2,794 research outputs found
A Note on Flips in Diagonal Rectangulations
Rectangulations are partitions of a square into axis-aligned rectangles. A
number of results provide bijections between combinatorial equivalence classes
of rectangulations and families of pattern-avoiding permutations. Other results
deal with local changes involving a single edge of a rectangulation, referred
to as flips, edge rotations, or edge pivoting. Such operations induce a graph
on equivalence classes of rectangulations, related to so-called flip graphs on
triangulations and other families of geometric partitions. In this note, we
consider a family of flip operations on the equivalence classes of diagonal
rectangulations, and their interpretation as transpositions in the associated
Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This
complements results from Law and Reading (JCTA, 2012) and provides a complete
characterization of flip operations on diagonal rectangulations, in both
geometric and combinatorial terms
Computing Optimal Shortcuts for Networks
We study augmenting a plane Euclidean network with a segment, called shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Questions of this type have received considerable attention recently, mostly for discrete variants of the problem. We study a fully continuous setting, where all points on the network and the inserted segment must be taken into account. We present the first results on the computation of optimal shortcuts for general networks in this model, together with several results for networks that are paths, restricted to two types of shortcuts: shortcuts with a fixed orientation and simple shortcuts
Continuous mean distance of a weighted graph
We study the concept of the continuous mean distance of a weighted graph. For
connected unweighted graphs, the mean distance can be defined as the arithmetic
mean of the distances between all pairs of vertices. This parameter provides a
natural measure of the compactness of the graph, and has been intensively
studied, together with several variants, including its version for weighted
graphs. The continuous analog of the (discrete) mean distance is the mean of
the distances between all pairs of points on the edges of the graph. Despite
being a very natural generalization, to the best of our knowledge this concept
has been barely studied, since the jump from discrete to continuous implies
having to deal with an infinite number of distances, something that increases
the difficulty of the parameter. In this paper we show that the continuous mean
distance of a weighted graph can be computed in time quadratic in the number of
edges, by two different methods that apply fundamental concepts in discrete
algorithms and computational geometry. We also present structural results that
allow a faster computation of this continuous parameter for several classes of
weighted graphs. Finally, we study the relation between the (discrete) mean
distance and its continuous counterpart, mainly focusing on the relevant
question of the convergence when iteratively subdividing the edges of the
weighted graph
Shortest Paths in Portalgons
Any surface that is intrinsically polyhedral can be represented by a collection of simple polygons (fragments), glued along pairs of equally long oriented edges, where each fragment is endowed with the geodesic metric arising from its Euclidean metric. We refer to such a representation as a portalgon, and we call two portalgons equivalent if the surfaces they represent are isometric.
We analyze the complexity of shortest paths. We call a fragment happy if any shortest path on the portalgon visits it at most a constant number of times. A portalgon is happy if all of its fragments are happy. We present an efficient algorithm to compute shortest paths on happy portalgons.
The number of times that a shortest path visits a fragment is unbounded in general. We contrast this by showing that the intrinsic Delaunay triangulation of any polyhedral surface corresponds to a happy portalgon. Since computing the intrinsic Delaunay triangulation may be inefficient, we provide an efficient algorithm to compute happy portalgons for a restricted class of portalgons
On approximating shortest paths in weighted triangular tessellations
We study the quality of weighted shortest paths when a continuous
2-dimensional space is discretized by a weighted triangular tessellation. In
order to evaluate how well the tessellation approximates the 2-dimensional
space, we study three types of shortest paths: a weighted shortest path~, which is a shortest path from to in the
space; a weighted shortest vertex path , which is a
shortest path where the vertices of the path are vertices of the tessellation;
and a weighted shortest grid path~, which is a shortest
path whose edges are edges of the tessellation. The ratios , , provide
estimates on the quality of the approximation.
Given any arbitrary weight assignment to the faces of a triangular
tessellation, we prove upper and lower bounds on the estimates that are
independent of the weight assignment. Our main result is that in the worst case, and this is tight.Comment: 17 pages, 10 figure
Matching points with disks with a common intersection
We consider matchings with diametral disks between two sets of points R and
B. More precisely, for each pair of matched points p in R and q in B, we
consider the disk through p and q with the smallest diameter. We prove that for
any R and B such that |R|=|B|, there exists a perfect matching such that the
diametral disks of the matched point pairs have a common intersection. In fact,
our result is stronger, and shows that a maximum weight perfect matching has
this property
On the number of higher order Delaunay triangulations
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