From 7ab8af7124b81fa35f54e8cf648faf1c461d4fb9 Mon Sep 17 00:00:00 2001 From: Aaron Huber Date: Thu, 17 Mar 2022 10:23:12 -0400 Subject: [PATCH] Switched appendix to \onecolumn; fixed nits @atri pointed out. --- acknowledgements.tex | 1 - app_approx-alg-analysis.tex | 2 +- app_onepass-analysis.tex | 2 +- app_set-to-bag-pdb.tex | 3 +- main.bbl | 748 +++++++++++++++++++++++++----------- main.bib | 16 +- main.out | 83 ++-- main.pdf | Bin 1017505 -> 1009700 bytes main.synctex.gz | Bin 831008 -> 809595 bytes 9 files changed, 571 insertions(+), 284 deletions(-) diff --git a/acknowledgements.tex b/acknowledgements.tex index c025e94..6fd24a9 100644 --- a/acknowledgements.tex +++ b/acknowledgements.tex @@ -1,3 +1,2 @@ %root: main.tex -\section{Acknowledgements} We thank Virginia Williams for showing us \Cref{eq:3p-3tri}, which greatly simplified our earlier proof of Lemma 3.8, and for graciously allowing us to use it. \ No newline at end of file diff --git a/app_approx-alg-analysis.tex b/app_approx-alg-analysis.tex index 3e4cac2..3f69ccc 100644 --- a/app_approx-alg-analysis.tex +++ b/app_approx-alg-analysis.tex @@ -94,7 +94,7 @@ For the claimed probability bound of $\probOf\left(\left|\vari{acc} - \rpoly(\pr This concludes the proof for the first claim of theorem~\ref{lem:mon-samp}. Next, we prove the claim on the runtime. \paragraph*{Run-time Analysis} -The runtime of the algorithm is dominated first by \Cref{alg:mon-sam-onepass} (which by \Cref{lem:one-pass} takes time $O\left({\size(\circuit)}\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$) and then by $\samplesize$ iterations of the loop in \Cref{alg:sampling-loop}. Each iteration's run time is dominated by the call to \sampmon in \Cref{alg:mon-sam-sample} (which by \Cref{lem:sample} takes $O\left(\log{k} \cdot k \cdot {\depth(\circuit)}\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$ +The runtime of the algorithm is dominated first by \Cref{alg:mon-sam-onepass} which has $O\left({\size(\circuit)}\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$ runtime by \Cref{lem:one-pass}. There are then $\samplesize$ iterations of the loop in \Cref{alg:sampling-loop}. Each iteration's run time is dominated by the call to \sampmon in \Cref{alg:mon-sam-sample} (which by \Cref{lem:sample} takes $O\left(\log{k} \cdot k \cdot {\depth(\circuit)}\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$ ) and the check \Cref{alg:check-duplicate-block}, which by the subsequent argument takes $O(k\log{k})$ time. We sort the $O(k)$ variables by their block IDs and then check if there is a duplicate block ID or not. Combining all the times discussed here gives us the desired overall runtime. \qed \end{proof} diff --git a/app_onepass-analysis.tex b/app_onepass-analysis.tex index afe6afe..9d2f898 100644 --- a/app_onepass-analysis.tex +++ b/app_onepass-analysis.tex @@ -90,7 +90,7 @@ When $\gate_{k+1}.\type = \circplus$, then by line~\ref{alg:one-pass-plus} $\gat When $\gate_{k+1}.\type = \circmult$, then line~\ref{alg:one-pass-mult} computes $\gate_{k+1}.\prt = \gate_{{k+1}_\lchild.\prt} \circmult \gate_{{k+1}_\rchild}.\prt$, which indeed by \cref{eq:T-all-ones} is correct. This concludes the proof of correctness. \paragraph*{Runtime Analysis} -It is known that $\topord(G)$ is computable in linear time. There are $\size(\circuit)$ iterations, each of which takes $O\left( \multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log\inparen{\size(\circuit)}}\right)$ time. This can be seen since each of all the numbers which the algorithm computes is at most $\abs{\circuit}(1,\dots,1)$. Hence, by definition each such operation takes $\multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log{\size(\circuit)}}$ time, which proves the claimed runtime. +It is known that $\topord(G)$ is computable in linear time. There are $\size(\circuit)$ iterations. Each iteration has runtime $O\left( \multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log\inparen{\size(\circuit)}}\right)$ time. This can be seen since each of all the numbers which the algorithm computes is at most $\abs{\circuit}(1,\dots,1)$. Hence, by definition each such operation takes $\multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log{\size(\circuit)}}$ time, which proves the claimed runtime. \qed \end{proof} diff --git a/app_set-to-bag-pdb.tex b/app_set-to-bag-pdb.tex index 59e3e6e..73cdc00 100644 --- a/app_set-to-bag-pdb.tex +++ b/app_set-to-bag-pdb.tex @@ -1,5 +1,6 @@ - +\onecolumn \section{Generalizing Beyond Set Inputs} + \label{sec:gener-results-beyond} \subsection{\abbrTIDB{}s} diff --git a/main.bbl b/main.bbl index 410f993..4fac178 100644 --- a/main.bbl +++ b/main.bbl @@ -1,285 +1,569 @@ -\begin{thebibliography}{10} +%%% -*-BibTeX-*- +%%% Do NOT edit. File created by BibTeX with style +%%% ACM-Reference-Format-Journals [18-Jan-2012]. -\bibitem{pdbench} -pdbench. -\newblock \url{http://pdbench.sourceforge.net/}. -\newblock Accessed: 2020-12-15. +\begin{thebibliography}{48} -\bibitem{DBLP:conf/vldb/AgrawalBSHNSW06} -Parag Agrawal, Omar Benjelloun, Anish~Das Sarma, Chris Hayworth, Shubha~U. - Nabar, Tomoe Sugihara, and Jennifer Widom. -\newblock Trio: A system for data, uncertainty, and lineage. -\newblock In {\em VLDB}, pages 1151--1154, 2006. +%%% ==================================================================== +%%% NOTE TO THE USER: you can override these defaults by providing +%%% customized versions of any of these macros before the \bibliography +%%% command. Each of them MUST provide its own final punctuation, +%%% except for \shownote{}, \showDOI{}, and \showURL{}. The latter two +%%% do not use final punctuation, in order to avoid confusing it with +%%% the Web address. +%%% +%%% To suppress output of a particular field, define its macro to expand +%%% to an empty string, or better, \unskip, like this: +%%% +%%% \newcommand{\showDOI}[1]{\unskip} % LaTeX syntax +%%% +%%% \def \showDOI #1{\unskip} % plain TeX syntax +%%% +%%% ==================================================================== -\bibitem{AB15} -Antoine Amarilli, Pierre Bourhis, and Pierre Senellart. -\newblock Probabilities and provenance via tree decompositions. -\newblock {\em PODS}, 2015. +\ifx \showCODEN \undefined \def \showCODEN #1{\unskip} \fi +\ifx \showDOI \undefined \def \showDOI #1{#1}\fi +\ifx \showISBNx \undefined \def \showISBNx #1{\unskip} \fi +\ifx \showISBNxiii \undefined \def \showISBNxiii #1{\unskip} \fi +\ifx \showISSN \undefined \def \showISSN #1{\unskip} \fi +\ifx \showLCCN \undefined \def \showLCCN #1{\unskip} \fi +\ifx \shownote \undefined \def \shownote #1{#1} \fi +\ifx \showarticletitle \undefined \def \showarticletitle #1{#1} \fi +\ifx \showURL \undefined \def \showURL {\relax} \fi +% The following commands are used for tagged output and should be +% invisible to TeX +\providecommand\bibfield[2]{#2} +\providecommand\bibinfo[2]{#2} +\providecommand\natexlab[1]{#1} +\providecommand\showeprint[2][]{arXiv:#2} -\bibitem{AD11d} -Yael Amsterdamer, Daniel Deutch, and Val Tannen. -\newblock Provenance for aggregate queries. -\newblock In {\em PODS}, pages 153--164, 2011. +\bibitem[pdb([n.\,d.])]% + {pdbench} + \bibinfo{year}{[n.\,d.]}\natexlab{}. +\newblock \bibinfo{title}{pdbench}. +\newblock \bibinfo{howpublished}{\url{http://pdbench.sourceforge.net/}}. +\newblock +\newblock +\shownote{Accessed: 2020-12-15}. -\bibitem{Antova_fastand} -Lyublena Antova, Thomas Jansen, Christoph Koch, and Dan Olteanu. -\newblock Fast and simple relational processing of uncertain data. -\bibitem{AGM} -Albert Atserias, Martin Grohe, and D{\'{a}}niel Marx. -\newblock Size bounds and query plans for relational joins. -\newblock {\em {SIAM} J. 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Ngo}, {and} \bibinfo{person}{Atri Rudra}.} + \bibinfo{year}{2016}\natexlab{}. +\newblock \showarticletitle{FAQ: Questions Asked Frequently}. In + \bibinfo{booktitle}{\emph{PODS}}. \bibinfo{pages}{13--28}. +\newblock + + +\bibitem[Kopelowitz and Williams(2020)]% + {triang-hard} +\bibfield{author}{\bibinfo{person}{Tsvi Kopelowitz} {and} + \bibinfo{person}{Virginia~Vassilevska Williams}.} + \bibinfo{year}{2020}\natexlab{}. +\newblock \showarticletitle{Towards Optimal Set-Disjointness and + Set-Intersection Data Structures}. 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Ilyas}, {and} \bibinfo{person}{Christopher + R{\'{e}}}.} \bibinfo{year}{2017}\natexlab{}. +\newblock \showarticletitle{HoloClean: Holistic Data Repairs with Probabilistic + Inference}. +\newblock \bibinfo{journal}{\emph{Proc. {VLDB} Endow.}} \bibinfo{volume}{10}, + \bibinfo{number}{11} (\bibinfo{year}{2017}), \bibinfo{pages}{1190--1201}. +\newblock + + +\bibitem[Roy et~al\mbox{.}(2011)]% + {roy-11-f} +\bibfield{author}{\bibinfo{person}{Sudeepa Roy}, \bibinfo{person}{Vittorio + Perduca}, {and} \bibinfo{person}{Val Tannen}.} + \bibinfo{year}{2011}\natexlab{}. +\newblock \showarticletitle{Faster query answering in probabilistic databases + using read-once functions}. In \bibinfo{booktitle}{\emph{ICDT}}. +\newblock + + +\bibitem[Ré and Suciu(2007)]% + {RS07} +\bibfield{author}{\bibinfo{person}{C. Ré} {and} \bibinfo{person}{D. Suciu}.} + \bibinfo{year}{2007}\natexlab{}. +\newblock \showarticletitle{Materialized views in probabilistic databases: for + information exchange and query optimization}. 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Valiant}.} + \bibinfo{year}{1979}\natexlab{}. +\newblock \showarticletitle{The Complexity of Enumeration and Reliability + Problems}. +\newblock \bibinfo{journal}{\emph{SIAM J. 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Ngo and diff --git a/main.out b/main.out index c131781..4e8a3b5 100644 --- a/main.out +++ b/main.out @@ -17,45 +17,44 @@ \BOOKMARK [1][-]{section.5}{\376\377\0005\000\040\000R\000e\000l\000a\000t\000e\000d\000\040\000W\000o\000r\000k}{}% 17 \BOOKMARK [1][-]{section.6}{\376\377\0006\000\040\000C\000o\000n\000c\000l\000u\000s\000i\000o\000n\000s\000\040\000a\000n\000d\000\040\000F\000u\000t\000u\000r\000e\000\040\000W\000o\000r\000k}{}% 18 \BOOKMARK [1][-]{section*.11}{\376\377\000A\000c\000k\000n\000o\000w\000l\000e\000d\000g\000m\000e\000n\000t\000s}{}% 19 -\BOOKMARK [1][-]{section.7}{\376\377\0007\000\040\000A\000c\000k\000n\000o\000w\000l\000e\000d\000g\000e\000m\000e\000n\000t\000s}{}% 20 -\BOOKMARK [1][-]{section*.13}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 21 -\BOOKMARK 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