Minor grammar fixes for S.3

mult_distinct_p.tex

@@ -85,7 +85,7 @@ needs time $\bigOmega{\kmatchtime}$, assuming $\kmatchtime\ge \omega\inparen{\ab
 \end{Theorem}
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-Note that the second row of \Cref{tab:lbs} follows from \Cref{prop:expection-of-polynom}, \Cref{thm:mult-p-hard-result}, \Cref{lem:tdet-om}, and \Cref{thm:k-match-hard} while the third row is proved by \Cref{prop:expection-of-polynom}, \Cref{thm:mult-p-hard-result}, \Cref{lem:tdet-om}, and \Cref{conj:known-algo-kmatch}. Since \Cref{conj:known-algo-kmatch} is non-standard, the latter hardness result should be interpreted as follows. Any substantial polynomial improvement for \Cref{prob:bag-pdb-poly-expected} (over the trivial algorithm that converts $\poly$ into SMB and then uses \Cref{cor:expct-sop} for \abbrStepTwo) would lead to an improvement over the state of the art {\em upper} bounds on  $\kmatchtime$. Finally, note that \Cref{thm:mult-p-hard-result} needs one to be able to compute the expected multiplicities over $(2k+1)$ distinct values of $p_i$, each of which corresponds to distinct $\bpd$ (for the same $\tupset$), which explain the `Multiple' entry in the second column in the second and third row in \Cref{tab:lbs}. Next, we argue how to get rid of this latter requirement.
+Note that the second row of \Cref{tab:lbs} follows from \Cref{prop:expection-of-polynom}, \Cref{thm:mult-p-hard-result}, \Cref{lem:tdet-om}, and \Cref{thm:k-match-hard} while the third row is proved by \Cref{prop:expection-of-polynom}, \Cref{thm:mult-p-hard-result}, \Cref{lem:tdet-om}, and \Cref{conj:known-algo-kmatch}. Since \Cref{conj:known-algo-kmatch} is non-standard, the latter hardness result should be interpreted as follows. Any substantial polynomial improvement for \Cref{prob:bag-pdb-poly-expected} (over the trivial algorithm that converts $\poly$ into SMB and then uses \Cref{cor:expct-sop} for \abbrStepTwo) would lead to an improvement over the state of the art {\em upper} bounds on  $\kmatchtime$. Finally, note that \Cref{thm:mult-p-hard-result} needs one to be able to compute the expected multiplicities over $(2k+1)$ distinct values of $p_i$, each of which corresponds to distinct $\bpd$ (for the same $\tupset$), which explain the `Multiple' entries in the second column of the second and third rows in \Cref{tab:lbs}. Next, we argue how to get rid of this latter requirement.
 
 
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single_p.tex

@@ -11,7 +11,7 @@ Fix $\prob\in (0,1)$. Then assuming \Cref{conj:graph} is true, any algorithm tha
 \end{Theorem}
 
 Note that \Cref{prop:expection-of-polynom} and \Cref{th:single-p-hard} above imply the hardness result in the first row of \Cref{tab:lbs}.
-We note that \Cref{thm:k-match-hard} and \Cref{conj:known-algo-kmatch} (and the lower bounds in the second and third row of Table~\ref{tab:lbs}) need $k$ to be large enough (in particular, we need a family of hard queries). But the above \Cref{th:single-p-hard} (and the lower bound in first row of Table~\ref{tab:lbs}) holds for $k=3$ (and hence for a fixed query).
+We note that \Cref{thm:k-match-hard} and \Cref{conj:known-algo-kmatch} (and the lower bounds in the second and third rows of Table~\ref{tab:lbs}) need $k$ to be large enough (in particular, we need a family of hard queries). But the above \Cref{th:single-p-hard} (and the lower bound in first row of Table~\ref{tab:lbs}) holds for $k=3$ (and hence for a fixed query).
 
 
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