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Approximating the Smallest k-Enclosing Geodesic Disc in a Simple Polygon

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Algorithms and Data Structures (WADS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14079))

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Abstract

We consider the problem of finding a geodesic disc of smallest radius containing at least k points from a set of n points in a simple polygon that has m vertices, r of which are reflex vertices. We refer to such a disc as a SKEG disc. We present an algorithm to compute a SKEG disc using higher-order geodesic Voronoi diagrams with worst-case time \(O(k^{2} n + k^{2} r + \min (kr, r(n-k)) + m)\) ignoring polylogarithmic factors.

We then present a 2-approximation algorithm that finds a geodesic disc containing at least k points whose radius is at most twice that of a SKEG disc. Our algorithm runs in \(O(n \log ^{2} n \log r + m)\) expected time using \(O(n + m)\) expected space if \(k \in O\left( n / \log n\right) \); if \(k \in \omega \left( n / \log n\right) \), the algorithm computes a 2-approximation solution with high probability in \(O(n \log ^{2} n \log r + m)\) worst-case time with \(O(n + m)\) space.

Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Notes

  1. 1.

    Also known as the minimum enclosing disc problem.

  2. 2.

    In this paper, we use the notation |Z| to denote the number of points in Z if Z is a point set, or the number of vertices of Z if Z is a face or a polygon.

  3. 3.

    A \(\beta \)-approximation means that the disc returned has a radius at most \(\beta \) times the radius of an optimal solution.

  4. 4.

    When we refer to a point p being in a polygon P, we mean that p is in the interior of P or on the boundary, \(\partial P\).

  5. 5.

    We say an event happens with high probability if the probability is at least \(1 - n^{-\lambda }\) for some constant \(\lambda \).

  6. 6.

    We note that depending on the relations of the values m, r, and n to each other, there may be situations in which our algorithms may be improved by polylogarithmic factors by using the k-nearest neighbour query data structure.

  7. 7.

    Stated briefly, the order-k Voronoi diagram is a generalization of the Voronoi diagram such that each face is the locus of points whose k nearest neighbours are the k points of S associated with (i.e., that define) the face.

  8. 8.

    We could identify convex subpolygons before we get to the triangles, but it would not improve the asymptotic runtime.

  9. 9.

    We say an event happens with high probability if the probability is at least \(1 - n^{-\lambda }\) for some constant \(\lambda \).

  10. 10.

    At the moment, removing points from \(\mathbb {S}\) is a practical consideration; no lower bound is clear on the number of elements from \(\mathbb {S}\) that can be discarded in each iteration, though if one were to find a constant fraction lower bound, one could shave a logarithmic factor off the runtime.

  11. 11.

    Sometimes, as in [31], this is called a cusp. In [31] the funnel is defined as beginning at the cusp and ending at the diagonal.

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Acknowledgements

The authors thank Pat Morin, Jean-Lou de Carufel, Michiel Smid, and Sasanka Roy for helpful discussions as well as anonymous reviewers.

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Authors and Affiliations

  1. School of Computer Science, Carleton University, Ottawa, Canada

    Prosenjit Bose

  2. Carleton University, Ottawa, Canada

    Anthony D’Angelo

  3. University of Manitoba, Winnipeg, Canada

    Stephane Durocher

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  1. Prosenjit Bose
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  2. Anthony D’Angelo
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  3. Stephane Durocher
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Correspondence to Anthony D’Angelo .

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Editors and Affiliations

  1. Carleton University, Ottawa, ON, Canada

    Pat Morin

  2. University of California, Santa Barbara, CA, USA

    Subhash Suri

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Bose, P., D’Angelo, A., Durocher, S. (2023). Approximating the Smallest k-Enclosing Geodesic Disc in a Simple Polygon. In: Morin, P., Suri, S. (eds) Algorithms and Data Structures. WADS 2023. Lecture Notes in Computer Science, vol 14079. Springer, Cham. https://doi.org/10.1007/978-3-031-38906-1_13

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  • DOI: https://doi.org/10.1007/978-3-031-38906-1_13

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