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Geometric Arrangements PDF

178 Pages·2008·1.35 MB·English
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Tel Aviv University Raymond and Beverly Sackler Faculty of Exact Sciences School of Computer Science Geometric Arrangements: Substructures and Algorithms Thesis submitted for the degree of “Doctor of Philosophy” by Esther Ezra Submitted to the Senate of Tel-Aviv University August 2007 The work on this thesis was carried out under the supervision of Prof. Micha Sharir iii iv To Yaron with much love. To my parents, for their unconditional love. To my brothers Alon and Hezi, my sister Shelly, and to Savta Shula. v vi Abstract In this thesis we study a variety of problems in computational and combinatorial geometry, which involve arrangements of geometric objects in the plane and in higher dimensions. Some of these problems involve the design and analysis of algorithms on arrangements and related structures, while othersestablish combinatorialbounds onthe complexity ofvarious substructures in arrangements. Informally, anarrangementisthesubdivision ofspaceinducedbyacollectionofgeomet- ric objects. For example, a collection of triangles in the plane subdivides it into polygonal regions, each being a maximal connected region contained in a fixed subset of the triangles and disjoint from all the others. This subdivision is the arrangement of the triangles, and each of these polygonal regions is a cell of the arrangement. A substructure in an arrange- ment is a collection of certain features of the arrangement. Two main substructures that we study in this thesis, under both combinatorial and algorithmic aspects, are the union of geometric objects and a single cell in an arrangement. This thesis consists of two major parts, where in the first we discuss several algorith- mic problems, and in the second we present combinatorial bounds on substructures in arrangements. Algorithmic problems. Constructing the union of geometric objects. A central problem in computational and combinatorial geometry, with various applications, concerns the union of geometric objects. Given a collection S of geometric objects in d-space, let U = U(S) denote their union. Informally, the combinatorial complexity of the union is the overall number of features of the arrangement of S that appear on its boundary. For example, if S is a set of triangles as above, then their union boundary consists of all vertices (intersections between a pair of triangle boundaries, or an original vertex of a triangle) and edges (a maximal connected portion of a boundary of a triangle, that does not contain any vertex of the arrangement in its relative interior) that are not contained in the interior of any of the triangles in S. It is well known that in the worst case the combinatorial complexity of the union can be (asymptotically) the same as that of the entire arrangement. However, there are various special cases for which this complexity is considerably smaller (by, roughly, vii one order of magnitude than the complexity of the full arrangement), and, as a result, the union boundary can be constructed much more efficiently (that is, without constructing the entire arrangement). In Chapter 2 we present a subquadratic “output-sensitive” algorithm to construct the boundary of the union of a set of triangles in the plane, under the assumption that there exists a (relatively small) subset of triangles (unknown to us) such that their union is equal to the union of the entire set. Our approach is fairly general, and we show that it can be extended to compute efficiently the union of simply shaped bodies of constant description complexity in d-space, when the union is determined by a small subset of the bodies. The solution is based on a variant of the Br¨onnimann-Goodrich technique [41] for obtaining an approximate solution to the hitting-set problem. Counting triple intersections among triangles in 3-space. Intersection problems are among the most basic problems in computational geometry, with many different appli- cation areas. The problem of reporting all intersections in a given collection of geometric objects has received considerable attention, and several efficient output-sensitive algorithms (algorithms whose running time depends on the output size) have been designed for this problem (mostly for planar instances). However, in some applications, we are only inter- ested in counting the overall number of intersections, without reporting them explicitly. In this case, one prefers an algorithm whose running time does not depend on the number of intersections (which, in the worst case, is proportional to the size of the arrangement induced by the input objects). In Chapter 3 we present an algorithm that efficiently counts all intersecting triples among a collection of n triangles in 3-space in nearly-quadratic time. This solves a problem posed by Pellegrini [120]. Using a variant of the technique, the algorithm can also represent the set of all triple intersections in a compact form, as the disjoint union of complete tripartite hypergraphs, which requires nearly-quadratic construction time and storage. A compactrepresentationofthisformisusefulfordrawingarandomvertex(tripleintersection point)ofthearrangement ofthegiven triangles. Ourapproachalsoappliestoanycollection of planar objects of constant description complexity in 3-space, with the same performance bounds. We also prove that this counting problem belongs to the “3sum-hard” family, and thus our algorithm is likely to be nearly optimal in the worst case. Analyzing the ICP algorithm. The matching and analysis of geometric patterns and shapes is an important problem that arises in various application areas. In a typical sce- nario, we aregiven two objects A andB, andwe wish todetermine how much they resemble each other (with respect to some cost function). Usually one of the objects may undergo certain transformations, like translation, rotation and/or scaling, in order to be matched with the other object as well as possible. In many cases, the objects are represented as finite sets of (sampled) points in two or three dimensions. viii A heuristic matching algorithm that is widely used, due to its simplicity (and its good performance in practice), is the Iterative Closest Point algorithm, or the ICP algorithm for short, of Besl and McKay [36]. Given two point sets A and B in d-space, we wish to minimize a cost function φ(A + t,B), over all translations t of A relative to B. The algorithm starts with an arbitrary translation that aligns A to B (suboptimally), and then repeatedly performs local improvements that keep re-aligning A to B, while decreasing the given cost function φ(A+t,B), until no improvement is possible. This algorithm has been identified and used as a practical heuristic solution over the past fifteen years. Many experimental reports on its performance, including additional heuristic enhancements of it have been published [36, 79, 122, 127]. Still, this technique has never before been subject to a serious and rigorous analysis of its worst-case behavior. In Chapter 4 we analyze the performance of this algorithm, and present several of its structural geometric properties. In particular, we present upper and lower bounds for the number of iterations that it performs, where we consider two standard measures of resem- blance that the algorithm attempts to optimize: The RMS (root mean squared distance) andthe(one-sided)Hausdorffdistance. Weshowthatinbothcasesthenumberofiterations performed by the algorithm is polynomial in the number of input points. In particular, the upper bound is quadratic in the one-dimensional case, under the RMS measure, for which we present a lower bound construction that requires Ω(nlogn) iterations, where n is the overall size of the input. Under the Hausdorff measure, this bound is only O(n) for input point setswhose spreadispolynomialinn, andthisistight intheworst case. Regarding the structural geometric properties of the algorithm, we show, for the RMS measure, that at each iteration of the algorithm the cost function monotonically and strictly decreases along the vector ∆t of the relative translation. As a result, we conclude that the polygonal path π, obtained by concatenating all the relative translations that are computed during the execution of the algorithm, does not intersect itself. In particular, in the one-dimensional problem all the relative translations of the ICP algorithm are in the same (left or right) direction. For the Hausdorff measure, some of these properties continue to hold (such as monotonicity in one dimension), whereas others do not. Combinatorial bounds on substructures in arrangements A single cell. A natural problem that relates to substructures in arrangements is to analyze thecomplexity of asinglecell in anarrangement of geometric objects. InChapter 5 we study the case where the input consists of k convex polyhedra in 3-space with n facets in total. We show that in this case the combinatorial complexity of a single cell in the arrangement of the polyhedra is O(nk1+ε), for any ε > 0, thus settling a conjecture of Aronov et al. [28], who presented a lower bound of Ω(nkα(k)), and conjectured that the upper bound is close to O(nk). We also design an efficient deterministic algorithm that constructs a single cell of the arrangement, whose running time matches the combinatorial bound up to a polylogarithmic factor. We note that a nearly-quadratic bound on the ix complexity of a single cell in 3-dimensional arrangements is well known, for fairly general types of arrangements [25, 86]. The novelty of our bound is its dependence on k, which makes it linear in n for any fixed k. Union of geometric objects. As discussed above, constructing the union of geometric objects in d-space is a central problem in computational geometry with many applications. On the combinatorial front, it is interesting to seek for natural examples, for which the combinatorial complexity of the union is considerably smaller than that of the full arrange- ment. Besides yielding more efficient algorithms that construct the union in these special cases, these problems involve intricate combinatorial techniques, which we believe to be of independent interest, and may find additional applications to related problems. In Chapter 6 we study the case where the input consists of “fat” tetrahedra in 3-space, and derive a nearly-quadratic bound on the complexity of their union. Our bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [114]. Our result extends, in a significant way, the result of Pach et al. [114] for the restricted case of nearly congruent cubes. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in 3-space, having arbitrary side lengths, is O(n2+ε), for any ε > 0 (again, significantly extending the result of [114]). Our analysis can easily be extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have arbitrary sizes) in 3-space. Finally, we also show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane (this latter bound is known, even in a slightly sharper form [109, 118], but the new proof is considerably simpler). In spite of the steady progress during the past decade in the study of the union of objects in 3-space, the case of arbitrary fat tetrahedra, considered one of the major basic instances, has remained open. Regular vertices. Another problem that belongs to this family is to obtain, for a collec- tion of n objects in the plane, a sharp bound on the number of regular vertices (intersection points of two object boundaries that intersect twice), which appear on the boundary of the union. In Chapter 7 we show that the number of regular vertices that appear on the bound- ary of the union of n compact convex sets in the plane, such that the boundaries of any pair of these sets intersect in at most some constant number s of points, is O(n4/3+ε), for any ε > 0. Our bound is nearly tight in the worst case (already for s = 4), and improves earlier bounds due to Aronov et al. [21]. x

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in total. We show that in this case the combinatorial complexity of a single cell in the . 4.3 The ICP Algorithm on the Line under the RMS Measure .
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