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Optimal strategies for the one-round discrete Voronoi game on a line

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Abstract

The one-round discrete Voronoi game, with respect to a n-point user set \(\mathcal {U}\), consists of two players Player 1 (P1) and Player 2 (P2). At first, P1 chooses a set \(\mathcal{F}_{1}\) of m facilities following which P2 chooses another set \(\mathcal{F}_{2}\) of m facilities, disjoint from \(\mathcal{F}_{1}\), where m(=O(1)) is a positive constant. The payoff of P2 is defined as the cardinality of the set of points in \(\mathcal{U}\) which are closer to a facility in \(\mathcal{F}_{2}\) than to every facility in \(\mathcal{F}_{1}\), and the payoff of P1 is the difference between the number of users in \(\mathcal{U}\) and the payoff of P2. The objective of both the players in the game is to maximize their respective payoffs. In this paper, we address the case where the points in \(\mathcal{U}\) are located along a line. We show that if the sorted order of the points in \(\mathcal{U}\) along the line is known, then the optimal strategy of P2, given any placement of facilities by P1, can be computed in O(n) time. We then prove that for m≥2 the optimal strategy of P1 in the one-round discrete Voronoi game, with the users on a line, can be computed in \(O(n^{m-\lambda_{m}})\) time, where 0<λ m <1, is a constant depending only on m.

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Acknowledgements

The authors wish to thank anonymous referees for valuable comments which improved the presentation of the paper.

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Correspondence to Bhaswar B. Bhattacharya.

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A preliminary version of this paper has appeared in the Proc. 17th Annual International Computing and Combinatorics Conference (COCOON), 2011, 213–224.

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Banik, A., Bhattacharya, B.B. & Das, S. Optimal strategies for the one-round discrete Voronoi game on a line. J Comb Optim 26, 655–669 (2013). https://doi.org/10.1007/s10878-011-9447-6

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