Switched appendix to \onecolumn; fixed nits @atri pointed out.
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%root: main.tex
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\section{Acknowledgements}
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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.
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@ -94,7 +94,7 @@ For the claimed probability bound of $\probOf\left(\left|\vari{acc} - \rpoly(\pr
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This concludes the proof for the first claim of theorem~\ref{lem:mon-samp}. Next, we prove the claim on the runtime.
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\paragraph*{Run-time Analysis}
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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 $O\left(\log{k} \cdot k \cdot {\depth(\circuit)}\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$
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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 $O\left(\log{k} \cdot k \cdot {\depth(\circuit)}\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$
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) 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.
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\qed
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\end{proof}
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@ -90,7 +90,7 @@ When $\gate_{k+1}.\type = \circplus$, then by line~\ref{alg:one-pass-plus} $\gat
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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.
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\paragraph*{Runtime Analysis}
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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.
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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.
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\qed
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\end{proof}
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@ -1,5 +1,6 @@
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\onecolumn
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\section{Generalizing Beyond Set Inputs}
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\label{sec:gener-results-beyond}
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\subsection{\abbrTIDB{}s}
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748
main.bbl
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main.bbl
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\begin{thebibliography}{10}
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%%% -*-BibTeX-*-
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%%% Do NOT edit. File created by BibTeX with style
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%%% ACM-Reference-Format-Journals [18-Jan-2012].
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\bibitem{pdbench}
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pdbench.
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\newblock \url{http://pdbench.sourceforge.net/}.
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\newblock Accessed: 2020-12-15.
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\begin{thebibliography}{48}
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\bibitem{DBLP:conf/vldb/AgrawalBSHNSW06}
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Parag Agrawal, Omar Benjelloun, Anish~Das Sarma, Chris Hayworth, Shubha~U.
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Nabar, Tomoe Sugihara, and Jennifer Widom.
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\newblock Trio: A system for data, uncertainty, and lineage.
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\newblock In {\em VLDB}, pages 1151--1154, 2006.
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\bibitem{AB15}
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Antoine Amarilli, Pierre Bourhis, and Pierre Senellart.
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\newblock {\em PODS}, 2015.
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\bibitem{AD11d}
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Yael Amsterdamer, Daniel Deutch, and Val Tannen.
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\newblock Provenance for aggregate queries.
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\newblock In {\em PODS}, pages 153--164, 2011.
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\bibitem[pdb([n.\,d.])]%
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{pdbench}
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\bibinfo{year}{[n.\,d.]}\natexlab{}.
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\newblock
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\newblock
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\shownote{Accessed: 2020-12-15}.
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\bibitem{Antova_fastand}
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Lyublena Antova, Thomas Jansen, Christoph Koch, and Dan Olteanu.
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\newblock Fast and simple relational processing of uncertain data.
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\bibitem{AGM}
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\newblock Size bounds and query plans for relational joins.
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\newblock {\em {SIAM} J. Comput.}, 42(4):1737--1767, 2013.
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\newblock \href {https://doi.org/10.1137/110859440}
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{\path{doi:10.1137/110859440}}.
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\bibitem[Agrawal et~al\mbox{.}(2006)]%
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{DBLP:conf/vldb/AgrawalBSHNSW06}
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\bibfield{author}{\bibinfo{person}{Parag Agrawal}, \bibinfo{person}{Omar
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Benjelloun}, \bibinfo{person}{Anish~Das Sarma}, \bibinfo{person}{Chris
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Hayworth}, \bibinfo{person}{Shubha~U. Nabar}, \bibinfo{person}{Tomoe
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Sugihara}, {and} \bibinfo{person}{Jennifer Widom}.}
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\bibinfo{year}{2006}\natexlab{}.
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\newblock \showarticletitle{Trio: A System for Data, Uncertainty, and Lineage}.
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In \bibinfo{booktitle}{\emph{VLDB}}. \bibinfo{pages}{1151--1154}.
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\newblock
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\bibitem{bahar-93-al}
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\newblock In {\em IEEE CAD}, 1993.
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\bibitem{DBLP:journals/pvldb/BeskalesIG10}
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\newblock {\em Proc. {VLDB} Endow.}, 3(1):197--207, 2010.
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\bibitem[Amarilli et~al\mbox{.}(2015)]%
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{AB15}
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\bibfield{author}{\bibinfo{person}{Antoine Amarilli}, \bibinfo{person}{Pierre
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Bourhis}, {and} \bibinfo{person}{Pierre Senellart}.}
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\bibinfo{year}{2015}\natexlab{}.
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\newblock \showarticletitle{Probabilities and provenance via tree
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decompositions}.
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\newblock \bibinfo{journal}{\emph{PODS}} (\bibinfo{year}{2015}).
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\newblock
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\bibitem{arith-complexity}
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\newblock {\em Algebraic complexity theory}, volume 315.
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{AD11d}
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\bibfield{author}{\bibinfo{person}{Yael Amsterdamer}, \bibinfo{person}{Daniel
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\bibinfo{year}{2011}\natexlab{}.
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\newblock \showarticletitle{Provenance for Aggregate Queries}. In
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\bibinfo{booktitle}{\emph{PODS}}. \bibinfo{pages}{153--164}.
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\newblock URL:
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\url{https://www.sciencedirect.com/science/article/pii/S0022000006000675},
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\href {https://doi.org/https://doi.org/10.1016/j.jcss.2006.04.007}
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{\path{doi:https://doi.org/10.1016/j.jcss.2006.04.007}}.
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\bibitem{k-match}
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\bibfield{author}{\bibinfo{person}{Albert Atserias}, \bibinfo{person}{Martin
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Grohe}, {and} \bibinfo{person}{D{\'{a}}niel Marx}.}
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\bibinfo{year}{2013}\natexlab{}.
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||||
\newblock \showarticletitle{Size Bounds and Query Plans for Relational Joins}.
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||||
\newblock \bibinfo{journal}{\emph{{SIAM} J. Comput.}} \bibinfo{volume}{42},
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\bibinfo{number}{4} (\bibinfo{year}{2013}), \bibinfo{pages}{1737--1767}.
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\newblock
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\urldef\tempurl%
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\url{https://doi.org/10.1137/110859440}
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\showDOI{\tempurl}
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||||
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||||
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||||
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||||
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||||
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||||
|
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|
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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||||
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||||
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|
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||||
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||||
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||||
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||||
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||||
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||||
{GT06}
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||||
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||||
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||||
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||||
{IL84a}
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||||
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||||
\bibinfo{person}{Witold Lipski~Jr}.} \bibinfo{year}{1984}\natexlab{}.
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||||
\newblock \showarticletitle{Incomplete Information in Relational Databases}.
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||||
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||||
{jha-12-pdwm}
|
||||
\bibfield{author}{\bibinfo{person}{Abhay~Kumar Jha} {and} \bibinfo{person}{Dan
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||||
Suciu}.} \bibinfo{year}{2012}\natexlab{}.
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||||
{DBLP:journals/jal/KarpLM89}
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||||
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|
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||||
Enumeration Problems}.
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||||
{kenig-13-nclexpdc}
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|
||||
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||||
Probabilistic Databases Computable in P-Time}. In
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\bibinfo{booktitle}{\emph{SUM}}, Vol.~\bibinfo{volume}{8078}.
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||||
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\bibinfo{person}{Hung~Q. Ngo}, {and} \bibinfo{person}{Atri Rudra}.}
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\bibinfo{booktitle}{\emph{PODS}}. \bibinfo{pages}{13--28}.
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||||
{triang-hard}
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||||
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\bibinfo{person}{Virginia~Vassilevska Williams}.}
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Set-Intersection Data Structures}. In \bibinfo{booktitle}{\emph{ICALP}},
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Vol.~\bibinfo{volume}{168}. \bibinfo{pages}{74:1--74:16}.
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||||
{kumari:2016:qdb:communicating}
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||||
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||||
Achmiz}, {and} \bibinfo{person}{Oliver Kennedy}.}
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{ngo-survey}
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||||
\bibfield{author}{\bibinfo{person}{Hung~Q. Ngo}.}
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||||
Results, and Open Problems}. In \bibinfo{booktitle}{\emph{PODS}}.
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||||
{skew}
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||||
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|
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||||
of join algorithms}.
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\bibinfo{number}{4} (\bibinfo{year}{2013}), \bibinfo{pages}{5--16}.
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||||
Huang}, {and} \bibinfo{person}{Christoph Koch}.}
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16
main.bib
16
main.bib
|
|
@ -1,3 +1,13 @@
|
|||
@INPROCEEDINGS{4497507,
|
||||
author={Antova, Lyublena and Jansen, Thomas and Koch, Christoph and Olteanu, Dan},
|
||||
booktitle={2008 IEEE 24th International Conference on Data Engineering},
|
||||
title={Fast and Simple Relational Processing of Uncertain Data},
|
||||
year={2008},
|
||||
volume={},
|
||||
number={},
|
||||
pages={983-992},
|
||||
doi={10.1109/ICDE.2008.4497507}}
|
||||
|
||||
@article{CHEN20061346,
|
||||
title = {Strong computational lower bounds via parameterized complexity},
|
||||
journal = {Journal of Computer and System Sciences},
|
||||
|
|
@ -119,12 +129,6 @@ Decompositions},
|
|||
year = {2007}
|
||||
}
|
||||
|
||||
@misc{Antova_fastand,
|
||||
author = {Lyublena Antova and Thomas Jansen and Christoph Koch and Dan Olteanu},
|
||||
title = {Fast and Simple Relational Processing of Uncertain Data},
|
||||
year = {}
|
||||
}
|
||||
|
||||
@inproceedings{DBLP:conf/pods/KhamisNR16,
|
||||
author = {Mahmoud Abo Khamis and
|
||||
Hung Q. Ngo and
|
||||
|
|
|
|||
83
main.out
83
main.out
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|
@ -17,45 +17,44 @@
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|||
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