Abstract
In 2011, Biro et al. [4] initiated the concept of beacon attraction trajectory motivated by routing messages in sensor network systems. Let P be a polygonal region such that there is a point particle at each point in P. When we activate a beacon at a point \(b\in P\), each particle in P greedily moves toward b. For a point \(p\in P\), if the particle at p reaches b, we say b attracts p. We call a point \(b\in P\) a beacon kernel point of P if a beacon at b attracts all points in P. The beacon kernel of P is defined as the set of all beacon kernel points of P. In 2013 [3] Biro presented a naive quadratic time algorithm to compute the beacon kernel of polygonal domains and showed that this bound is tight. But, obtaining a sub-quadratic time algorithm for computing the beacon kernel of simple polygons remained open. In this paper, we answer to this open problem by presenting an \(O(n^{1.5}\log ^2 n)\) time algorithm for computing the beacon kernel of simple polygons.
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Notes
- 1.
In the original paper, this time complexity is \(O(m^{0.5}2^{O(\log ^* m)})\). For the sake of simplicity, here we use \(O(m^{0.5}\log m)\).
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Bhattacharya, B., Mozafari, A., Shermer, T.C. (2024). A Sub-quadratic Time Algorithm for Computing the Beacon Kernel of Simple Polygons. In: Wu, W., Tong, G. (eds) Computing and Combinatorics. COCOON 2023. Lecture Notes in Computer Science, vol 14423. Springer, Cham. https://doi.org/10.1007/978-3-031-49193-1_6
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