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.ITCS.2021.70
URN: urn:nbn:de:0030-drops-136098
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Rossman, Benjamin

Shrinkage of Decision Lists and DNF Formulas

LIPIcs-ITCS-2021-70.pdf (0.6 MB)


We establish nearly tight bounds on the expected shrinkage of decision lists and DNF formulas under the p-random restriction R_p for all values of p ∈ [0,1]. For a function f with domain {0,1}ⁿ, let DL(f) denote the minimum size of a decision list that computes f. We show that E[DL(f ↾ R_p)] ≤ DL(f)^log_{2/(1-p)}((1+p)/(1-p)). For example, this bound is √{DL(f)} when p = √5-2 ≈ 0.24. For Boolean functions f, we obtain the same shrinkage bound with respect to DNF formula size plus 1 (i.e., replacing DL(⋅) with DNF(⋅)+1 on both sides of the inequality).

BibTeX - Entry

  author =	{Benjamin Rossman},
  title =	{{Shrinkage of Decision Lists and DNF Formulas}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{70:1--70:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{James R. Lee},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-136098},
  doi =		{10.4230/LIPIcs.ITCS.2021.70},
  annote =	{Keywords: shrinkage, decision lists, DNF formulas}

Keywords: shrinkage, decision lists, DNF formulas
Collection: 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)
Issue Date: 2021
Date of publication: 04.02.2021

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