Optimal Strategies for the One-Round Discrete Voronoi Game on a Line

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.