Done with S5

circuits-model-runtime.tex

@@ -22,7 +22,7 @@ In~\Cref{sec:results-circuits} we argue why results from earlier sections also h
 \subsubsection{Extending our results to lineage circuits}
 \label{sec:results-circuits}
 
-We first note that since expression trees are a special case of them, all of our hardness results in~\Cref{sec:hard} are still valid for lineage circuits.
+We first note that since expression trees are a special case of linear circuits, all of our hardness results in~\Cref{sec:hard} are still valid for the latter.
 
 Observe that \textsc{Approx}\textsc{imate}$\rpoly$ (\Cref{alg:mon-sam} in \Cref{sec:algo}) works for lineage circuits as long as the same guarantees on $\onepass$ and $\sampmon$ (\Cref{lem:one-pass} and \Cref{lem:sample} respectively) hold for lineage circuits as well. 
 It turns out that this is the case, simply because both algorithms rely on only one property of expression trees: that each node has two children; 
@@ -76,7 +76,8 @@ For the specifics on how lineage circuits are translated to represent polynomial
 \subsubsection{Circuit size vs. runtime}
 \label{sec:circuit-runtime}
 
-We now connect the size of a lineage circuit (where the size of a lineage circuit is the number of vertices in the corresponding DAG\footnote{since each node has indegree at most two, this also is the same up to constants to counting the number of edges in the DAG.}) for a given SPJU query $Q$ to its $\qruntime{Q}$.  We do this formally by showing that the size of the lineage circuit is asymptotically no worse than the corresponding runtime of a large class of deterministic query processing algorithms.
+We now connect the size of a lineage circuit (where the size of a lineage circuit is the number of vertices in the corresponding DAG %\footnote{since each node has indegree at most two, this also is the same up to constants to counting the number of edges in the DAG.})
+ for a given SPJU query $Q$ to its $\qruntime{Q}$.  We do this formally by showing that the size of the lineage circuit is asymptotically no worse than the corresponding runtime of a large class of deterministic query processing algorithms.
 
 \begin{lemma}
 \label{lem:circuits-model-runtime}
@@ -89,7 +90,7 @@ We now have all the pieces to argue the following, which formally states that ou
 Given an SPJU query $Q$ for a TIDB, we can present $(1\pm\eps)$ approximation to the expectation of each output tuple with probability at least $1-\delta$ in time $O_k\left(\frac 1{\eps^2}\cdot\qruntime{Q}\cdot \log{\frac{1}{\conf}}\cdot \log(n)\right)$.
 \end{Corollary}
 \begin{proof}
-This follows from~\Cref{lem:circuits-model-runtime} and (the lineage circuit counterpart-- see~\Cref{sec:results-circuits} of)~\Cref{cor:approx-algo-const-p} (where the latter is used with $\delta$ being substituted\footnote{Recall that~\Cref{cor:approx-algo-const-p} is stated for a single output tuple so to get the required guarantee for all (at most $n^k$) output tuples of $Q$ we get at most $\frac \delta{n^k}$ probability of failure for each output tuple and then just a union bound over all output tuples. } with $\frac \delta{n^k}$).
+This follows from~\Cref{lem:circuits-model-runtime} and (the lineage circuit counterpart-- see~\Cref{sec:results-circuits})~\Cref{cor:approx-algo-const-p} (where the latter is used with $\delta$ being substituted\footnote{Recall that~\Cref{cor:approx-algo-const-p} is stated for a single output tuple so to get the required guarantee for all (at most $n^k$) output tuples of $Q$ we get at most $\frac \delta{n^k}$ probability of failure for each output tuple and then just a union bound over all output tuples. } with $\frac \delta{n^k}$).
 \end{proof}
 
 \subsection{Higher moments}
@@ -101,4 +102,4 @@ In addition, we could e.g. prove bounds of probability of the multiplicity being
 While we do not have a good approximation algorithm for this problem, we can make some progress as follows:
 Note that for any positive integer $m$ we can compute the expectation $\poly^m$ (since this only changes the degree of the corresponding lineage polynomial by a factor of $m$). 
 In other words, we can compute the $m$-th moment of the multiplicities as well allowing us to e.g. to use Chebyschev inequality or other high moment based probability bounds on the events we might be interested in. 
-However, we leave the question of coming up with a wider range of approximation algorithms for future work.
+However, we leave the question of coming up with a more accurate approximation algorithms for future work.