Small changes, fixing bugs and typos.

circuits-model-runtime.tex

@@ -45,7 +45,7 @@ For these algorithms, $\jointime{R_1, \ldots, R_n}$ is linear in the {\em AGM bo
 &\begin{aligned}
     &\qruntimenoopt{R,\gentupset,\bound} = |\gentupset.R| 
     & 
-    &\qquad\qquad\qruntimenoopt{\sigma \query, \gentupset,\bound} = \qruntimenoopt{\query,\gentupset}\\
+    &\qquad\qquad\qruntimenoopt{\sigma \query, \gentupset,\bound} = \qruntimenoopt{\query,\gentupset,\bound}\\
 \end{aligned}&\\
 %\vspace{-.6cm}
  &\qruntimenoopt{\pi \query, \gentupset,\bound} = \qruntimenoopt{\query,\gentupset,\bound} +\abs{\query(\gentupset)}&\\

mult_distinct_p.tex

@@ -51,7 +51,7 @@ For any graph $G=(V,\edgeSet)$ and $\kElem\ge 1$, define
 \end{Definition}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\noindent Returning to \Cref{fig:two-step}, it can be seen that $\poly_{G}^\kElem(\vct{X})$ is the lineage polynomial from query $\qhard^k$, which we define next ($\query_2$ from~\Cref{sec:intro} is the same query with $k=2$). Let us alias 
+\noindent Returning to \Cref{fig:two-step}, it can be seen that $\poly_{G}^\kElem(\vct{X})$ is the lineage polynomial from query $\qhard^k$, which we define next ($\query_1$ from~\Cref{sec:intro} is the same query with $k=1$). Let us alias 
 \begin{lstlisting}
 SELECT DISTINCT 1 FROM T $t_1$, R r, T $t_2$
 WHERE $t_1$.Point = r.Point$_1$ AND $t_2$.Point = r.Point$_2$
@@ -70,7 +70,7 @@ Note that this implies that $\poly_{G}^\kElem$ is indeed a $1$-\abbrTIDB lineage
 Next, we note that the runtime for answering $\qhard^k$ on deterministic database $\tupset$, as defined above, is $O_k\inparen{\numedge}$ (i.e. deterministic query processing is `easy' for this query):
 \begin{Lemma}\label{lem:tdet-om}
 Let $\qhard^k$ and $\tupset$ be as defined above. Then
-$\qruntimenoopt{\qhard^k, \tupset}$ is $O_k\inparen{\numedge}$.
+$\qruntimenoopt{\qhard^k, \tupset, \bound}$ is $O_k\inparen{\numedge}$.
 \end{Lemma}
 
 \subsection{Multiple Distinct $\prob$ Values}