Algorithms for fat objects : decompositions and applications

Computational geometry is the branch of theoretical computer science that deals with algorithms and data structures for geometric objects. The most basic geometric objects include points, lines, polygons, and polyhedra. Computational geometry has applications in many areas of computer science, including computer graphics, robotics, and geographic information systems. In many computational-geometry problems, the theoretical worst case is achieved by input that is in some way "unrealistic". This causes situations where the theoretical running time is not a good predictor of the running time in practice. In addition, algorithms must also be designed with the worst-case examples in mind, which causes them to be needlessly complicated. In recent years, realistic input models have been proposed in an attempt to deal with this problem. The usual form such solutions take is to limit some geometric property of the input to a constant. We examine a specific realistic input model in this thesis: the model where objects are restricted to be fat. Intuitively, objects that are more like a ball are more fat, and objects that are more like a long pole are less fat. We look at fat objects in the context of five different problems—two related to decompositions of input objects and three problems suggested by computer graphics. Decompositions of geometric objects are important because they are often used as a preliminary step in other algorithms, since many algorithms can only handle geometric objects that are convex and preferably of low complexity. The two main issues in developing decomposition algorithms are to keep the number of pieces produced by the decomposition small and to compute the decomposition quickly. The main question we address is the following: is it possible to obtain better decompositions for fat objects than for general objects, and/or is it possible to obtain decompositions quickly? These questions are also interesting because most research into fat objects has concerned objects that are convex. We begin by triangulating fat polygons. The problem of triangulating polygons—that is, partitioning them into triangles without adding any vertices—has been solved already, but the only linear-time algorithm is so complicated that it has never been implemented. We propose two algorithms for triangulating fat polygons in linear time that are much simpler. They make use of the observation that a small set of guards placed at points inside a (certain type of) fat polygon is sufficient to see the boundary of such a polygon. We then look at decompositions of fat polyhedra in three dimensions. We show that polyhedra can be decomposed into a linear number of convex pieces if certain fatness restrictions aremet. We also show that if these restrictions are notmet, a quadratic number of pieces may be needed. We also show that if we wish the output to be fat and convex, the restrictions must be much tighter. We then study three computational-geometry problems inspired by computer graphics. First, we study ray-shooting amidst fat objects from two perspectives. This is the problem of preprocessing data into a data structure that can answer which object is first hit by a query ray in a given direction from a given point. We present a new data structure for answering vertical ray-shooting queries—that is, queries where the ray’s direction is fixed—as well as a data structure for answering ray-shooting queries for rays with arbitrary direction. Both structures improve the best known results on these problems. Another problem that is studied in the field of computer graphics is the depth-order problem. We study it in the context of computational geometry. This is the problem of finding an ordering of the objects in the scene from "top" to "bottom", where one object is above the other if they share a point in the projection to the xy-plane and the first object has a higher z-value at that point. We give an algorithm for finding the depth order of a group of fat objects and an algorithm for verifying if a depth order of a group of fat objects is correct. The latter algorithm is useful because the former can return an incorrect order if the objects do not have a depth order (this can happen if the above/below relationship has a cycle in it). The first algorithm improves on the results previously known for fat objects; the second is the first algorithm for verifying depth orders of fat objects. The final problem that we study is the hidden-surface removal problem. In this problem, we wish to find and report the visible portions of a scene from a given viewpoint—this is called the visibility map. The main difficulty in this problem is to find an algorithm whose running time depends in part on the complexity of the output. For example, if all but one of the objects in the input scene are hidden behind one large object, then our algorithm should have a faster running time than if all of the objects are visible and have borders that overlap. We give such an algorithm that improves on the running time of previous algorithms for fat objects. Furthermore, our algorithm is able to handle curved objects and situations where the objects do not have a depth order—two features missing from most other algorithms that perform hidden surface removal

A technique for adding range restrictions to generalized searching problems

In a generalized searching problem, a set $S$ of $n$ colored geometric objects has to be stored in a data structure, such that for any given query object $q$, the distinct colors of the objects of $S$ intersected by $q$ can be reported efficiently. In this paper, a general technique is presented for adding a range restriction to such a problem. The technique is applied to the problem of querying a set of colored points (resp.\ fat triangles) with a fat triangle (resp.\ point). For both problems, a data structure is obtained having size $O(n^{1+\epsilon})$ and query time $O((\log n)^2 + C)$. Here, $C$ denotes the number of colors reported by the query, and $\epsilon$ is an arbitrarily small positive constant

Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles

In this paper we study two geometric data structure problems in the special case when input objects or queries are fat rectangles. We show that in this case a significant improvement compared to the general case can be achieved. We describe data structures that answer two- and three-dimensional orthogonal range reporting queries in the case when the query range is a \emph{fat} rectangle. Our two-dimensional data structure uses $O(n)$ words and supports queries in $O(\log\log U +k)$ time, where $n$ is the number of points in the data structure, $U$ is the size of the universe and $k$ is the number of points in the query range. Our three-dimensional data structure needs $O(n\log^{\varepsilon}U)$ words of space and answers queries in $O(\log \log U + k)$ time. We also consider the rectangle stabbing problem on a set of three-dimensional fat rectangles. Our data structure uses $O(n)$ space and answers stabbing queries in $O(\log U\log\log U +k)$ time.Comment: extended version of a WADS'19 pape

Computing the visibility map of fat objects

AbstractWe give an output-sensitive algorithm for computing the visibility map of a set of n constant-complexity convex fat polyhedra or curved objects in 3-space. Our algorithm runs in O((n+k) polylog n) time, where k is the combinatorial complexity of the visibility map. This is the first algorithm for computing the visibility map of fat objects that does not require a depth order on the objects and is faster than the best known algorithm for general objects. It is also the first output-sensitive algorithm for curved objects that does not require a depth order

Vertical ray shooting and computing depth orders of fat objects

We present new results for three problems dealing with a set $\mathcal{P}$ of $n$ convex constant-complexity fat polyhedra in 3-space. (i) We describe a data structure for vertical ray shooting in $\mathcal{P}$ that has $O(\log^2 n)$ query time and uses $O(n\log^2 n)$ storage. (ii) We give an algorithm to compute in $O(n\log^3 n)$ time a depth order on $\mathcal{P}$ if it exists. (iii) We give an algorithm to verify in $O(n\log^3 n)$ time whether a given order on $\mathcal{P}$ is a valid depth order. All three results improve on previous results

Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

Effizient algorithms for generalized intersection searching on non-iso-oriented objects

In a generalized intersection searching problem, a set $S$ of colored geometric objects is to be preprocessed so that, given a query object $q$, the distinct colors of the objects of $S$ that are intersected by $q$ can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and are rich in applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented, i.e., axes-parallel, or where the color classes satisfy additional properties, e.g., connectedness. In this paper, efficient algorithms are given for several generalized problems involving non-iso-oriented objects. These problems include: generalized halfspace range searching in ${\cal R}^d$, for any fixed $d \geq 2$, segment intersection searching, triangle stabbing, and triangle range searching in ${\cal R}^2$. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search