Improved Bounds on the Union Complexity of Fat Objects

Abstract

We introduce a new class of fat, not necessarily convex or polygonal, objects in the plane, namely locally γ -fat objects. We prove that the union complexity of any set of n such objects is O(λ s+2 (n) log 2 n). This improves the best known bound, and extends it to a more general class of objects.

Better bounds on the union complexity of locally fat objects

“…He proved that the union complexity of n socalled (α, β)-covered objects is O(λ t+2 (n) log 2 n log log n). Recently this was further generalized and slightly improved by De Berg [4], who showed that the union complexity of n so-called locally γ-fat objects is O(λ t+2 (n) log 2 n). The class of locally γ-fat objects-see Section 2 for a formal definition-is the most general class of fat objects for which near-linear union-bounds are known, and it is also the class we will consider in this paper.…”

“…For non-convex objects, local fatness is a stronger condition than fatness. This stronger condition is necessary, as there are sets of n objects that are fat under the definition of Van der Stappen (but not locally fat) whose union complexity is Θ(n 2 ) [4].…”

“…We define D o to be the connected component that contains the center of D. Definition 1. [4] 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).…”

A note about weak ε-nets for axis-parallel boxes in d-space

“…Concerning the extension to fat objects, there are several different definitions of fatness in the geometry literature (see, e.g., [11,9]). We adopt the following definition presented by Chan [5], which is the most appropriate for our purposes.…”

Star-quadtrees and guard-quadtrees: I/O-efficient indexes for fat triangulations and low-density planar subdivisions

“…The most common assumption in such models is that the objects are fat, that is, they are not arbitrarily long and narrow. Fatness has been studied extensively in computational geometry in recent years, and solutions to many fundamental problems have been improved by exploiting fatness and related notions; see [11,13] and references therein.…”