Added label to Cor 4.9

approx_alg.tex

@@ -151,6 +151,7 @@ Note that the above implies that with the assumption $\prob_0>0$ and $\gamma<1$
 
 Finally, note that by \Cref{prop:circuit-depth} and \Cref{lem:circ-model-runtime} for any $\raPlus$ query $\query$, there exists a circuit $\circuit^*$ such that $\depth(\circuit^*)\le O_{|Q|}(\log{n})$ and $\size(\circuit)\le O_k\inparen{\qruntime{\query, \dbbase}}$. Using this along with \Cref{lem:val-ub}, \Cref{cor:approx-algo-const-p} and the fact that $n\le \qruntime{\query, \dbbase}$, we answer \Cref{prob:big-o-joint-steps} in the affirmative as follows:
 \begin{Corollary}
+\label{cor:approx-algo-punchline}
 Let $\query$ be an $\raPlus$ query and $\pdb$ be an \abbrBIDB with $p_0>0$ and $\gamma<1$ (where $p_0,\gamma$ as in \Cref{cor:approx-algo-const-p}) are absolute constants. Let $\poly(\vct{X})=\apolyqdt$ for any result tuple $\tup$ with $\deg(\poly)=k$. Then one can compute an approximation satisfying \Cref{eq:approx-algo-bound-main} in time $O_{k,|Q|,\error',\conf}\inparen{\qruntime{\query, \dbbase}}$ (given $\query,\dbbase$ and $p_i$ for each $i\in [n]$ that defines $\pd$).
 %Let $\poly(\vct{X})$ be a \abbrBIDB-lineage polynomial correspoding to an \abbrBIDB circuit $\circuit$ that satisfies the specific conditions in \Cref{lem:val-ub}. Then one can compute an approximation satisfying \Cref{eq:approx-algo-bound-main} in time
 % $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$. % for the case when $\circuit$ satisfies the specific conditions in \Cref{lem:val-ub}. 

rebuttal.tex

@@ -2,22 +2,29 @@
 %!TEX root=./main.tex
 \definecolor{GrayRew}{gray}{0.85}
 \newcommand{\RCOMMENT}[1]{\medskip\noindent \begin{tabular}{|p{\linewidth-3ex}}\rowcolor{GrayRew} #1 \end{tabular}\smallskip\\}
-\section{Rebuttal}
-This paper is a resubmission; As such, we use this section to document the changes that have been made since our prior submission, and in particular, how we have addressed reviewer critiques.
+\section{Response to first cycle reviewer comments}
+This paper is a resubmission of our submmission to the ICDT first cycle. We thank the reviewers for their insightful comments, which we believe has helped improve the presentation of the paper tremendously.
+We use this section to document the changes that have been made since our prior submission, and in particular, how we have addressed reviewer comments (reviewer comments are shaded followed by our responses).
 
 \subsection{Meta Review}
 \RCOMMENT{Problem definition not stated rigorously nor motivated. Discussion needed on the standard PDB approach vs your approach.}
-We made the decision to rewrite \Cref{sec:intro} to specifically address this concern.  The opening paragraph precisely and formally states the query evaluation problem in \abbrBPDB\xplural.  We use a series of problem statements to clearly define the problem we are addressing as it relates to the query evaluation problem.  We have included significant discussion of the standard approach, e.g. see the paragraph \textbf{Relationship to Set-Probabilistic Query Evaluation} on page 4.
+We rewrote \Cref{sec:intro} to specifically address this concern.  The opening paragraph precisely and formally states the query evaluation problem in \abbrBPDB\xplural.  We use a series of problem statements to clearly define the problem we are addressing as it relates to the query evaluation problem.  
+ We  made the concrete problem statements more precise by more clearly formalizing $\qruntime{Q, \dbbase}$ and stating our runtime objectives relative to it \AR{Dangling notation or Cref not being able to figure out `Sections'?}(\Cref{prob:informal,prob:big-o-joint-steps,prob:intro-stmt})
+%Notably, explicit discussion of provenance polynomials is limited to the proofs in the appendices.
+
+We have included ia discussion of the standard approach, e.g. see the paragraph \textbf{Relationship to Set-Probabilistic Query Evaluation} on page 4.
 
 \RCOMMENT{Definition 2.6 on reduced BIDB polynomials seem not the right tool for the studied problem.}
-We have chosen to stick with a less formal, ad-hoc definition (please see \Cref{def:reduced-poly} and \Cref{def:reduced-bi-poly}) of the general problem as suggested by both Reviewer 1 and Reviewer 2, and made the concrete problem statement more precise by more clearly formalizing $\qruntime{Q, \dbbase}$ and stating our runtime objectives relative to it (\Cref{prob:informal,prob:big-o-joint-steps,prob:intro-stmt})
-Notably, explicit discussion of provenance polynomials is limited to the proofs in the appendices.
+We have chosen to stick with a less formal, ad-hoc definition (please see \Cref{def:reduced-poly} and \Cref{def:reduced-bi-poly}) of the general problem as suggested by both Reviewer 1 and Reviewer 2. Our earlier proof of the current \Cref{lem:exp-poly-rpoly} (in the appendix) had a small bug, which also has been fixed.
 
 \RCOMMENT{The paper is very difficult to read. Improvements are needed in particular for the presentation of the approximation results and their proofs. Also for the notation. Missing definitions for used notions need to be added. Ideally use one instead of three query languages (UCQ, RA+, SPJU).}
-\AH{How have we handled the presentation of the approximation results and their proofs?}
+%\AH{How have we handled the presentation of the approximation results and their proofs?}
+%\AR{Added text at the end of the answer
 We have chosen one specific query language throughout the paper ($\raPlus$) and made a concerted effort to use clean, defined, non-ambiguous notation.
 We have also simplified the notation by limiting the paper's use of provenance semirings (which are needed solely for proofs) to the appendix.
-To the best of our examination, all notation conflicts have been addressed and definitions for used notions are added (see e.g. \Cref{def:Gk} appears before \Cref{lem:3m-G2} and \Cref{lem:lin-sys}.
+To the best of our examination, all notation conflicts have been addressed and definitions for used notions are added (see e.g. \Cref{def:Gk} appears before \Cref{lem:3m-G2} and \Cref{lem:lin-sys}).
+
+After the rewrite of \Cref{sec:intro}, we had even less space for \Cref{sec:algo}. However, we have modified \Cref{sec:algo} so that it flows better. In particular, we start off with the algorithm idea first (paragraph \textbf{Overview of our Techniques} in \Cref{sec:intro} also has more details on the intuition behind the approximation algorithm) and then state the results (with more details on how we argue the claimed runtime). Finally, we clearly state \Cref{cor:approx-algo-punchline} for which queries our linear-time approximation result holds.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Reviewer 1}