NDLI: Approximate Query Complexity

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Approximate Query Complexity

Content Provider	ACM Digital Library
Author	Smyth, Clifford
Copyright Year	2011
Abstract	Let $\textit{f}$ : {0, $1}\textit{n}$ → {0, 1}. Let $\textit{μ}$ be a product probability measure on {0, $1}\textit{n}.$ For $\textit{ε}$ ≥ 0, we define $\textit{Dε}(\textit{f}),$ the $\textit{ε-approximate}$ decision tree complexity of $\textit{f},$ to be the minimum depth of a decision tree $\textit{T}$ with $\textit{μ}(\textit{T}(\textit{x})$ ≠ $\textit{f}(\textit{x}))$ ≤ $\textit{ε}.$ For $\textit{j}$ = 0 or 1 and for $\textit{Δ}$ ≥ 0, we define $\textit{Cj,Δ}(\textit{f}),$ the $\textit{Δ-approximate}$ $\textit{j}-certificate$ complexity of $\textit{f},$ to be the minimum certificate complexity of a set $\textit{A}$ ⊆ $\textit{Ω}$ with $μ(AΔf^{™1}(j))$ ≤ $\textit{ε}.$ Note that if $\textit{μ}(\textit{x})$ > 0 for all $\textit{x}$ then $D_{0}(f)$ = $\textit{D}(\textit{f})$ and $Cj_{,0}(f)$ = $\textit{Cj}(\textit{f})$ are the ordinary decision tree and $\textit{j}-certificate$ complexities of $\textit{f},$ respectively. We extend the well-known result, $\textit{D}(\textit{f})$ ≤ $C_{1}(f)C_{0}(f)$ [Blum and Impagliazzo 1987; Hartmanis and Hemachandra 1991; Tardos 1989], proving that for all $\textit{ε}$ > 0 there exists a $\textit{Δ}$ > 0 and a constant $\textit{K}$ = $\textit{K}(\textit{ε},$ $\textit{Δ})$ > 0 such that for all $\textit{n},$ $\textit{μ},$ $\textit{f},$ $\textit{Dε}(\textit{f})$ ≤ K $C_{1,}Δ(f)C_{0,}Δ$ $(\textit{f}).$ We also give a partial answer to a related question on query complexity raised by Tardos [1989]. We prove generalizations of these results to general product probability spaces.
Starting Page	1
Ending Page	11
Page Count	11
File Format	PDF
ISSN	19423454
e-ISSN	19423462
DOI	10.1145/2003685.2003688
Volume Number	3
Issue Number	1
Journal	ACM Transactions on Computation Theory (TOCT)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2011-08-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Approximate certificate complexity Approximate decision tree complexity Certificate complexity Communication complexity Decision tree complexity
Content Type	Text
Resource Type	Article
Subject	Computational Theory and Mathematics Theoretical Computer Science

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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