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On the Perimeter of Fat Objects

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On the Perimeter of Fat Objects CCCG 2010, Winnipeg MB, August 9{11, 2010 On the perimeter of fat objects Prosenjit Bose∗ Otfried Cheongy Vida Dujmovi´c∗ Abstract An "-good object O, introduced by Valtr [10], has the property that every point p 2 O can see a constant We show that the size of the perimeter of (α; β)-covered fraction of the area of O. We require this only of the objects is a linear function of the diameter. Specifically, points p 2 @O, and call such objects "-area-good (this diam(O) for an (α; β)-covered object O, per(O) ≤ c αβ sin2 α , for a is a strictly larger class of objects). Kirkpatrick [8] in- positive constant c. One easy consequence of the result troduced a similar class of "-boundary-good objects O, is that every point on the boundary of such an object where every point p 2 @O can see a constant fraction of sees a constant fraction of the boundary. Locally γ{fat the length of the boundary of O. objects are a generalization of (α; β){covered objects. Table 1 shows the relations between these four classes We show that no such relationship between perimeter of objects. The table can be interpreted as follows. A and diameter can hold for locally γ-fat objects. YES entry in the table means that the class of objects quantified by a constant c in the row of Table 1 belongs to the class of objects quantified by c in the column of 1 Introduction 0 Table 1, where c0 is some function of c. The entry NO means that no such constant exists. Trivially, all the Often, the worst case lower bound for a geometric al- entries on the diagonal of the table are YES. The four gorithm occurs when the input consists of `long' and NO entries that are not justified by any references are `skinny' objects. However, such artificial configurations implied by the fact that thin triangles are "-area-good do not occur in many `realistic' inputs. This motivates and "-boundary-good objects with " = 1, yet they are the study of objects considered to be more likely to oc- neither (α; β)-covered, nor locally γ-fat for any α; β; γ. cur in real-life applications. A number of realistic mod- els have been introduced and studied in the literature object class (α ; β )-covered locally γ -fat (see [5] for a survey). We consider four such classes 0 0 0 (α; β)-covered YES YES [4] of objects, namely (α; β)-covered objects, locally γ-fat locally γ-fat NO [4] (Fig. 2(b)) YES objects, "-area-good objects, and "-boundary-good ob- "-area-good NO NO jects. (α; β)-covered objects and locally γ-fat objects "-boundary-good NO NO are classes of fat objects, that is, objects that can- not be arbitrarily long and skinny. "-area-good and "- object class "0-area-good "0-boundary-good boundary-good objects are `realistic' in a different sense (α; β)-covered YES (Obs. 1) YES (Cor. 1) which is expressed in terms of visibility. We briefly de- locally γ-fat NO (Fig. 2(b)) NO (Fig. 2(b)) scribe the four models. "-area-good YES NO (Obs. 2) (α; β)-covered objects were introduced by Efrat [6] as "-boundary-good NO (Obs. 2) YES a generalization of convex fat objects to a class of non- convex objects. Roughly speaking, an object O is (α; β)- Table 1: Pairwise relationships between four classes of covered if for every point p on the boundary, @O, of O objects. there is a large fat triangle contained in O that has p as a vertex. Locally γ-fat objects were introduced by de Various properties have been proven for realistic ob- Berg [4] as a generalization of (α; β)-covered objects. An jects under various models, in particular good union object O is locally γ-fat if for any disk D with center p complexity, good guardability, and good perimeter- in O and not completely containing O, the connected length. component of D \ O containing p has area at least γ The complexity of the union of n general (constant- times the area of D. complexity) objects can be Θ(n2), but this can only be ∗School of Computer Science, Carleton University, Ottawa, achieved with objects that are long and skinny. The Canada fjit,[email protected]. Research supported in union complexity of a set of n pseudo-disks is O(n) [7], part by NSERC. and we consider a class of objects to have good union y Department of Computer Science, Korea Advanced In- complexity if the union of n objects from that class has stitute of Science and Technology, Daejeon, South Korea [email protected]. Research supported by the Korea Science near-linear size. and Engineering Foundation Grant R01-2008-000-11607-0. Fat objects behave more like disks as opposed to line 22nd Canadian Conference on Computational Geometry, 2010 segments, and indeed the union complexity of n fat tri- objects have good perimeter-length, see the last column angles|that is, triangles having no angle smaller than of Table 2. some constant α|is O(n log log n) [9]. Efrat general- ized this result by showing that the union complexity of 2 Perimeter of (α; β){covered objects n (α; β)-covered objects (of constant description com- 2 plexity) is at most O(λs(n) log n log log n) [6]. This re- An object∗ O is (α; β){covered if for each point p 2 @O sult was later improved and generalized to locally γ{fat there exists a triangle T (p), called a good triangle of p, objects by de Berg [4]. On the other hand, since ev- such that: ery convex object is "-area-good and "-boundary-good, these classes do not have good union complexity. This 1. p is a vertex of T (p), T (p) ⊆ O, and is summarized in the second column of Table 2. 2. each angle of T is at least α, and the length of each object class good union-complexity edge of T is at least β · diam(O). (α; β)-covered YES [6] Notice that this immediately implies that α ≤ π=3. locally γ-fat YES [4] For a point p in the plane, a ray at p is a half-line with "-area-good NO endpoint at p that is oriented away from p. The direc- "-boundary{good NO tion of a ray R is the counter-clockwise angle between object class good good the positive x-axis (rooted at p) and ray R. guardability perim.-length Consider a parallelogram having as one of its ver- (α; β)-covered YES [2, 1], YES (Thm. 1) α tices a point p in the plane and having angle 2 at p, also [8] & Cor. 1 0 < α ≤ π=3. Let d be the direction of the ray at p locally γ-fat NO (Fig. 2(b)) NO (Thm. 2) that bisects the angle at p . We call such a parallel- "-area-good YES [10, 1] ? ogram a d-directional tile at p. If the four edges of a "-bound.{good YES [8, 1] ? d-directional tile at p have the same length r|that is, it is a rhombus|then we call it a d-directional r-long Table 2: Properties of the four classes of objects. tile at p. Let Γ denote the set of directions, Γ = fdi := α 10π 1 i 5 ji 2 Z; 0 ≤ i < α g. Let r(β) := 2 β sin α · diam(O). A convex object can be guarded with a single guard, and we say that a class of objects has good guardabil- Lemma 1 Let O be an (α; β){covered object. For every ity if any object in the class can be guarded by a con- point p 2 @O, there exists a direction d 2 Γ such that stant number of guards. Valtr [10] showed that "-good the d-directional r(β)-long tile at p is contained in fO n objects have good guardability. His proof, together @Og [ fpg. with the main theorem of Addario-Berry et al. [1], im- plies the same result for "-area-good objects. Kirk- Proof. Let T (p) be a good triangle at p. Since each patrick [8] showed that "-boundary-good objects have angle of T (p) is at least α, there are three consecutive good guardability. Locally γ-fat objects are known not directions di; di+1; di+2 in Γ such that for each of the to have good guardability, while (α; β)-covered objects three directions, the ray at p with that direction inter- have good guardability, see the third column of Table 2. sects the interior of T (p). Consider the intersection, I, The perimeter of a convex object is at most π times its of the disk of radius 2r(β) centered at p and the region diameter, while for general objects, the perimeter length of the plane bounded by two rays at p with directions cannot be bounded by a function of the diameter. We di and di+2. The height of T (p) is greater than 2r(β). consider a class of objects to have good perimeter-length Thus I ⊂ fT (p) n @T (p)g [ fpg. This completes the if the length of the perimeter of an object is at most a ∗In this paper, by an object we mean a closed region, O, of 2 2 constant times its diameter.
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