External polygon containment problems

AbstractGiven a convex polygonal object P with k vertices and an environment consisting of polygonal obstacles with a total of n corners, we seek a placement for the largest copy of P that does not intersect any of the obstacles, allowing translation, rotation and scaling. We employ the parametric search technique of Megiddo (1983), and the fixed size polygon placement algorithms developed by Leven and Sharir (1987), to obtain an algorithm that runs in time O(k2nλ6(kn)log3(kn)loglog(kn)). We also present several other efficient algorithms for restricted variants of the extremal polygon containment problem, using the same ideas. These variants include: placement of the largest homothetic copies of one or two convex polygons in another convex polygon and placement of the largest similar copy of a triangle in a convex polygon

Happy Ending: An Empty Hexagon in Every Set of 30 Points

Satisfiability solving has been used to tackle a range of long-standing open math problems in recent years. We add another success by solving a geometry problem that originated a century ago. In the 1930s, Esther Klein's exploration of unavoidable shapes in planar point sets in general position showed that every set of five points includes four points in convex position. For a long time, it was open if an empty hexagon, i.e., six points in convex position without a point inside, can be avoided. In 2006, Gerken and Nicol\'as independently proved that the answer is no. We establish the exact bound: Every 30-point set in the plane in general position contains an empty hexagon. Our key contributions include an effective, compact encoding and a search-space partitioning strategy enabling linear-time speedups even when using thousands of cores

Morphing Contact Representations of Graphs

We consider the problem of morphing between contact representations of a plane graph. In a contact representation of a plane graph, vertices are realized by internally disjoint elements from a family of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in the graph. In a morph between two contact representations we insist that at each time step (continuously throughout the morph) we have a contact representation of the same type. We focus on the case when the geometric objects are triangles that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs. We study piecewise linear morphs, where each step is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. We provide a polynomial-time algorithm that decides whether there is a piecewise linear morph between two RT-representations of a plane triangulation, and, if so, computes a morph with a quadratic number of linear morphs. As a direct consequence, we obtain that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the "top-most" triangle in both representations corresponds to the same vertex. This shows that the realization space of such RT-representations of any 4-connected plane triangulation forms a connected set

Graph Theory

This is the report on an Oberwolfach conference on graph theory, held 16-22 January 2005. There were three main components to the event: 5-minute presentations, lectures, and workshops. All participants were asked to give a 5-minute presentation of their interests on the first day, and subsequent days were divided into lectures and workshops. The latter ranged over many different topics, but the main three topics were: infinite graphs, topological methods and their use to prove theorems in graph theory, and Rota’s conjecture for matroids

Oriented Spanners

Given a point set $P$ in the Euclidean plane and a parameter $t$, we define an \emph{oriented $t$-spanner} as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in $G$ through those points is at most a factor $t$ longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a $1$-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in $O(n^8)$ time for $n$ points, and a greedy algorithm that computes a $5$-spanner in $O(n\log n)$ time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented $O(1)$-spanner.Comment: conference version: ESA '2

Simultaneously Embedding Planar Graphs at Fixed Vertex Locations

We discuss the problem of embedding planar graphs onto the plane with pre-specified vertex locations. In particular, we introduce a method for constructing such an embedding for both the case where the mapping from the vertices onto the vertex locations is fixed and the case where this mapping can be chosen. Moreover, the technique we present is sufficiently abstract to generalize to a method for constructing simultaneous planar embeddings with fixed vertex locations. In all cases, we are concerned with minimizing the number of bends per edge in the embeddings we produce. In the case where the mapping is fixed, our technique guarantees embeddings with at most 8n - 7 bends per edge in the worst case and, on average, at most 16/3n - 1 bends per edge. This result improves previously known techniques by a significant constant factor. When the mapping is not pre-specified, our technique guarantees embeddings with at most O(n^(1 - 2^(1-k))) bends per edge in the worst case and, on average, at most O(n^(1 - 1/k)) bends per edge, where k is the number of graphs in the simultaneous embedding. This improves upon the previously known O(n) bound on the number of bends per edge for k at least 2. Moreover, we give an average-case lower bound on the number of bends that has similar asymptotic behaviour to our upper bound

Development of a text reading system on video images

Since the early days of computer science researchers sought to devise a machine which could automatically read text to help people with visual impairments. The problem of extracting and recognising text on document images has been largely resolved, but reading text from images of natural scenes remains a challenge. Scene text can present uneven lighting, complex backgrounds or perspective and lens distortion; it usually appears as short sentences or isolated words and shows a very diverse set of typefaces. However, video sequences of natural scenes provide a temporal redundancy that can be exploited to compensate for some of these deficiencies. Here we present a complete end-to-end, real-time scene text reading system on video images based on perspective aware text tracking. The main contribution of this work is a system that automatically detects, recognises and tracks text in videos of natural scenes in real-time. The focus of our method is on large text found in outdoor environments, such as shop signs, street names and billboards. We introduce novel efficient techniques for text detection, text aggregation and text perspective estimation. Furthermore, we propose using a set of Unscented Kalman Filters (UKF) to maintain each text region¿s identity and to continuously track the homography transformation of the text into a fronto-parallel view, thereby being resilient to erratic camera motion and wide baseline changes in orientation. The orientation of each text line is estimated using a method that relies on the geometry of the characters themselves to estimate a rectifying homography. This is done irrespective of the view of the text over a large range of orientations. We also demonstrate a wearable head-mounted device for text reading that encases a camera for image acquisition and a pair of headphones for synthesized speech output. Our system is designed for continuous and unsupervised operation over long periods of time. It is completely automatic and features quick failure recovery and interactive text reading. It is also highly parallelised in order to maximize the usage of available processing power and to achieve real-time operation. We show comparative results that improve the current state-of-the-art when correcting perspective deformation of scene text. The end-to-end system performance is demonstrated on sequences recorded in outdoor scenarios. Finally, we also release a dataset of text tracking videos along with the annotated ground-truth of text regions

Network Algorithms for Complex Systems with Applications to Non-linear Oscillators and Genome Assembly

Network and complex system models are useful for studying a wide range of phenomena, from disease spread to traffic flow. Because of the broad applicability of the framework it is important to develop effective simulations and algorithms for complex networks. This dissertation presents contributions to two applied problems in this area First, we study an electro-optical, nonlinear, and time-delayed feedback loop commonly used in applications that require a broad range of chaotic behavior. For this system we detail a discrete-time simulation model, exploring the model's synchronization behavior under specific coupling conditions. Expanding upon already published results that investigated changes in feedback strength, we explore how both time-delay and nonlinear sensitivity impact synchronization. We also relax the requirement of strictly identical systems components to study how synchronization regions are affected when coupled systems have non-identical components (parameters). Last, we allow wider variance in coupling strengths, including unique strengths to each system, to identify a rich synchronization region not previously seen. In our second application, we take a complex networks approach to improving genome assembly algorithms. One key part of sequencing a genome is solving the orientation problem. The orientation problem is finding the relative orientations for each data fragment generated during sequencing. By viewing the genomic data as a network we can apply standard analysis techniques for community finding and utilize the significantly modular structure of the data. This structure informs development and application of two new heuristics based on (A) genetic algorithms and (B) hierarchical clustering for solving the orientation problem. Genetic algorithms allow us to preserve some internal structure while quickly exploring a large solution space. We present studies using a multi-scale genetic algorithm to solve the orientation problem. We show that this approach can be used in conjunction with currently used methods to identify a better solution to the orientation problem. Our hierarchical algorithm further utilizes the modular structure of the data. By progressively solving and merging sub-problems together we pick optimal `local' solutions while allowing more global corrections to occur later. Our results show significant improvements over current techniques for both generated data and real assembly data

Solving Correspondences for Non-Rigid Deformations

Projecte final de carrera realitzat en col.laboració amb l'IR