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 a player i is defined as the cardinality of the set of points in \(\mathcal U\) which are closer to a point in \(\mathcal F_i\) than to every point in \(\mathcal F_j\), for i ≠ j. 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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Banik, A., Bhattacharya, B.B., Das, S. (2011). Optimal Strategies for the One-Round Discrete Voronoi Game on a Line. In: Fu, B., Du, DZ. (eds) Computing and Combinatorics. COCOON 2011. Lecture Notes in Computer Science, vol 6842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22685-4_19
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DOI: https://doi.org/10.1007/978-3-642-22685-4_19
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