7 research outputs found
Computing Delaunay triangulation with imprecise input data
Published versio
Complexity and Algorithms for the Discrete Fr\'echet Distance Upper Bound with Imprecise Input
We study the problem of computing the upper bound of the discrete Fr\'{e}chet
distance for imprecise input, and prove that the problem is NP-hard. This
solves an open problem posed in 2010 by Ahn \emph{et al}. If shortcuts are
allowed, we show that the upper bound of the discrete Fr\'{e}chet distance with
shortcuts for imprecise input can be computed in polynomial time and we present
several efficient algorithms.Comment: 15 pages, 8 figure
Delaunay triangulation of imprecise points in linear time after preprocessing
An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of disjoint unit disks in the plane in time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay
Computing Delaunay triangulation with imprecise input data
The key step in the construction of the Delaunay triangulation of a finite set of planar points is to establish correctly whether a given point of this set is inside oroutside the circle determined by any other three points. We address the problem of formulating the in-circle testwhen the coordinates of the planar points are given only up to a given precision, which is usually the case in practice. By modelling imprecise points as rectangles, and using the idea of partial disc, we construct a reliable in-circle test that provides the best possible Delaunay triangulation with the imprecise input data given by rectangles
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by ErdËťos
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version