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Improved Bounds on the Union Complexity of Fat Objects Discrete Comput Geom (2008) 40: 127–140 DOI 10.1007/s00454-007-9029-7 Improved Bounds on the Union Complexity of Fat Objects Mark de Berg Received: 20 October 2006 / Revised: 9 July 2007 / Published online: 15 September 2007 © Springer Science+Business Media, LLC 2007 Abstract We introduce a new class of fat, not necessarily convex or polygonal, ob- jects in the plane, namely locally γ -fat objects. We prove that the union complexity of 2 any set of n such objects is O(λs+2(n) log n). This improves the best known bound, and extends it to a more general class of objects. Keywords Fat Objects · Union Complexity · Density Lemma 1 Introduction The running time of geometric algorithms and the amount of storage used by geomet- ric data structures often depend on the combinatorial complexity of certain geometric structures. Hence, the study of the combinatorial complexity of geometric structures is an important and active area within computational geometry. In this paper we study the combinatorial complexity of the union of a set F of n objects in the plane. This is relevant because there are many geometric algorithms and data structures whose performance depends on the union complexity of certain collections of planar ob- jects. Examples are algorithms for hidden-surface removal [13], data structures for ray shooting [4, 5, 12], algorithms for computing depth orders [1, 5, 12], and algo- rithms for motion planning [14, 20]. In the worst case the complexity of the union of n constant-complexity objects in the plane can be as high as (n2), a bound which is for example achieved by a set of n long and thin rectangles arranged in a grid-like pattern. In many applications, The research was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301. M. de Berg () Department of Computing Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands e-mail: [email protected] 128 Discrete Comput Geom (2008) 40: 127–140 however, one would expect that the objects have some favorable properties and that the union complexity is much lower. One such property that has received considerable attention is fatness. Intuitively, an object is called fat if it is not arbitrarily long and skinny—see Sect. 2 for precise definitions. There are many algorithmic results for fat objects, several of which depend on the union complexity of fat objects in the plane. Hence, the union complexity of fat objects in the plane has been studied extensively. One of the first results on the union complexity of fat objects was for fat wedges, that is, wedges whose interior angle is bounded from below by a constant. For this case it has been shown [2, 10] that the union complexity is O(n). Matoušek et al. [15] considered the case of δ-fat triangles, that is, triangles all of whose angles are at least δ for some fixed constant δ. They proved that the union complexity of n such triangles is O((1/δ3)n log log n). Later this bound was improved by Pach and Tardos [17]to O((1/δ log(1/δ))n log log n). Several people have worked on extending these results to more general types of fat objects, in particular to curved and/or non-convex objects [8, 9, 11, 21] and to higher dimensions [3, 16]. The most general result for planar objects to date is by Efrat [8], who considered so-called (α, β)-covered objects—see the next section for a definition. Efrat proved that the union complexity of n constant-complexity (α, β)- 2 covered objects is bounded by O(λs+2(n) log n log log n), where s is the maximum number of intersections between any pair of object boundaries and λt (n) denotes the maximum length of an (n, t) Davenport–Schinzel sequence; λt (n) is near-linear for any constant t [18]. (The constant hidden in the O-notation depends on the constants α and β.) We introduce in Sect. 2 a new class of fat objects in the plane, namely locally γ -fat objects. This class is more general than the class of (α, β)-covered objects. We prove that the union complexity of n constant-complexity locally γ -fat objects 2 is O(λs+2(n) log n), thus not only generalizing the result of Efrat but also slightly improving the bound. One of the novel ingredients of our proof is a lemma—we call it the Density Lemma—stating that the set of boundary pieces of the union of fat objects in the plane has low density. This is a powerful tool, which we also used elsewhere [4], and which we believe will find other applications in the study of unions of fat objects in the plane. The Density Lemma implies that the union complexity of two sets of locally fat objects can be bounded by the sum of the union complexities of the two sets (with a constant multiplicative factor that depends on the fatness). It then fol- lows for example that the union of a set of fat triangles and disks in the plane has O(nlog log n) complexity, which was, to the best of our knowledge, not known. The Density Lemma allows us to reduce the problem of bounding the union com- plexity of locally γ -fat objects to the problem of bounding the union complexity of so-called consistently oriented fat quasi-triangles; these are almost triangular fat shapes with two edges in a fixed orientation. We then give a simple proof that the 2 union complexity of such shapes is O(λs+2(n) log n). An interesting feature of our proof is that, unlike Efrat’s proof, it does not rely on the result of Matoušek et al. [15] for fat triangles. Discrete Comput Geom (2008) 40: 127–140 129 2 Preliminaries Let F := {o1,...,on} be a set of objects in the plane. From now on, we assume that each object is compact, that is, bounded and closed. We also assume that each object has constant complexity; in particular we assume that the boundary of each object consists of O(1) algebraic curves of constant maximum degree. Hence, any two object boundaries intersect at most s times for some constant s. We denote the boundary of an object o by ∂o. Fatness and low density We first define (α, β)-covered objects, as introduced by Efrat [8]. Figure 1(i) illustrates the definition. Definition 2.1 A planar object o is called (α, β)-covered if for every point p on the boundary of o one can place a triangle tp with the following properties: (i) tp is contained inside o; (ii) p is a vertex of tp; (iii) tp is an α-fat triangle, that is, all its angles are at least α; (iv) the length of each edge of tp is at least β ·diam(o), where diam(o) is the diameter of o. Next we introduce a new characterization of fatness. Let area(o) denote the area of an object o. Consider an object o and a disk D whose center lies inside o.Ifo is non-convex D ∩ o may consist of several connected components. We define D o to be the connected component that contains the center of D. Definition 2.2 Let o be an object in the plane and γ a parameter with 0 ≤ γ ≤ 1. We say that o is locally γ -fat if, for any disk D whose center lies in o and that does not fully contain o in its interior, we have area(D o) ≥ γ · area(D). This definition is illustrated in Fig. 1(ii). It is similar to the fatness definition in- troduced by Van der Stappen et al. [19, 20], except that we use area(D o) instead of area(D ∩ o). Thus for convex objects, where D o = D ∩ o, the definitions are identical. The following proposition shows the relation between (α, β)-covered objects and locally γ -fat objects. Proposition 2.3 (i) Any (α, β)-covered object is locally γ -fat for some γ = (αβ2). (ii) There are no constants α, β that depend only on γ such that any locally γ -fat object is an (α, β)-covered object. Fig. 1 Illustration of the definition of (α, β)-covered object and the definition of locally γ -fat object 130 Discrete Comput Geom (2008) 40: 127–140 Fig. 2 (i) A fat object that is not locally fat. (ii)Thefirsttwo objects in a construction showing that objects that are fat according to the definition of Van der Stappen et al. [19, 20] can have a quadratic union Proof (i) Let o be an (α, β)-covered object and let D be a disk centered at a point p ∈ o and not containing o in its interior. First assume p ∈ ∂o. Then there is an α-fat triangle tp ⊂ o with p as a vertex all of whose edges have length at least β · diam(o). Clearly (D ∩ tp) ⊂ (D o).Iftp is not fully contained in D then area(D ∩ tp) = (α · area(D)) because all angles of tp are at least α. If, on the other hand, tp ⊂ D then 2 2 area(D o) ≥ area(D ∩ tp) = area(tp) = α · (β · diam(o)) = αβ · area(D) , where the last equality follows because D does not fully contain o. Now assume p lies in the interior of o.Letp be a point on ∂o with mini- mum distance to p.Ifdist(p, p) ≥ radius(D)/2 then area(D o) ≥ area(D)/4.
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