Abstract
In the planar one-round discrete Voronoi game, two players \(\mathcal {P}\) and \(\mathcal {Q}\) compete over a set V of n voters represented by points in \(\mathbb {R}^2\). First, \(\mathcal {P}\) places a set P of k points, then \(\mathcal {Q}\) places a set Q of \(\ell \) points, and then each voter \(v\in V\) is won by the player who has placed a point closest to v. It is well known that if \(k=\ell =1\), then \(\mathcal {P}\) can always win n/3 voters and that this is worst-case optimal. We study the setting where \(k>1\) and \(\ell =1\). We present lower bounds on the number of voters that \(\mathcal {P}\) can always win, which improve the existing bounds for all \(k\geqslant 4\). As a by-product, we obtain improved bounds on small \(\varepsilon \)-nets for convex ranges for even numbers of points in general position.
MdB is supported by the Dutch Research Council (NWO) through Gravitation grant NETWORKS-024.002.003.
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Notes
- 1.
Our definition of \(\varepsilon \)-net is slightly weaker than usual, since a range missing the \(\varepsilon \)-net may contain up to \(\left\lceil \varepsilon n \right\rceil \) points, instead of \(\lfloor \varepsilon n\rfloor \) points.
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de Berg, M., van Wordragen, G. (2023). Improved Bounds for Discrete Voronoi Games. 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_20
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