On finding widest empty curved corridors

Open archive-ElsevierAn α-siphon of width w is the locus of points in the plane that are at the same distance w from a 1-corner polygonal chain C such that α is the interior angle of C. Given a set P of n points in the plane and a fixed angle α, we want to compute the widest empty α-siphon that splits P into two non-empty sets.We present an efficient O(n log3 n)-time algorithm for computing the widest oriented α-siphon through P such that the orientation of a half-line of C is known.We also propose an O(n3 log2 n)-time algorithm for the widest arbitrarily-oriented version and an (nlog n)-time algorithm for the widest arbitrarily-oriented α-siphon anchored at a given point

Small permutation classes

We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number $\kappa$, approximately 2.20557, for which there are only countably many permutation classes of growth rate (Stanley-Wilf limit) less than $\kappa$ but uncountably many permutation classes of growth rate $\kappa$, answering a question of Klazar. We go on to completely characterize the possible sub-$\kappa$ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced herein), partial well-order, and atomicity (also known as the joint embedding property)

Reconstructing Generalized Staircase Polygons with Uniform Step Length

Visibility graph reconstruction, which asks us to construct a polygon that has a given visibility graph, is a fundamental problem with unknown complexity (although visibility graph recognition is known to be in PSPACE). We show that two classes of uniform step length polygons can be reconstructed efficiently by finding and removing rectangles formed between consecutive convex boundary vertices called tabs. In particular, we give an $O(n^2m)$-time reconstruction algorithm for orthogonally convex polygons, where $n$ and $m$ are the number of vertices and edges in the visibility graph, respectively. We further show that reconstructing a monotone chain of staircases (a histogram) is fixed-parameter tractable, when parameterized on the number of tabs, and polynomially solvable in time $O(n^2m)$ under reasonable alignment restrictions.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Extensions of the Maximum Bichromatic Separating Rectangle Problem

In this paper, we study two extensions of the maximum bichromatic separating rectangle (MBSR) problem introduced in \cite{Armaselu-CCCG, Armaselu-arXiv}. One of the extensions, introduced in \cite{Armaselu-FWCG}, is called \textit{MBSR with outliers} or MBSR-O, and is a more general version of the MBSR problem in which the optimal rectangle is allowed to contain up to $k$ outliers, where $k$ is given as part of the input. For MBSR-O, we improve the previous known running time bounds of $O(k^7 m \log m + n)$ to $O(k^3 m + m \log m + n)$. The other extension is called \textit{MBSR among circles} or MBSR-C and asks for the largest axis-aligned rectangle separating red points from blue unit circles. For MBSR-C, we provide an algorithm that runs in $O(m^2 + n)$ time.Comment: 14 pages, 14 figures, full version of CCCG pape

Optimally fast incremental Manhattan plane embedding and planar tight span construction

We describe a data structure, a rectangular complex, that can be used to represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane. We show how to use this data structure to construct the tight span of a metric space given as an n x n distance matrix, when the tight span is homeomorphic to a subset of the plane, in time O(n^2), and to add a single point to a planar tight span in time O(n). As an application of this construction, we show how to test whether a given finite metric space embeds isometrically into the Manhattan plane in time O(n^2), and add a single point to the space and re-test whether it has such an embedding in time O(n).Comment: 39 pages, 15 figure