1,303 research outputs found
The one-round Voronoi game replayed
We consider the one-round Voronoi game, where player one (``White'', called
``Wilma'') places a set of n points in a rectangular area of aspect ratio r
<=1, followed by the second player (``Black'', called ``Barney''), who places
the same number of points. Each player wins the fraction of the board closest
to one of his points, and the goal is to win more than half of the total area.
This problem has been studied by Cheong et al., who showed that for large
enough and r=1, Barney has a strategy that guarantees a fraction of 1/2+a,
for some small fixed a.
We resolve a number of open problems raised by that paper. In particular, we
give a precise characterization of the outcome of the game for optimal play: We
show that Barney has a winning strategy for n>2 and r>sqrt{2}/n, and for n=2
and r>sqrt{3}/2. Wilma wins in all remaining cases, i.e., for n>=3 and
r<=sqrt{2}/n, for n=2 and r<=sqrt{3}/2, and for n=1. We also discuss complexity
aspects of the game on more general boards, by proving that for a polygon with
holes, it is NP-hard to maximize the area Barney can win against a given set of
points by Wilma.Comment: 14 pages, 6 figures, Latex; revised for journal version, to appear in
Computational Geometry: Theory and Applications. Extended abstract version
appeared in Workshop on Algorithms and Data Structures, Springer Lecture
Notes in Computer Science, vol.2748, 2003, pp. 150-16
The still-Life density problem and its generalizations
A "still Life" is a subset S of the square lattice Z^2 fixed under the
transition rule of Conway's Game of Life, i.e. a subset satisfying the
following three conditions:
1. No element of Z^2-S has exactly three neighbors in S;
2. Every element of S has at least two neighbors in S;
3. Every element of S has at most three neighbors in S.
Here a ``neighbor'' of any x \in Z^2 is one of the eight lattice points
closest to x other than x itself. The "still-Life conjecture" is the assertion
that a still Life cannot have density greater than 1/2 (a bound easily
attained, for instance by {(x,y): x is even}). We prove this conjecture,
showing that in fact condition 3 alone ensures that S has density at most 1/2.
We then consider variations of the problem such as changing the number of
allowed neighbors or the definition of neighborhoods; using a variety of
methods we find some partial results and many new open problems and
conjectures.Comment: 29 pages, including many figures drawn as LaTeX "pictures
On the computation of zone and double zone diagrams
Classical objects in computational geometry are defined by explicit
relations. Several years ago the pioneering works of T. Asano, J. Matousek and
T. Tokuyama introduced "implicit computational geometry", in which the
geometric objects are defined by implicit relations involving sets. An
important member in this family is called "a zone diagram". The implicit nature
of zone diagrams implies, as already observed in the original works, that their
computation is a challenging task. In a continuous setting this task has been
addressed (briefly) only by these authors in the Euclidean plane with point
sites. We discuss the possibility to compute zone diagrams in a wide class of
spaces and also shed new light on their computation in the original setting.
The class of spaces, which is introduced here, includes, in particular,
Euclidean spheres and finite dimensional strictly convex normed spaces. Sites
of a general form are allowed and it is shown that a generalization of the
iterative method suggested by Asano, Matousek and Tokuyama converges to a
double zone diagram, another implicit geometric object whose existence is known
in general. Occasionally a zone diagram can be obtained from this procedure.
The actual (approximate) computation of the iterations is based on a simple
algorithm which enables the approximate computation of Voronoi diagrams in a
general setting. Our analysis also yields a few byproducts of independent
interest, such as certain topological properties of Voronoi cells (e.g., that
in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI;
Ref [51] points to a freely available computer application which implements
the algorithms; to appear in Discrete & Computational Geometry (available
online
The Discrete Voronoi game in ℝ\u3csup\u3e2\u3c/sup\u3e
In this paper we study the last round of the discrete Voronoi game in ℝ2, a problem which is also of independent interest in competitive facility location. The game consists of two players P1 and P2, and a finite set U of users in the plane. The players have already placed two disjoint sets of facilities F and S, respectively, in the plane. The game begins with P1 placing a new facility followed by P2 placing another facility, and the objective of both the players is to maximize their own total payoffs. In this paper we propose polynomial time algorithms for determining the optimal strategies of both the players for arbitrarily located existing facilities F and S. We show that in the L1 and the L∞ metrics, the optimal strategy of P2, given any placement of P1, can be found in O(n log n) time, and the optimal strategy of P1 can be found in O(n5 log n) time. In the L2 metric, the optimal strategies of P2 and P1 can be obtained in O(n2) and O(n2) and O(n8) times, respectively
The projector algorithm: a simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs
The Voronoi diagram is a certain geometric data structure which has numerous
applications in various scientific and technological fields. The theory of
algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich
and useful, with several different and important algorithms. However, this
theory has been quite steady during the last few decades in the sense that no
essentially new algorithms have entered the game. In addition, most of the
known algorithms are serial in nature and hence cast inherent difficulties on
the possibility to compute the diagram in parallel. In this paper we present
the projector algorithm: a new and simple algorithm which enables the
(combinatorial) computation of 2D Voronoi diagrams. The algorithm is
significantly different from previous ones and some of the involved concepts in
it are in the spirit of linear programming and optics. Parallel implementation
is naturally supported since each Voronoi cell can be computed independently of
the other cells. A new combinatorial structure for representing the cells (and
any convex polytope) is described along the way and the computation of the
induced Delaunay graph is obtained almost automatically.Comment: This is a major revision; re-organization and better presentation of
some parts; correction of several inaccuracies; improvement of some proofs
and figures; added references; modification of the title; the paper is long
but more than half of it is composed of proofs and references: it is
sufficient to look at pages 5, 7--11 in order to understand the algorith
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