One-round discrete Voronoi game, with the users on a line, can be computed in $O(n^)$ time, where $0< \lambda_m < 1$, is a constant depending only on $m$. We then prove that for $m \geq 2$ the optimal strategy of P1 in the 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. In this paper, we address the case where the points in $\mathcal U$ are located along a line. The objective of both the players in the game is to maximize their respective payoffs. 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. 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 one-round discrete Voronoi game, with respect to a $n$-point user set $\mathcal U$,Ĭonsists of two players Player 1 (P1) and Player 2 (P2).
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