Fixed small bug with compilation.

approx_alg.tex

@@ -3,7 +3,8 @@
 
 \section{$1 \pm \epsilon$ Approximation Algorithm}\label{sec:algo}
 
-In \Cref{sec:hard}, we showed that computing the expected multiplicity of a compressed lineage polynomial for \ti (even just based on project-join queries), and by extension \bi (or any $\semNX$-PDB) is unlikely to be possible in linear time (\Cref{thm:mult-p-hard-result}), even if all tuples have the same probability  (\Cref{th:single-p-hard}).
+In \Cref{sec:hard}, we showed that computing the expected multiplicity of a compressed lineage polynomial for \ti (even just based on project-join queries), and by extension \bi (or more general \abbrPDB models) %any $\semNX$-PDB) 
+is unlikely to be possible in linear time (\Cref{thm:mult-p-hard-result}), even if all tuples have the same probability  (\Cref{th:single-p-hard}).
 Given this, we now design an approximation algorithm for our problem that runs in {\em linear time}.\footnote{For a very broad class of circuits: please see the discussion after \Cref{lem:val-ub} for more.}
 The folowing approximation algorithm applies to \bi, though our bounds are more meaningful for a non-trivial subclass of \bis that contains both \tis, as well as the PDBench benchmark~\cite{pdbench}.
 %it is then desirable to have an algorithm to approximate the multiplicity in linear time, which is what we describe next.

intro-rewrite-070921.tex

@@ -32,7 +32,7 @@ Note that, if $\apolyqdt$ is given, then \Cref{prob:bag-pdb-query-eval} reduces
 However, as we will show, these results also have implications for \cref{prob:bag-pdb-query-eval} when considering the cost of generating polynomials of query result tuples.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mypar{\abbrTIDBs}
+\mypar{\abbrTIDB\xplural}
 %Solving~\cref{prob:bag-pdb-query-eval} for arbitrary $\pd$ is hopeless since we need exponential space to repreent an arbitrary $\pd$.
 We initially focus on tuple-independent probabilistic bag-databases (\abbrTIDB),\BG{cite} a compressed encoding of probabilistic databases where the presence of each individual tuple (out of a total of $\numvar$ input tuples) in a possible world is modeled as an independent probabilistic event\footnote{
 This model corresponds to the classical set-relational approach to \abbrTIDB{}s, where we can handle the case of each input tuple having its own multiplicity by replacing each input tuple with as many copies as its multiplicity. To make each duplicate tuple unique in a set-\abbrTIDB we can assign unique keys across all duplicates. This increases the size of the input but this overhead is negligible when each input tuple has constant multiplicity. %$\tup$ in $\pdb$.

main.tex

@@ -84,7 +84,7 @@ sensitive=true
 
 
 
-\title{Fine-Grained Analysis of Query Evaluation Over Bag PDBs}
+\title{Parameterized and Fine-Grained Analysis of Query Evaluation Over Bag PDBs}
 \titlerunning{Bag PDB Queries}
 
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