VC-dimension of perimeter visibility domains

https://doi.org/10.1016/j.ipl.2014.06.011Get rights and content

Highlights

  • We give an overview over the results concerning VC-dimension of range spaces defined by visibility.

  • We examine the intersections of Visibility Domains on the Perimeter of a Simple Polygon.

  • We obtain a new upper bound for the VC-dimension of Perimeter Visibility Domains.

Abstract

We obtain an upper bound of 7 for the VC-dimension of Perimeter Visibility Domains in simple polygons.

Introduction

The VC-dimension is a fundamental parameter of a range space that, intuitively speaking, measures how differently the ranges intersect with subsets of the ground set. Besides its importance in machine learning, VC-dimension became significant to computational geometry, chiefly through its role in the Epsilon Net Theorem by Haussler and Welzl [4]. This theorem states that whenever a range space has got finite VC-dimension d then there exists an ε-net for this space of size Cd1εlog1ε for some small constant C.

For many important geometric range spaces such as rectangles, circles or halfspaces, the VC-dimension is not hard to estimate. The VC-dimension of range spaces of visibility domains was first considered by Kalai and Matoušek [6]. As an application, the Epsilon Net Theorem gives an upper bound of O(rlogr) on the number of guards needed to guard a polygonal art gallery where every point sees at least an r-th part of the entire polygon. The VC-dimension of the set of visibility polygons inside polygons contributes a constant factor to this upper bound. Therefore, better upper bounds on the VC-dimension immediately yield better upper bounds on the number of guards needed. Kalai and Matoušek [6] showed that the VC-dimension of visibility polygons in a simple polygon is finite. They also gave an example of a gallery with VC-dimension 5. Furthermore, they showed that there is no constant that bounds the VC-dimension for polygons with holes. For simple polygons, Valtr [11] gave an example of a gallery with VC-dimension 6 and proved an upper bound of 23. In the same paper he showed an upper bound for the VC-dimension of a gallery with holes of O(logh) where h is the number of holes, and art galleries with holes that have VC-dimensions of this size can also be constructed. These results for galleries with holes easily carry over to the case of Perimeter Visibility Domains. In [3] Gilbers and Klein show that the VC-dimension of Visibility Polygons of a Simple Polygon is at most 14 (an extended abstract of this paper appeared in [2]). Isler et al. [5] examined the case of exterior visibility. In this setting the points of S lie on the boundary of a polygon P and the ranges are sets of the form vis(v) where v is a point outside the convex hull of P. They showed that the VC-dimension is 5. They also considered a more restricted version of exterior visibility where the view points v all must lie on a circle around P, with VC-dimension 2. For a 3-dimensional version of exterior visibility with S on the boundary of a polyhedron Q they found that the VC-dimension is in O(logn) where n is the number of vertices of Q. King [7] examined the VC-dimension of visibility regions on polygonal terrains. For 1.5-dimensional terrains he proved that the VC-dimension equals 4 and on 2.5-dimensional terrains there is no constant bound. Kirkpatrick [9] showed that it is possible to guard a polygon with O(rloglogr) many perimeter guards (i.e. guards on the boundary of the polygon) if every point on the boundary sees at least an r-th part of the boundary. In [8] King and Kirkpatrick extended this work and obtained an O(loglogOPT)-approximation algorithm for finding the minimum number of guards on the perimeter that guard the polygon. As an open question they asked whether it is easier to find the VC-dimension in the case of perimeter guards than for general visibility polygons. They show that the corresponding VC-dimension is at least 5. In this paper we show that in the case of simple polygons one can obtain an upper bound of 7 for this VC-dimension, by extending the technique from Gilbers and Klein [1].

Section snippets

VC-dimension

The following definition of VC-dimension is adopted from [10].

Definition 1

Let F be a set system on a set X. A subset SX is said to be shattered by F if each of the subsets of S can be obtained as the intersection of some FF with S. We define the VC-dimension of F, denoted by dim(F), as the supremum of the sizes of all finite shattered subsets of X. If arbitrarily large subsets can be shattered, the VC-dimension is ∞.

If a finite subset YX with |Y|=n is shattered by F, then the set ΠF(Y)={YF|FF} has 2n

Perimeter Visibility Domains

Let P be a simple polygon with boundary B. As usual, for a point pP its Visibility Polygon vis(p) is the set of points v such that the whole segment pv¯ is contained in P. We restrict our attention to the portions of visibility polygons on the boundary, vis(p)B. For every pB we will call this boundary portion its Perimeter Visibility Domain and denote it by V(p). As we are only concerned with this kind of visibility domains in this paper, we will refer to them simply as Visibility Domains,

The upper bound for the VC-dimension

Theorem 1

The VC-dimension of visibility domains on the boundary is at most 7.

Proof

We will show that there can be no set of eight boundary points that is shattered by sets of V.

Let S={s1,,s8}B (where the points are in counterclockwise order along the boundary) is a set on the boundary of P. Assume to a contradiction that S is shattered by the set of visibility domains V of points in B.

Let us define the special points l=s1 and r=s5. Let C the part of the boundary between l and r traversed in counterclockwise

Conclusion

We have shown that the VC-dimension of Perimeter Visibility Domains is at most 7. With the best known lower bound it follows that this VC-dimension is between 5 and 7.

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