Minor tweaks to S.4

approx_alg.tex

@@ -5,7 +5,7 @@
 We showed in~\Cref{sec:hard} that a runtime of $\bigO{\qruntime{\optquery{\query},\tupset,\bound}}$ cannot be acheived for~\Cref{prob:bag-pdb-poly-expected}.  In light of this, we desire to produce an approximation algorithm that runs in time $\bigO{\qruntime{\optquery{\query},\tupset,\bound}}$.  We do this by showing the result via circuits,
 such that our approximation algorithm for this problem runs in $\bigO{\abs{\circuit}}$ for a very broad class of circuits, (thus affirming~\Cref{prob:intro-stmt});  see the discussion after \Cref{lem:val-ub} for more.
 The following approximation algorithm applies to bag query semantics over both
-\abbrCTIDB lineage polynomials and general \abbrBIDB lineage polynomials in practice, where for the latter we note that a $1$-\abbrTIDB is equivalently a \abbrBIDB (blocks are size $1$).  Our experimental results (see~\Cref{app:subsec:experiment}) which use queries from the PDBench benchmark~\cite{pdbench} show a low $\gamma$ (see~\Cref{def:param-gamma}) supporting the notion that our bounds hold for general \abbrBIDB in practice.
+\abbrCTIDB lineage polynomials and general \abbrBIDB lineage polynomials in practice, where for the latter we note that a $1$-\abbrTIDB is equivalently a \abbrBIDB (blocks are size $1$).  Our experimental results (see~\Cref{app:subsec:experiment}) which use queries from the PDBench benchmark~\cite{pdbench} show a low $\gamma$ (see~\Cref{def:param-gamma}) supporting the notion that our bounds hold for general \abbrBIDB\xplural in practice.
 
 Corresponding proofs and pseudocode for all formal statements and algorithms
   can be found in \Cref{sec:proofs-approx-alg}.
@@ -70,7 +70,7 @@ We solve~\Cref{prob:intro-stmt} for any fixed $\epsilon > 0$ in what follows.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mypar{Algorithm Idea}
-Our approximation algorithm (\approxq pseudo code in \Cref{sec:proof-lem-approx-alg})
+Our approximation algorithm (\approxq) % pseudo code in \Cref{sec:proof-lem-approx-alg})
 is based on the following observation.
 Given a lineage polynomial $\poly(\vct{X})=\polyf(\circuit)$ for circuit \circuit over 
 \abbrOneBIDB (recall that all \abbrCTIDB can be reduced to \abbrOneBIDB by~\Cref{prop:ctidb-reduct}), we have: 
@@ -113,7 +113,7 @@ In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the abo
 
 The restriction on $\gamma$ is satisfied by any
 $1$-\abbrTIDB (where $\gamma=0$ in the equivalent $1$-\abbrBIDB of~\Cref{prop:ctidb-reduct})
-as well as for all three queries of the PDBench \abbrBIDB benchmark (see \Cref{app:subsec:experiment} for experimental results). Further, we can also argue the following result, recalling from~\Cref{sec:intro} for \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, where $\tupset$ is the set of possible tuples across all possible worlds of $\pdb$.
+as well as for all three queries of the PDBench \abbrBIDB benchmark (\Cref{app:subsec:experiment}). Further, we can also argue the following result, recalling from~\Cref{sec:intro} for \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, where $\tupset$ is the set of possible tuples across all possible worlds of $\pdb$.
 
 \begin{Lemma}
 \label{lem:ctidb-gamma}
@@ -136,14 +136,14 @@ 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 $\circuit^*$ for $\apolyqdt$ such that $\depth(\circuit^*)\le O_{|Q|}(\log{n})$ and $\size(\circuit)\le O_k\inparen{\qruntime{\query, \tupset, \bound}}$. Using this along with \Cref{lem:val-ub}, \Cref{cor:approx-algo-const-p} and the fact that $n\le \qruntime{\query, \tupset, \bound}$, we have the following corollary:
 \begin{Corollary}
 \label{cor:approx-algo-punchline}
-Let $\query$ be an $\raPlus$ query and $\pdb$ be a \emph{\abbrOneBIDB} 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 $O_{k,|Q|,\error',\conf}\inparen{\qruntime{\optquery{\query}, \tupset, \bound}}$ (given $\query,\tupset$ and $p_i$ for each $i\in [n]$ that defines $\pd$).
+Let $\query$ be an $\raPlus$ query and $\pdb$ be a \emph{\abbrOneBIDB} 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 $O_{k,|Q|,\error',\conf}\inparen{\qruntime{\optquery{\query}, \tupset, \bound}}$ (given $\query,\tupset$ and $p_i$ for each $i\in [n]$ that defines $\pd$).
 \end{Corollary}
 
 Next, we note that the above result along with \Cref{lem:ctidb-gamma}
 answers \Cref{prob:big-o-joint-steps} in the affirmative as follows:
 \begin{Corollary}
 \label{cor:approx-algo-punchline-ctidb}
-Let $\query$ be an $\raPlus$ query and $\pdb$ be a \abbrCTIDB with $p_0>0$ (where $p_0$ as in \Cref{cor:approx-algo-const-p}) is an absolute constant. 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 $O_{k,|Q|,\error',\conf,\bound}\inparen{\qruntime{\optquery{\query}, \tupset, \bound}}$ (given $\query,\tupset$ and $\prob_{\tup, j}$ for each $\tup\in\tupset,~j\in\pbox{\bound}$ that defines $\bpd$).
+Let $\query$ be an $\raPlus$ query and $\pdb$ be a \abbrCTIDB with $p_0>0$, where $p_0$ as in \Cref{cor:approx-algo-const-p}, is an absolute constant. 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 $O_{k,|Q|,\error',\conf,\bound}\inparen{\qruntime{\optquery{\query}, \tupset, \bound}}$ (given $\query,\tupset$ and $\prob_{\tup, j}$ for each $\tup\in\tupset,~j\in\pbox{\bound}$ that defines $\bpd$).
 \end{Corollary}
 \begin{proof}[Proof of~\Cref{cor:approx-algo-punchline-ctidb}]
 By~\Cref{lem:ctidb-gamma} and~\Cref{cor:approx-algo-punchline}, the proof follows.