License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2018.82
URN: urn:nbn:de:0030-drops-90862
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9086/
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Kane, Daniel M. ; Lovett, Shachar ; Moran, Shay

Generalized Comparison Trees for Point-Location Problems

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LIPIcs-ICALP-2018-82.pdf (0.5 MB)

Abstract

Let H be an arbitrary family of hyper-planes in d-dimensions. We show that the point-location problem for H can be solved by a linear decision tree that only uses a special type of queries called generalized comparison queries. These queries correspond to hyperplanes that can be written as a linear combination of two hyperplanes from H; in particular, if all hyperplanes in H are k-sparse then generalized comparisons are 2k-sparse. The depth of the obtained linear decision tree is polynomial in d and logarithmic in |H|, which is comparable to previous results in the literature that use general linear queries.
This extends the study of comparison trees from a previous work by the authors [Kane {et al.}, FOCS 2017]. The main benefit is that using generalized comparison queries allows to overcome limitations that apply for the more restricted type of comparison queries.
Our analysis combines a seminal result of Forster regarding sets in isotropic position [Forster, JCSS 2002], the margin-based inference dimension analysis for comparison queries from [Kane {et al.}, FOCS 2017], and compactness arguments.

BibTeX - Entry

@InProceedings{kane_et_al:LIPIcs:2018:9086,
  author =	{Daniel M. Kane and Shachar Lovett and Shay Moran},
  title =	{{Generalized Comparison Trees for Point-Location Problems}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{82:1--82:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9086},
  URN =		{urn:nbn:de:0030-drops-90862},
  doi =		{10.4230/LIPIcs.ICALP.2018.82},
  annote =	{Keywords: linear decision trees, comparison queries, point location problems}
}

Keywords: linear decision trees, comparison queries, point location problems
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018

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