The One-Round Voronoi Game

LetQ = {q1, q2,..., qn} be a set ofn points on the plane. The largest empty circle (LEG) problem consists in finding the largest circleC with center in the convex hull ofQ such that no pointqi eQ lies in the interior ofC. Shamos recently outlined anO(n logn) algorithm for solving this problem.(9) In this paper it is shown that this algorithm does not always work correctly. A different approach is proposed here and shown to also result in anO(n logn) algorithm. The new approach has the advantage that it can also solve more general problems. In particular, it is shown that if the center ofC is constrained to lie in an arbitrary convexn-gon, an0(n logn) algorithm can still be obtained. Finally, an0(n logn +k logn) algorithm is given for solving this problem when the center ofC is constrained to lie in an arbitrary simplen-gonP. wherek denotes the number of intersections occurring between edges ofP and edges of the Voronoi diagram ofQ andk ⩽O(n2).

Monte-Carlo Tree Search (MCTS) is a powerful tool in games with a finite branching factor. The paper describes an artificial player playing the Voronoi game, a game with an infinite branching factor. First, it shows how to use MCTS on a discretization of the Voronoi game, and the effects of enhancements such as RAVE and Gaussian processes (GP). Then a set of experimental results shows that MCTS with UCB+RAVE or with UCB+GP are good first solutions for playing the Voronoi game without domain-dependent knowledge. Moreover, the paper shows how the playing level can be greatly improved by using geometrical knowledge about Voronoi diagrams. The balance of diagrams is the key concept. A new set of experimental results shows that a player using MCTS and geometrical knowledge outperforms a player without knowledge.

Thek-Delaunay tree extends the Delaunay tree introduced in [1], and [2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the order≤k Voronoi diagrams, can be done inO(n logn+k 3n) expected time and O(k2n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k ⌈(d+1)/2⌉+1 n ⌊(d+1)/2⌋) and space complexityO(k ⌈(d+1)/2⌉ n ⌊(d+1)/2⌋). The algorithm is simple and experimental results are given.

We study competitive location problems in a continuous setting, in which facilities have to be placed in a rectangular domain R of normalized dimensions of 1 and rho ge 1, and distances are measured according to the Manhattan metric. We show that the family of balanced facility configurations (in which the Voronoi cells of individual facilities are equalized with respect to a number of geometric properties) is considerably richer in this metric than for Euclidean distances. Our main result considers the One-Round Voronoi Game with Manhattan distances, in which first player White and then player Black each place n points in R; each player scores the area for which one of its facilities is closer than the facilities of the opponent. We give a tight characterization: White has a winning strategy if and only if rho ge n; for all other cases, we present a winning strategy for Black.

We study novel variations of Voronoi games and associated random processes that we call Voronoi choice games. These games provide a rich framework for studying questions regarding the power of small numbers of choices in multi-player, competitive scenarios, and they further lead to many interesting, non-trivial random processes that appear worthy of study. As an example of the type of problem we study, suppose a group of n miners (or players) are staking land claims through the following process: each miner has m associated points independently and uniformly distributed on an underlying space (such as the unit circle, the unit square, or the unit torus), so the kth miner will have associated points p_{k1}, p_{k2}, ..., p_{km}. We generally here think of m as being a small constant, such as 2. Each miner chooses one of these points as the base point for their claim. Each miner obtains mining rights for the area of the square that is closest to their chosen base; that is, they obtain the Voronoi cell corresponding to their chosen point in the Voronoi diagram of the n chosen points. Each player's goal is simply to maximize the amount of land under their control. What can we say about the players’ strategy and the equilibria of such games? In our main result, we derive bounds on the expected number of pure Nash equilibria for a variation of the 1-dimensional game on the circle where a player owns the arc starting from their point and moving clockwise to the next point. This result uses interesting properties of random arc lengths on circles, and demonstrates the challenges in analyzing these kinds of problems. We also provide several other related results. In particular, for the 1-dimensional game on the circle, we show that a pure Nash equilibrium always exists when each player owns the part of the circle nearest to their point, but it is NP-hard to determine whether a pure Nash equilibrium exists in the variant when each player owns the arc starting from their point clockwise to the next point. This last result, in part, motivates our examination of the random setting.