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Sweeping Points

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Abstract

Given a set of points in the plane, and a sweep-line as a tool, what is best way to move the points to a target point using a sequence of sweeps? In a sweep, the sweep-line is placed at a start position somewhere in the plane, then moved orthogonally and continuously to another parallel end position, and then lifted from the plane. The cost of a sequence of sweeps is the total length of the sweeps. Another parameter of interest is the number of sweeps. Four variants are discussed, depending on whether the target is a hole or a pile, and whether the target is specified or freely selected by the algorithm. Here we present a ratio 4/π≈1.27 approximation algorithm in the length measure, which performs at most four sweeps. We also prove that, for the two constrained variants, there are sets of n points for which any sequence of minimum cost requires 3n/2−O(1) sweeps.

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Authors and Affiliations

  1. Department of Computer Science, University of Wisconsin–Milwaukee, Milwaukee, WI, 53201-0784, USA

    Adrian Dumitrescu

  2. Department of Computer Science, Utah State University, Logan, UT, 84322-4205, USA

    Minghui Jiang

Authors
  1. Adrian Dumitrescu
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  2. Minghui Jiang
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Corresponding author

Correspondence to Adrian Dumitrescu.

Additional information

A preliminary version of this paper appeared in the Proceedings of the 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2008), MIT, Boston, USA, August 2008; LNCS, Vol. 5171, pp. 63–76, Springer, Berlin.

A. Dumitrescu was supported in part by NSF CAREER grant CCF-0444188.

M. Jiang was supported in part by NSF grant DBI-0743670.

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Dumitrescu, A., Jiang, M. Sweeping Points. Algorithmica 60, 703–717 (2011). https://doi.org/10.1007/s00453-009-9364-6

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  • DOI: https://doi.org/10.1007/s00453-009-9364-6

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