Output-Sensitive Algorithms for Uniform Partitions of Points
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 , 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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