Equal Entries in Totally Positive Matrices

We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) $n\textrm{-by-}n$ matrix is $\Theta(n^{4/3})$ (resp. $\Theta(n^{3/2}$)). Relationships with point-line incidences in the plane, Bruhat order of permutations, and $TP$ completability are also presented. We also examine the number and positionings of equal $2\textrm{-by-}2$ minors in a $2\textrm{-by-}n$ $TP$ matrix, and give a relationship between the location of equal $2\textrm{-by-}2$ minors and outerplanar graphs.Comment: 15 page

Many-to-One Boundary Labeling with Backbones

In this paper we study \emph{many-to-one boundary labeling with backbone leaders}. In this new many-to-one model, a horizontal backbone reaches out of each label into the feature-enclosing rectangle. Feature points that need to be connected to this label are linked via vertical line segments to the backbone. We present dynamic programming algorithms for label number and total leader length minimization of crossing-free backbone labelings. When crossings are allowed, we aim to obtain solutions with the minimum number of crossings. This can be achieved efficiently in the case of fixed label order, however, in the case of flexible label order we show that minimizing the number of leader crossings is NP-hard.Comment: 23 pages, 10 figures, this is the full version of a paper that is about to appear in GD'1

A LiDAR Point Cloud Generator: from a Virtual World to Autonomous Driving

3D LiDAR scanners are playing an increasingly important role in autonomous driving as they can generate depth information of the environment. However, creating large 3D LiDAR point cloud datasets with point-level labels requires a significant amount of manual annotation. This jeopardizes the efficient development of supervised deep learning algorithms which are often data-hungry. We present a framework to rapidly create point clouds with accurate point-level labels from a computer game. The framework supports data collection from both auto-driving scenes and user-configured scenes. Point clouds from auto-driving scenes can be used as training data for deep learning algorithms, while point clouds from user-configured scenes can be used to systematically test the vulnerability of a neural network, and use the falsifying examples to make the neural network more robust through retraining. In addition, the scene images can be captured simultaneously in order for sensor fusion tasks, with a method proposed to do automatic calibration between the point clouds and captured scene images. We show a significant improvement in accuracy (+9%) in point cloud segmentation by augmenting the training dataset with the generated synthesized data. Our experiments also show by testing and retraining the network using point clouds from user-configured scenes, the weakness/blind spots of the neural network can be fixed

Langford sequences and a product of digraphs

Skolem and Langford sequences and their many generalizations have applications in numerous areas. The $\otimes_h$-product is a generalization of the direct product of digraphs. In this paper we use the $\otimes_h$-product and super edge-magic digraphs to construct an exponential number of Langford sequences with certain order and defect. We also apply this procedure to extended Skolem sequences.Comment: 10 pages, 6 figures, to appear in European Journal of Combinatoric

Multi-Tier Annotations in the Verbmobil Corpus

In very large and diverse scientific projects where as different groups as linguists and engineers with different intentions work on the same signal data or its orthographic transcript and annotate new valuable information, it will not be easy to build a homogeneous corpus. We will describe how this can be achieved, considering the fact that some of these annotations have not been updated properly, or are based on erroneous or deliberately changed versions of the basis transcription. We used an algorithm similar to dynamic programming to detect differences between the transcription on which the annotation depends and the reference transcription for the whole corpus. These differences are automatically mapped on a set of repair operations for the transcriptions such as splitting compound words and merging neighbouring words. On the basis of these operations the correction process in the annotation is carried out. It always depends on the type of the annotation as well as on the position and the nature of the difference, whether a correction can be carried out automatically or has to be fixed manually. Finally we present a investigation in which we exploit the multi-tier annotations of the Verbmobil corpus to find out how breathing is correlated with prosodic-syntactic boundaries and dialog acts. 1

Area-Universal Rectangular Layouts

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. Rectangular layouts appear in various applications: as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it might hence be desirable if one rectangular layout can represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. More generally, given any rectangular layout L and any assignment of areas to its regions, we show that there can be at most one layout (up to horizontal and vertical scaling) which is combinatorially equivalent to L and achieves a given area assignment. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.Comment: 19 pages, 16 figure