Abstract
This paper discusses the problem of covering and hitting a set of line segments \(\mathcal L\) in \(\mathbb {R}^2\) by a pair of axis-parallel squares such that the side length of the larger of the two squares is minimized. We also discuss the restricted version of covering, where each line segment in \(\mathcal L\) is to be covered completely by at least one square. The proposed algorithm for the covering problem reports the optimum result by executing only two passes of reading the input data sequentially. The algorithm proposed for the hitting and restricted covering problems produces optimum result in \(O(n\log n)\) time. All the proposed algorithms are in-place, and they use only O(1) extra space. The solution of these problems also give a \(\sqrt{2}\) approximation for covering and hitting those line segments \(\mathcal L\) by two congruent disks of minimum radius with same computational complexity.
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Notes
- 1.
\(L_\infty \) distance between two points a and b is given by \(\max (|x(a)-x(b)|,|y(a)-y(b)|)\).
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Sadhu, S., Roy, S., Nandy, S.C., Roy, S. (2017). Optimal Covering and Hitting of Line Segments by Two Axis-Parallel Squares. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_38
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DOI: https://doi.org/10.1007/978-3-319-62389-4_38
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