Abstract
Let s be a point in a polygonal domain \({\mathcal {P}}\) of \(h-1\) holes and n vertices. We consider a quickest visibility query problem. Given a query point q in \({\mathcal {P}}\), the goal is to find a shortest path in \({\mathcal {P}}\) to move from s to seeq as quickly as possible. Previously, Arkin et al. (SoCG 2015) built a data structure of size \(O(n^22^{\alpha (n)}\log n)\) that can answer each query in \(O(K\log ^2 n)\) time, where \(\alpha (n)\) is the inverse Ackermann function and K is the size of the visibility polygon of q in \({\mathcal {P}}\) (and K can be \(\varTheta (n)\) in the worst case). In this paper, we present a new data structure of size \(O(n\log h + h^2)\) that can answer each query in \(O(h\log h\log n)\) time. Our result improves the previous work when h is relatively small. In particular, if h is a constant, then our result even matches the best result for the simple polygon case (i.e., \(h=1\)), which is optimal. As a by-product, we also have a new algorithm for a shortest-path-to-segment query problem. Given a query line segment \(\tau \) in \({\mathcal {P}}\), the query seeks a shortest path from s to all points of \(\tau \). Previously, Arkin et al. gave a data structure of size \(O(n^22^{\alpha (n)}\log n)\) that can answer each query in \(O(\log ^2 n)\) time, and another data structure of size \(O(n^3\log n)\) with \(O(\log n)\) query time. We present a data structure of size O(n) with query time \(O\big (h\log \frac{n}{h}\big )\), which also favors small values of h and is optimal when \(h=O(1)\).
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Notes
In fact, since each bisector edge of \(\mathrm{SPM}(s)\) is a convex curve, C is naturally a splinegon [24].
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A preliminary version of this paper appeared in the Proceedings of the 33rd International Symposium on Computational Geometry (SoCG 2017). This research was supported in part by NSF under Grant CCF-1317143.
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Wang, H. Quickest Visibility Queries in Polygonal Domains. Discrete Comput Geom 62, 374–432 (2019). https://doi.org/10.1007/s00454-019-00108-8
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DOI: https://doi.org/10.1007/s00454-019-00108-8