Changed \idb to overline{Omega} to avoid clash with asymptotic notation

intro-rewrite-070921.tex

@@ -222,6 +222,8 @@ The second step,  \termStepTwo (\abbrStepTwo) consists of computing $\expct\pbox
 For \abbrBPDB $\pdb$, query $\query$, let $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit)$ denote the runtime of \abbrStepOne, when it outputs $\circuit$ (which is a representation of $\poly$-- more on this representation shortly).
 Let us denote by $\timeOf{\abbrStepTwo}(\circuit)$ (recall $\circuit$ is the output of \abbrStepOne) the runtime of \abbrStepTwo, allowing us to formally define our objective:
 
+\input{two-step-model}
+
 \begin{Problem}\label{prob:big-o-joint-steps}
 Given \abbrBPDB $\pdb$, $\raPlus$ query $\query$ and output tuple $\tup$, 
 does there exist a $(1\pm\epsilon)$-approximation of $\expct_{\db\sim\pd}\pbox{\query\inparen{\db}\inparen{\tup}}$ (for all resuult tuples $\tup$) for some $\circuit$  such that
@@ -229,10 +231,14 @@ $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit) + \timeOf{\abbrStepTwo}(\circuit) \le
 \end{Problem}
 Note that if the answer to the above problem is yes, then we have shown that the answer to \Cref{prob:informal} is yes (when we are interested in approximating the expected muktiplities).
 
-We show in \Cref{sec:circuit-runtime}\OK{confirm this ref} an $O(\qruntime{Q, \dbbase})$ algorithm for constructing the lineage polynomial of the singleton result tuple of a count query.
+We show in \Cref{sec:gen}
+%\OK{confirm this ref}
+%Atri: fixed the ref
+ an $O(\qruntime{Q, \dbbase})$ algorithm for constructing the lineage polynomial of result tuples of an $\raPlus$ query $\query$ (or more preicsely its representation $\circuit$).
 % , and by extension the first step is in \sharpwonehard\AH{\sharpwonehard is not defined.}.
-A key insight of this paper is that the representation matters;
-One can have compact representations of $\poly(\vct{X})$ (e.g., resulting from optimizations like projection push-down~\cite{DBLP:books/daglib/0020812}, which produce factorized representations of $\poly(\vct{X})$.
+A key insight of this paper is that the representation of $\circuit$ matters. For example if we insist that $\circuit$ represent the lineage polynomial in the standard monomial basis (henceforth, \abbrSMB)\footnote{This is the representation where the polynomial is reresented as sum of `pure' products-- see \Cref{def:smb} for a formal definition.}, the answer to the above question in general is no, since then we will need $\abs{\circuit}\ge \Omega\inparen{\inparen{\qruntime{Q, \dbbase}}^k}$, and hence, just $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit)$ will be too large.
+
+However, one can have compact representations of $\poly(\vct{X})$ (e.g., resulting from optimizations like projection push-down~\cite{DBLP:books/daglib/0020812}, which produce factorized representations of $\poly(\vct{X})$.
 For example, in~\Cref{fig:two-step}, $B(Y+Z)$ is a factorized representation of the SMB-form $BY+BZ$.  To capture such factorizations, this work uses (arithmetic) circuits\footnote{An arithmetic circuit has variable and/or numeric inputs, with internal nodes representing either an addition or multiplication operator.}
 as the representation system of $\poly(\vct{X})$.
 
@@ -248,7 +254,6 @@ as the representation system of $\poly(\vct{X})$.
 % In this case, we have for any output tuple $\tup$, $\expct\pbox{\poly(\vct{W})}=\Phi(1,\dots,1)$.
 % Thus, we have another case where $\timeOf{\abbrStepTwo}(Q,\pdb)$ is $\bigO{\timeOf{\abbrStepOne}(Q,\pdb)}$ and we again achieve deterministic query runtime for $\query\inparen{\pdb}$ (up to a constant factor).  These observations introduce our first formalization of~\Cref{prob:informal}:
 
-\input{two-step-model}
 
 Given $\timeOf{\abbrStepOne}(Q,\pdb) = O(\qruntime{Q, \dbbase})$, we can now focus on the complexity of \abbrStepTwo.
 We can represent the factorized lineage polynomial by the size of its correspoding arithmetic circuit $\circuit$ (which we denote by $|\circuit|$).

macros.tex

@@ -109,7 +109,7 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % Incomplete DB/PDBs           															      %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\newcommand{\idb}{\Omega}
+\newcommand{\idb}{{\overline{\Omega}}}
 \newcommand{\pd}{{\mathcal{P}_{\idb}}}%pd for probability distribution
 \newcommand{\pdassign}{\mathcal{P}}
 \newcommand{\pdb}{\mathcal{D}}