On the VC-dimension of half-spaces with respect to convex sets
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Date
2021
Publication Type
Journal Article
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Abstract
A family S of convex sets in the plane defines a hypergraph H = (S, epsilon) with S as a vertex set and epsilon as the set of hyperedges as follows. Every subfamily S' subset of S defines a hyperedge in epsilon if and only if there exists a halfspace h that fully contains S', and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in R-d, for d >= 3. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.
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published
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Volume
23 (3)
Pages / Article No.
2
Publisher
LORIA
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Subject
VC-dimension; epsilon nets; convex sets; halfplanes
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03457 - Welzl, Emo (emeritus) / Welzl, Emo (emeritus)