Back down to 15

intro-rewrite-070921.tex

@@ -223,7 +223,7 @@ In the remainder of this work, we demonstrate that a $(1\pm\epsilon)$ (multiplic
 
 \input{two-step-model}
 
-Like set-probabilistic databases, our approach adopts the two-step  intensional model of query evaluation, as illustrated in \Cref{fig:two-step}:
+We adopt the two-step intensional model of query evaluation used in set-\abbrPDB\xplural, as illustrated in \Cref{fig:two-step}:
 (i) \termStepOne (\abbrStepOne): Given input $\dbbase$ and $\query$, output every tuple $\tup$ that possibly satisfies $\query$, annotated with its lineage polynomial ($\poly(\vct{X})=\apolyqdt\inparen{\vct{X}}$);
 (ii) \termStepTwo (\abbrStepTwo): Given $\poly(\vct{X})$ for each tuple, compute $\expct\pbox{\poly(\vct{\randWorld})}$.
 Let $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit)$ denote the runtime of \abbrStepOne when it outputs $\circuit$ (which is a representation of $\poly$ as an arithmetic circuit --- more on this representation shortly).

mult_distinct_p.tex

@@ -30,7 +30,7 @@ There exists a constant $\eps_0>0$ such that given an undirected graph $G=(\vset
 \end{hypo}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %
-Based on the so called {\em Triangle detection hypothesis} (cf.~\cite{triang-hard}), which states that detection of whether $G$ has a triangle or not takes time $\Omega\inparen{|\edgeSet|^{4/3}}$, implies that in Conjecture~\ref{conj:graph} we can take $\eps_0\ge \frac 13$.
+The so called {\em Triangle detection hypothesis} (cf.~\cite{triang-hard}), which states that detecting the presence of triangles in $G$ takes time $\Omega\inparen{|\edgeSet|^{4/3}}$, implies that in Conjecture~\ref{conj:graph} we can take $\eps_0\ge \frac 13$.
 
 All of our hardness results rely on a simple lineage polynomial encoding of the edges of a graph.
 To prove our hardness result, consider a graph $G=(\vset, \edgeSet)$, where $|\edgeSet| = m$, $\vset = [\numvar]$. Our lineage polynomial has a variable $X_i$ for every $i$ in $[\numvar]$.