A straight line drawing of a graph is an open weak rectangle-of-influence (RI) drawing, if there is no vertex in the relative interior of the axis parallel rectangle induced by the end points of each edge. Despite recent interest of the graph drawing community in rectangle-of-influence drawings, no algorithm is known to test whether a graph has a planar open weak RI-drawing, not even for inner triangulated graphs. In this thesis, we have two major contributions. First we study open weak RI-drawings of plane graphs that must have a non-aligned frame, i.e., the graph obtained from removing the interior of every filled triangle is drawn such that no two vertices have the same coordinate. We introduce a new way to assign labels to angles, i.e., instances of vertices on faces. Using this labeling, we provide necessary and sufficient conditions characterizing those plane graphs that have open weak RI-drawings with non-aligned frame. We also give a polynomial algorithm to construct such a drawing if one exists. Our second major result is a negative result: deciding if a planar graph (i.e., one where we can choose the planar embedding) has an open weak RI-drawing is NP-complete. NP-completeness holds even for open weak RI-drawings with non-aligned frames.

Geometric computations are essential in many real-world problems. One important issue in geometric computations is that the geometric models in these problems can be so large that computations on them have infeasible storage or computation time requirements. Decomposition is a technique commonly used to partition complex models into simpler components. Whereas decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of components. In this work, we have developed an approximate technique, called Approximate Convex Decomposition (ACD), which decomposes a given polygon or polyhedron into "approximately convex" pieces that may provide similar benefits as convex components, while the resulting decomposition is both significantly smaller (typically by orders of magnitude) and can be computed more efficently. Indeed, for many applications, an ACD can represent the important structural features of the model more accurately by providing a mechanism for ignoring less significant features, such as wrinkles and surface texture. Our study of a wide range of applications shows that in addition to providing computational efficiency, ACD also provides natural multi-resolution or hierarchical representations. In this dissertation, we provide some examples of ACD's many potential applications, such as particle simulation, mesh generation, motion planning, and skeleton extraction.

No description available.

A straight line drawing of a graph is an open weak rectangle-of-influence (RI) drawing, if there is no vertex in the relative interior of the axis parallel rectangle induced by the end points of each edge. Despite recent interest of the graph drawing community in rectangle-of-influence drawings, no algorithm is known to test whether a graph has a planar open weak RI-drawing, not even for inner triangulated graphs. In this thesis, we have two major contributions. First we study open weak RI-drawings of plane graphs that must have a non-aligned frame, i.e., the graph obtained from removing the interior of every filled triangle is drawn such that no two vertices have the same coordinate. We introduce a new way to assign labels to angles, i.e., instances of vertices on faces. Using this labeling, we provide necessary and sufficient conditions characterizing those plane graphs that have open weak RI-drawings with non-aligned frame. We also give a polynomial algorithm to construct such a drawing if one exists. Our second major result is a negative result: deciding if a planar graph (i.e., one where we can choose the planar embedding) has an open weak RI-drawing is NP-complete. NP-completeness holds even for open weak RI-drawings with non-aligned frames.

This thesis involves the study of a range of geometric covering and piercing problems, where the unifying thread is approximation using disks. While some of the problems addressed in this work are solved exactly with polynomial time algorithms, many problems are shown to be at least NP-hard. For the latter, approximation algorithms are the best that we can do in polynomial time assuming that P is not equal to NP. One of the best known problems involving unit disks is the Discrete Unit Disk Cover (DUDC) problem, in which the input consists of a set of points P and a set of unit disks in the plane D, and the objective is to compute a subset of the disks of minimum cardinality which covers all of the points. Another perspective on the problem is to consider the centre points (denoted Q) of the disks D as an approximating set of points for P. An optimal solution to DUDC provides a minimal cardinality subset Q*, a subset of Q, so that each point in P is within unit distance of a point in Q*. In order to approximate the general DUDC problem, we also examine several restricted variants. In the Line-Separable Discrete Unit Disk Cover (LSDUDC) problem, P and Q are separated by a line in the plane. We write that l^- is the half-plane defined by l containing P, and l^+ is the half-plane containing Q. LSDUDC may be solved exactly in O(m^2n) time using a greedy algorithm. We augment this result by describing a 2-approximate solution for the Assisted LSDUDC problem, where the union of all disks centred in l^+ covers all points in P, but we consider using disks centred in l^- as well to try to improve the solution. Next, we describe the Within-Strip Discrete Unit Disk Cover (WSDUDC) problem, where P and Q are confined to a strip of the plane of height h. We show that this problem is NP-complete, and we provide a range of approximation algorithms for the problem with trade-offs between the approximation factor and running time. We outline approximation algorithms for the general DUDC problem which make use of the algorithms for LSDUDC and WSDUDC. These results provide the fastest known approximation algorithms for DUDC. As with the WSDUDC results, we present a set of algorithms in which better approximation factors may be had at the expense of greater running time, ranging from a 15-approximate algorithm which runs in O(mn + m log m + n log n) time to a 18-approximate algorithm which runs in O(m^6n+n log n) time. The next problems that we study are Hausdorff Core problems. These problems accept an input polygon P, and we seek a convex polygon Q which is fully contained in P and minimizes the Hausdorff distance between P and Q. Interestingly, we show that this problem may be reduced to that of computing the minimum radius of disk, call it k_opt, so that a convex polygon Q contained in P intersects all disks of radius k_opt centred on the vertices of P. We begin by describing a polynomial time algorithm for the simple case where P has only a single reflex vertex. On general polygons, we provide a parameterized algorithm which performs a parametric search on the possible values of k_opt. The solution to the decision version of the problem, i.e. determining whether there exists a Hausdorff Core for P given k_opt, requires some novel insights. We also describe an FPTAS for the decision version of the Hausdorff Core problem. Finally, we study Generalized Minimum Spanning Tree (GMST) problems, where the input consists of imprecise vertices, and the objective is to select a single point from each imprecise vertex in order to optimize the weight of the MST over the points. In keeping with one of the themes of the thesis, we begin by using disks as the imprecise vertices. We show that the minimization and maximization versions of this problem are NP-hard, and we describe some parameterized and approximation algorithms. Finally, we look at the case where the imprecise vertices consist of just two vertices each, and we show that the minimization version of the problem (which we call 2-GMST) remains NP-hard, even in the plane. We also provide an algorithm to solve the 2-GMST problem exactly if the combinatorial structure of the optimal solution is known. We identify a number of open problems in this thesis that are worthy of further study. Among them: Is the Assisted LSDUDC problem NP-complete? Can the WSDUDC results be used to obtain an improved PTAS for DUDC? Are there classes of polygons for which the determination of the Hausdorff Core is easy? Is there a PTAS for the maximum weight GMST problem on (unit) disks? Is there a combinatorial approximation algorithm for the 2-GMST problem (particularly with an approximation factor under 4)?

In this doctoral thesis, four questions related to computational geometry are considered. The first is an extremal combinatorics question regarding triangles with vertices taken from a set of n points in convex position. More precisely, two such triangles can exhibit eight distinct configurations and, for each subset of these configurations, we are interested in the asymptotics of how many triangles one can have while avoiding configurations in the subset (as a function of n). For most of these subsets, we answer this question optimally up to a logarithmic factor in the form of several Turán-type theorems. The answers for the remaining few were in turn tied to that of a long-standing open problem which appeared in the literature in the contexts of monotone matrices, tripod packing and 2-comparable sets. The second problem, called Line Segment Covering (LSC), is about covering the cells of an arrangement of line segments with these line segments, where a segment covers the cells it is incident to. Recently, a PTAS, an APX -hardness proof and a FPT algorithm for variants of this problem have been shown. This paper and a new slight generalization of one of its results is included as a chapter. The third problem has been posed in the Sixth Annual Workshop on Geometry and Graphs and concerns the design of road networks to minimize the maximum travel time between two point sets in the plane. Traveling outside the roads costs more time per unit of distance than traveling on the roads and the total length of the roads can not exceed a budget. When the point sets are the opposing sides of a unit square and the budget is at most √2, we were able to come up with a few network designs that cover all possible cases and are provably optimal. Furthermore, when both point sets are the boundary of a unit circle, we managed to disprove the natural conjecture that a concentric circle is an optimal design. Finally, we consider collision-avoiding schedules of unit-velocity axis-aligned trains departing and arriving from points in the integer lattice. We prove a few surprising results on the existence of constant upper bounds on the maximum delay that are independent of the train network. In particular, these upper bounds are shown to always exist in two dimensions and to exist in three dimensions for unit-length trains. We also showed computationally that, for several scenarios, these upper bounds are tight.

No description available.

The axial map of a town is one of the key components of the space syntax method – a tool for analysing urban layout. It is derived by placing the longest and fewest lines, called axial lines, to cross the adjacencies between convex polygons in a convex map of a town. Previous research has shown that placing axial lines to cross the adjacencies between a collection of convex polygons is NP-complete, even when the convex polygons are restricted to rectangles and the axial lines to have orthogonal orientation. In this document, we show that placing orthogonal axial lines in orthogonal rectangles where the adjacencies between the rectangles are restricted to be crossed only once (ALPSC- OLOR) is NP-complete. As a result, we infer the single adjacency crossing version of the general axial line placement problem is NP-complete. The transformation of NPcompleteness of ALP-SC-OLOR is from vertex cover for biconnected planar graphs. A heuristic is then presented that gives a reasonable approximate solution to ALP-SC-OLOR based on a greedy method.

Este trabalho aborda dois problemas de geometria computacional: união de círculos e união de (vários) polígonos. Para o problema da união de círculos, os principais algoritmos da literatura são revisados e um algoritmo simples, porém ineficiente, é introduzido. Este algoritmo é então adaptado para resolver o problema da união de polígonos, produzindo um algoritmo que é competitivo com o estado da arte e, dependendo da aplicação, utiliza menos armazenamento. / This work deals with two problems from the field of computational geometry: union of circles and union of (many) polygons. For the union of circles problem, the main algorithms in the literature are revised and a simple, albeit inefficient, algorithm is introduced. This algorithm is then adapted to solve the union of polygons problem, resulting in an algorithm that is competitive with the state of the art and, depending on the application, makes use of less storage.

Cerebral aneurysm is a pathology of the circulatory system in the brain in which an arterial wall balloons into a blood-filled sac. If the aneurysm ruptures, stroke can occur and has a high probability of causing permanent disability or death. Aneurysm surgery carries a high rate of morbidity and mortality compared to the natural rate of aneurysm rupture, so physicians must take care in recommending surgery for an aneurysm patient. However, very little is known about the etiology of brain aneurysm rupture and what prognostics exist. The International Study for Intracranial Aneurysms suggested that large aneurysm size and posterior location are important factors in identifying high rupture risk. However, many small aneurysms and aneurysms in other portions of the circulation still rupture. Many studies have assessed morphological traits, identified from aneurysm appearance on diagnostic medical images, and found such traits to be different in aneurysms that ruptured and aneurysms that did not rupture. In fact, more than 50 such morphological indices have been introduced in the literature, and many of them redundantly quantify particular morphological characteristics. In order to demonstrate the prognostic ability of morphology as an indicator of rupture risk, however, a large longitudinal cohort study must be carried out. A study such as this is time-consuming and expensive, and each additional hypothesis that a particular morphological index is predictive of rupture risk would require increasing the study population size in order to fulfill the necessary statistical power requirements for a rigorous test. Thus, a minimal set of physically meaningful, independent metrics that fully describe the aneurysm morphology is needed. In this dissertation an automated protocol was developed to process segmented medical images and extract an exhaustive set of morphological indices that quantify all relevant morphological features. Each morphological index was then analyzed for robustness to inter-user variability and for sensitivity to the particular morphological characteristic that it was designed to quantify. A factor analysis was then performed using the most robust, sensitive metrics on a population of unruptured aneurysms from five data centers and 276 patient-specific aneurysms. The results from the factor analysis were utilized to ascertain what morphological features those metrics truly described, if there were any redundancies in definition, and the variance each morphological trait described in the population. Four underlying morphological constructs were uncovered through the factor analysis. The first factor, sac size, was highly aligned with morphological indices that measured volume and one-dimensional size measurements. Sac size described 50% of the variance in the data set. The second factor, sac irregularity, was highly aligned with morphological indices that described various aspects of irregular shape. A set of variables that all were implicated in causing irregular shape, but in reality measured sac-neck size relation, also merited inclusion of a second metric to describe the variance seen in the second factor. Sac irregularity described 20% of the variance in the data set. The third factor, sac ellipticity, aligned highly with morphological indices that described the overarching ellipticity of the aneurysm sac independent of other non-spherical characteristics. Sac ellipticity described 13% of the variance in the data set. The fourth factor, sac-vessel size relation, aligned highly with morphological indices that described the size of the aneurysm sac in relation to its parent vessel. Sac-vessel size relation described 7% of the variance in the data set. All four of these factors in combination described 91% of the variance in the data set. Five morphological indices – non-planar isolation sac volume (Vnp), Voronoi diagram core evolution irregularity index (IRRvdc), tissue stretch ratio (TSR), Voronoi diagram core evolution ellipticity index (EIvdc) and size ratio (SRang) were determined to be the key indices for describing aneurysm morphology. Finally, the proposed metrics were used to test the hypothesis that aneurysms that are chosen for untreated observation are morphologically different than those that are treated – commonly referred to as selection bias. Study population was 27 patient-specific aneurysms that were placed on untreated observation (observation group) and 27 patient specific aneurysms that were size- and location-matched but were chosen for treatment (treated group). A significant difference was found in the morphological index that measured ellipticity between the two groups, indicating that physicians already commonly select highly elliptical aneurysms for treatment. This result may give insight into physicians’ choices, and merits further investigation with a larger data set for confirmation. Additionally, because the same result was replicated in both of the metrics chosen to quantify ellipticity (for both manual and automated methods), this highlighted the use of the morphological factors in determining an minimal set of independent, robust morphological indices.