TY - GEN

T1 - Small-size relative (p, ε)-approximations for well-behaved range spaces

AU - Ezra, Esther

PY - 2013

Y1 - 2013

N2 - We present improved upper bounds for the size of relative (p, ε)-approximation for range spaces with the following property: For any (finite) range space projected onto (that is, restricted to) a ground set of size n and for any parameter 1 < k < n, the number of ranges of size at most k is only nearly-linear in n and polynomial in k. Such range spaces are called "well behaved". Our bound is an improvement over the bound O (log(1/p)/ε2p) introduced by Li et al. [17] for the general case (where this bound has been shown to be tight in the worst case), when p << ε. We also show that such small size relative (p, ε)-approximations can be constructed in expected polynomial time. Our bound also has an interesting interpretation in the context of "p-nets": As observed by Har-Peled and Sharir [13], p-nets are special cases of relative (p, ε)-approximations. Specifically, when ε is a constant smaller than 1, the analysis in [13, 17] implies that there are p-nets of size O(log (1/p)/p) that are also relative approximations. In this context our construction significantly improves this bound for well-behaved range spaces. Despite the progress in the theory of p-nets and the existence of improved bounds corresponding to the cases that we study, these bounds do not necessarily guarantee a bounded relative error. Lastly, we present several geometric scenarios of well-behaved range spaces, and show the resulting bound for each of these cases obtained as a consequence of our analysis. In particular, when ε is a constant smaller than 1, our bound for points and axis-parallel boxes in two and three dimensions, as well as points and "fat" triangles in the plane, matches the optimal bound for p-nets introduced in [3, 25].

AB - We present improved upper bounds for the size of relative (p, ε)-approximation for range spaces with the following property: For any (finite) range space projected onto (that is, restricted to) a ground set of size n and for any parameter 1 < k < n, the number of ranges of size at most k is only nearly-linear in n and polynomial in k. Such range spaces are called "well behaved". Our bound is an improvement over the bound O (log(1/p)/ε2p) introduced by Li et al. [17] for the general case (where this bound has been shown to be tight in the worst case), when p << ε. We also show that such small size relative (p, ε)-approximations can be constructed in expected polynomial time. Our bound also has an interesting interpretation in the context of "p-nets": As observed by Har-Peled and Sharir [13], p-nets are special cases of relative (p, ε)-approximations. Specifically, when ε is a constant smaller than 1, the analysis in [13, 17] implies that there are p-nets of size O(log (1/p)/p) that are also relative approximations. In this context our construction significantly improves this bound for well-behaved range spaces. Despite the progress in the theory of p-nets and the existence of improved bounds corresponding to the cases that we study, these bounds do not necessarily guarantee a bounded relative error. Lastly, we present several geometric scenarios of well-behaved range spaces, and show the resulting bound for each of these cases obtained as a consequence of our analysis. In particular, when ε is a constant smaller than 1, our bound for points and axis-parallel boxes in two and three dimensions, as well as points and "fat" triangles in the plane, matches the optimal bound for p-nets introduced in [3, 25].

KW - Lovász Local Lemma

KW - Relative approximations

KW - Well-behaved range spaces

UR - http://www.scopus.com/inward/record.url?scp=84879654049&partnerID=8YFLogxK

U2 - 10.1145/2462356.2462363

DO - 10.1145/2462356.2462363

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AN - SCOPUS:84879654049

SN - 9781450320313

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 233

EP - 242

BT - Proceedings of the 29th Annual Symposium on Computational Geometry, SoCG 2013

PB - Association for Computing Machinery

T2 - 29th Annual Symposium on Computational Geometry, SoCG 2013

Y2 - 17 June 2013 through 20 June 2013

ER -