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New Algorithms and Bounds for Halving Pseudolines

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Algorithms and Discrete Applied Mathematics (CALDAM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12016))

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Abstract

Let P be a set of points in general position in the plane. A halving line of P is a line passing through two points of P and cutting the remaining \(n-2\) points in a half (almost half if n is odd). Generalized configurations of points and their representations using allowable sequences are useful for bounding the number of halving lines.

We study a problem of finding generalized configurations of points maximizing the number of halving pseudolines. We develop algorithms for optimizing generalized configurations of points using the new notion of partial allowable sequence and the problem of computing a partial allowable sequence maximizing the number of k-transpositions. It can be viewed as a sorting problem using transpositions of adjacent elements and maximizing the number of transpositions at position k.

We show that this problem can be solved in \(O(nk^n)\) time for any \(k>2\), and in \(O(n^k)\) time for \(k=1, 2\). We develop an approach for optimizing allowable sequences. Using this approach, we find new bounds for halving pseudolines for even n, \(n\le 100\).

The research is supported in part by NSF award CCF-1718994.

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Notes

  1. 1.

    This improves the space by a factor of \(k^{k-1}/(k-1)!\). For example, if \(k=5\) this a factor of 26.041.

  2. 2.

    The blocks in this section are different from alternating blocks used in the proof of Theorem 6.

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Authors and Affiliations

  1. University of Texas at Dallas, Richardson, TX, 75080, USA

    Sergey Bereg & Mohammadreza Haghpanah

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  1. Sergey Bereg
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  2. Mohammadreza Haghpanah
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Correspondence to Sergey Bereg .

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Editors and Affiliations

  1. University of Kerala, Thiruvananthapuram, India

    Manoj Changat

  2. Indian Statistical Institute, Kolkata, India

    Sandip Das

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Bereg, S., Haghpanah, M. (2020). New Algorithms and Bounds for Halving Pseudolines. In: Changat, M., Das, S. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2020. Lecture Notes in Computer Science(), vol 12016. Springer, Cham. https://doi.org/10.1007/978-3-030-39219-2_37

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  • DOI: https://doi.org/10.1007/978-3-030-39219-2_37

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