870724, and CCR-9732787, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by the second author was supported by an NSERC grant. Part of this work was done while the last two authors were visiting Department of Computer Science, University of Newcastle, Australia.