Improved 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 lie 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(b⁴(Δ²+log n)+n) . For the two-dimensional problem, we present a Monte Carlo algorithm that runs in subquadratic time for small 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. We also present a subquadratic algorithm for the special case of the two-dimensional problem when b=2.