19 research outputs found
Searching edges in the overlap of two plane graphs
Consider a pair of plane straight-line graphs, whose edges are colored red
and blue, respectively, and let n be the total complexity of both graphs. We
present a O(n log n)-time O(n)-space technique to preprocess such pair of
graphs, that enables efficient searches among the red-blue intersections along
edges of one of the graphs. Our technique has a number of applications to
geometric problems. This includes: (1) a solution to the batched red-blue
search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an
algorithm to compute the maximum vertical distance between a pair of 3D
polyhedral terrains one of which is convex in O(n log n) time, where n is the
total complexity of both terrains; (3) an algorithm to construct the Hausdorff
Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n)
time and O(n+m) space, where n is the total number of points in all clusters
and m is the number of crossings between all clusters; (4) an algorithm to
construct the farthest-color Voronoi diagram of the corners of n axis-aligned
rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle
problem for n parallel line segments in the plane in optimal O(n log n) time.
All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure
Nonrepetitive Colouring via Entropy Compression
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path
whose first half receives the same sequence of colours as the second half. A
graph is nonrepetitively -choosable if given lists of at least colours
at each vertex, there is a nonrepetitive colouring such that each vertex is
coloured from its own list. It is known that every graph with maximum degree
is -choosable, for some constant . We prove this result
with (ignoring lower order terms). We then prove that every subdivision
of a graph with sufficiently many division vertices per edge is nonrepetitively
5-choosable. The proofs of both these results are based on the Moser-Tardos
entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek
for the nonrepetitive choosability of paths. Finally, we prove that every graph
with pathwidth is nonrepetitively -colourable.Comment: v4: Minor changes made following helpful comments by the referee
Ordered coloring of grids and related graphs
AbstractWe investigate a coloring problem, called ordered coloring, in grids and some other families of grid-like graphs. Ordered coloring (also known as vertex ranking) has applications, among other areas, in efficient solving of sparse linear systems of equations and scheduling parallel assembly of products. Our main technical results improve upper and lower bounds for the ordered chromatic number of grids and related graphs
Stathis zachos at 70!
This year we are celebrating the 70th birthday of Stathis! We take this chance to recall some of his remarkable contributions to Computer Science. © Springer International Publishing AG 2017
Voronoi diagram of orthogonal polyhedra in two and three dimensions
International audienceVoronoi diagrams are a fundamental geometric data structure for obtaining proximity relations. We consider collections of axis-aligned orthogonal polyhedra in two and three-dimensional space under the max-norm, which is a particularly useful scenario in certain application domains. We construct the exact Voronoi diagram inside an orthogonal polyhedron with holes defined by such polyhedra. Our approach avoids creating full-dimensional elements on the Voronoi diagram and yields a skeletal representation of the input object. We introduce a complete algorithm in 2D and 3D that follows the subdivision paradigm relying on a bounding-volume hierarchy; this is an original approach to the problem. The complexity is adaptive and comparable to that of previous methods. Under a mild assumption it is O(n/∆) in 2D or O(n.α^2 /∆^2) in 3D, where n is the number of sites, namely edges or facets resp., ∆ is the maximum cell size for the subdivision to stop, and α bounds vertex cardinality per facet. We also provide a numerically stable, open-source implementation in Julia, illustrating the practical nature of our algorithm