Started rebuttal doc. Started adjusting S3 in appendix.

app_hardness-results.tex

@@ -168,9 +168,9 @@ The number of triangles in $\graph{\ell}$ for $\ell \geq 2$ will always be $0$ f
 \subsubsection{Proof of \Cref{lem:lin-sys}}
 \input{lin_sys}
 
-\subsubsection{Proof of \Cref{cor:bounds-tlc}}
-\begin{proof}
-We start by showing that there exists an algorithm that computes $\poly$ in $\bigO{\numedge}$ time for our hard query in \Cref{def:qk}.  Assume that the edges $E$ of graph $G$ are encoded in a relation $R$.  Then a simple table scan of $\rel$ will iterate through the entire set of $\numedge$ edges computing summation of $\poly$ in $\numedge$ steps, and we can replicate this sum $k$ times to output the final $\poly$, in at most $\numedge + k = \bigO{\numedge}$ steps.
-
-This implies that $\numedge \geq \Omega\inparen{\timeOf{\abbrStepOne}}$, and since the results of \Cref{thm:mult-p-hard-result} and \Cref{th:single-p-hard} are in the number of edges $\numedge$, then it follows that our lower bounds hold with respect to $\timeOf{\abbrStepOne}$.
-\end{proof}
+%\subsubsection{Proof of \Cref{cor:bounds-tlc}}
+%\begin{proof}
+%We start by showing that there exists an algorithm that computes $\poly$ in $\bigO{\numedge}$ time for our hard query in \Cref{def:qk}.  Assume that the edges $E$ of graph $G$ are encoded in a relation $R$.  Then a simple table scan of $\rel$ will iterate through the entire set of $\numedge$ edges computing summation of $\poly$ in $\numedge$ steps, and we can replicate this sum $k$ times to output the final $\poly$, in at most $\numedge + k = \bigO{\numedge}$ steps.
+%
+%This implies that $\numedge \geq \Omega\inparen{\timeOf{\abbrStepOne}}$, and since the results of \Cref{thm:mult-p-hard-result} and \Cref{th:single-p-hard} are in the number of edges $\numedge$, then it follows that our lower bounds hold with respect to $\timeOf{\abbrStepOne}$.
+%\end{proof}

app_single-p-proof-defs.tex

@@ -1,12 +1,6 @@
 %root: main.tex
-\subsection{Proof of \Cref{th:single-p-hard}}
-\begin{proof}
-For the sake of contradiction, assume that for any $G$, we can compute $\rpoly_{G}^3(\prob,\dots,\prob)$ in $o\inparen{m^{1+\eps_0}}$ time.
-Let $G$ be the input graph. It is easy to see that one can compute the expression tree for $\poly_{G}^3(\vct{X})$ in $O(m)$ time. Then by \Cref{th:single-p} we can compute $\numocc{G}{\tri}$ in further time $o\inparen{m^{1+\eps_0}}+O(m)$. Thus, the overall, reduction takes $o\inparen{m^{1+\eps_0}}+O(m)= o\inparen{m^{1+\eps_0}}$ time, which violates \Cref{conj:graph}.
-\qed
-\end{proof}
 
-\subsection{Tools to prove \Cref{lem:lin-sys}}
+\subsection{Tools to prove \Cref{th:single-p-hard}}%\Cref{lem:lin-sys}}
 
 Note that $\rpoly_{G}^3(\prob,\ldots, \prob)$ as a polynomial in $\prob$ has degree at most six. Next, we figure out the exact coefficients since this would be useful in our arguments:
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -41,13 +35,19 @@ Since $e\ne e'$, this case produces the following edge patterns: $\twopath, \two
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 Since $\prob$ is fixed, \Cref{lem:qE3-exp} gives us one linear equation in $\numocc{G}{\tri}$ and $\numocc{G}{\threedis}$ (we can handle the other counts due to equations (\ref{eq:1e})-(\ref{eq:3p-3tri})). However, we need to generate one more independent linear equation in these two variables. Towards this end we generate another graph related to $G$:
+\begin{Definition}\label{def:Gk}
+For $\ell \geq 1$, let graph $\graph{\ell}$ be a graph generated from an arbitrary graph $G$, by replacing every edge $e$ of $G$ with a $\ell$-path, such that all inner vertexes of an $\ell$-path replacement edge are disjoint from all other vertexes.\footnote{Note that $G\equiv \graph{1}$.}% of any other $\ell$-path replacement edge. % in the sense that they only intersect at the original intersection endpoints as seen in $\graph{1}$.
+\end{Definition}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+We will prove \Cref{th:single-p-hard} by the following reduction:
+\begin{Theorem}\label{th:single-p}
+Fix $\prob\in (0,1)$. Let $G$ be a graph on $\numedge$ edges.
+If we can compute $\rpoly_{G}^3(\prob,\dots,\prob)$ exactly in $T(\numedge)$ time, then we can exactly compute $\numocc{G}{\tri}$ %count the number of triangles, 3-paths, and 3-matchings in $G$
+in $O\inparen{T(\numedge) + \numedge}$ time.
+\end{Theorem}
+For clarity, we repeat the notion of $\numocc{G}{H}$ to mean the count of subgraphs in $G$ isomorphic to $H$.
+The following lemmas relate these counts in $\graph{2}$ to $\graph{1}$ ($G$).  The lemmas are used to prove \Cref{lem:lin-sys}.
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-Next, we relate the various sub-graph counts in $\graph{2}$ to $\graph{1}$ ($G$).
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Lemma}\label{lem:3m-G2}
 The $3$-matchings in graph $\graph{2}$ satisfy the identity:
 \begin{align*}
@@ -55,12 +55,27 @@ The $3$-matchings in graph $\graph{2}$ satisfy the identity:
 &+ 4 \cdot \numocc{\graph{1}}{\oneint} + 4 \cdot \numocc{\graph{1}}{\threepath} + 2 \cdot \numocc{\graph{1}}{\tri}.
 \end{align*}
 \end{Lemma}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Lemma}\label{lem:tri}
 For $\ell > 1$ and any graph $\graph{\ell}$, $\numocc{\graph{\ell}}{\tri} = 0$.
 \end{Lemma}
+
+Finally, the following result immediately implies \Cref{th:single-p}:
+\begin{Lemma}\label{lem:lin-sys}
+Fix $\prob\in (0,1)$. Given $\rpoly_{\graph{\ell}}^3(\prob,\dots,\prob)$ for $\ell\in [2]$, we can compute in $O(m)$ time a vector $\vct{b}\in\mathbb{R}^3$ such that
+\[ \begin{pmatrix}
+1 - 3p                                     &       -(3\prob^2 - \prob^3)\\
+10(3\prob^2 - \prob^3)		&       10(3\prob^2 - \prob^3)
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+\numocc{G}{\tri}]\\
+\numocc{G}{\threedis}
+\end{pmatrix}
+=\vct{b},
+\]
+allowing us to compute $\numocc{G}{\tri}$ and $\numocc{G}{\threedis}$ in $O(1)$ time.
+\end{Lemma}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Proof of \Cref{th:single-p}}
 \begin{proof}

intro-rewrite-070921.tex

@@ -99,7 +99,7 @@ We initially focus on tuple-independent probabilistic bag-databases\footnote{See
   Mirroring the implementation of bag relations in production database systems (e.g., Postgresql, DB2), tuple multiplicities are modeled by retaining copies of each tuple (up to its largest possible multiplicity).
   % To make each duplicate tuple unique in a set-\abbrTIDB we can assign unique keys across all duplicates.
   When each input tuple has constant multiplicity,
-  the increased input size is negligible.
+  the increased input size is negligible.\label{footnote:set-not-limit}
    %$\tup$ in $\pdb$.
   %This typically has an $\bigO{c}$ increase in size, for $c = \max_{\tup \in \db}\db\inparen{\tup}$, where $\db\inparen{\tup}$ denotes $\tup$'s multiplicity in the encoding.
   % We further generalize this model in \Cref{sec:background} and beyond.
@@ -191,7 +191,7 @@ As mentioned before, under set semantics, $\apolyqdt\inparen{\vct{X}}$ is a prop
 %Atri: If we get a reviewer who does not know what a propositional formula is then we are in trouble-- I did move some of the footnote text to the main part though
 %\footnote{To be precise, $\poly_\tup\inparen{\vct{X}}$ is a propositional formula composed of boolean variables and the logical disjunction and conjunction connectives.  Evaluating such a formula follows the standard semantics of the said operators on boolean variables ($\semB$-semiring semantics).}
 % whose evaluation follows the standard Boolean semi-ring semantics (i.e. addition is logical OR and multiplication is logical AND), denoting the presence or absence of $\tup$.
-and  $\expct\pbox{\apolyqdt\inparen{\vct{\randWorld}}}$ is the marginal probability of $\tup$ appearing in the output.  We note that the answer to \Cref{prob:informal} for set-sematics is also no. Dalvi and Suicu \cite{10.1145/1265530.1265571} showed that the complexity of the query evaluation problem over set-\abbrTIDB\xplural is \sharpphard 
+and  $\expct\pbox{\apolyqdt\inparen{\vct{\randWorld}}}$ is the marginal probability of $\tup$ appearing in the output.  We note that the answer to \Cref{prob:informal} for set-sematics is also no. Dalvi and Suicu \cite{10.1145/1265530.1265571} showed that the (data) complexity of the query evaluation problem over set-\abbrTIDB\xplural is \sharpphard 
 %OK: The former ---v
 %\AR{This is result for TIDBs for general set-PDBs?}
 %Atri: Again if we have a reviewer who does not know what \sharpp is then we are in trouble
@@ -372,20 +372,16 @@ Note that we have argued that for our specific example the expectation that we w
 \Cref{lem:tidb-reduce-poly} generalizes the equivalence to {\em all} $\raPlus$ queries on \abbrTIDB\xplural (proof in \Cref{subsec:proof-exp-poly-rpoly}).
 \begin{Lemma}\label{lem:tidb-reduce-poly}
 Let $\pdb$ be a \abbrTIDB over $n$ input tuples
-%\OK{Should this be $\vct{W}$?} $\vct{X} = \{X_1,\ldots,X_\numvar\}$
  such that the probability distribution $\pdassign$ over $\vct{W}\in\{0,1\}^\numvar$ (the set of possible worlds) is induced by the probability vector $\probAllTup = \inparen{\prob_1,\ldots,\prob_\numvar}$ where $\prob_i=\probOf\pbox{W_i=1}$.
-% $\probAllTup$ consists of each individual tuple's marginal probability across $\idb$.
 For any \abbrTIDB-lineage polynomial $\poly\inparen{\vct{X}}=\apolyqdt(\vct{X})$, it holds that $
-%\begin{equation*}
 	\expct_{\vct{W} \sim \pdassign}\pbox{\poly\inparen{\vct{W}}} = \rpoly\inparen{\probAllTup}.
-%\end{equation*}
 $
 \end{Lemma}
 
 To prove our hardness result we show that for the same $Q$ from the example above, for an arbitrary `product width' $k$, the query $Q^k$ is able to encode various hard graph-counting problems (assuming $\bigO{\numvar}$ tuples rather than the $O(1)$ tuples in \Cref{fig:two-step}).  
-We do so by considering an arbitrary graph $G$ (analogous to the $Route$ relation of $\query$) and analyzing how the coefficients in the (univariate) polynomial $\widetilde{\poly}\left(p,\dots,p\right)$ relate to counts of subgraphs in $G$ that are isomorphic to various graphs with $k$ edges. E.g., we exploit the fact that the leading coefficient in $\poly$ corresponding to $\query^k$ is proportional to the number of $k$-matchings in $G$, a known hard problem in parameterized/fine-grained complexity literature.
+We do so by considering an arbitrary graph $G$ (analogous to the $Route$ relation of $\query$) and analyzing how the coefficients in the (univariate) polynomial $\widetilde{\poly}\left(p,\dots,p\right)$ relate to counts of subgraphs in $G$ that are isomorphic to various graphs with $k$ edges. E.g., we exploit the fact that the leading coefficient in $\poly$ corresponding to $\query^k$ is proportional to the number of $k$-matchings in $G$, a known hard problem in parameterized/fine-grained complexity literature.  
 
-For an upper bound on approximating the expected count, it is easy to check that if all the probabilties are constant then ${\poly}\left(\prob_1,\dots, \prob_n\right)$ (i.e. evaluating the original lineage polynomial over the probability values) is a constant factor approximation.  For example, using $\query^2$ from above, using $\prob_A$ to denote $\probOf\pbox{A = 1}$ (and similarly for the other variables), we can see that
+For an upper bound on approximating the expected count, it is easy to check that if all the probabilties are constant then $\poly\left(\prob_1,\dots, \prob_n\right)$ (i.e. evaluating the original lineage polynomial over the probability values) is a constant factor approximation.  For example, using $\query^2$ from above, using $\prob_A$ to denote $\probOf\pbox{A = 1}$ (and similarly for the other variables), we can see that
 \begin{footnotesize}
 \begin{align*}
 \hspace*{-3mm}

lin_sys.tex

@@ -59,3 +59,11 @@ From  \Cref{eq:det-final} it can easily be seen that the roots of $\dtrm{\vct{M}
 \qed
 \end{proof}
 
+\subsection{Proof of \Cref{th:single-p-hard}}
+\begin{proof}
+For the sake of contradiction, assume that for any $G$, we can compute $\rpoly_{G}^3(\prob,\dots,\prob)$ in $o\inparen{m^{1+\eps_0}}$ time.
+Let $G$ be the input graph. It is easy to see that one can compute the expression tree for $\poly_{G}^3(\vct{X})$ in $O(m)$ time. Then by \Cref{th:single-p} we can compute $\numocc{G}{\tri}$ in further time $o\inparen{m^{1+\eps_0}}+O(m)$. Thus, the overall, reduction takes $o\inparen{m^{1+\eps_0}}+O(m)= o\inparen{m^{1+\eps_0}}$ time, which violates \Cref{conj:graph}.
+\qed
+\end{proof}
+
+

main.tex

@@ -4,9 +4,9 @@
 %\usepackage{cellspace}
 %\setlength\cellspacetoplimit{50pt}
 %\setlength\cellspacebottomlimit{50pt}
+\usepackage[table]{xcolor}%for rebuttal document, in particular \rowcolor
 \usepackage{caption}%caption for table
 \usepackage{cellspace}%padding of tabular cells
-\usepackage{booktabs}
 \usepackage{bm}%for math mode bold font
 \usepackage{relsize}%\mathlarger
 
@@ -58,6 +58,7 @@ sensitive=true
 \usepackage{url}
 \usepackage{cleveref}
 \usepackage{color}
+
 \graphicspath{ {figures/} }
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

rebuttal.tex

@@ -1,3 +1,30 @@
 %root: main.tex
+\definecolor{GrayRew}{gray}{0.85}
+\newcommand{\RCOMMENT}[1]{\medskip\noindent \begin{tabular}{|p{\linewidth-3ex}}\rowcolor{GrayRew} #1 \end{tabular}\smallskip\\}
 \section{Rebuttal}
-This paper is a resubmission, and being such, we use this section to document the changes that have been made since our prior submission, and in particular, how we have addressed reviewer critiques.
+This paper is a resubmission, and being such, we use this section to document the changes that have been made since our prior submission, and in particular, how we have addressed reviewer critiques.
+
+\subsection{Meta Review}
+\RCOMMENT{Problem definition not stated rigorously nor motivated. Discussion needed on the standard PDB approach vs your approach.}
+We made the decision to rewrite \Cref{sec:intro} to specifically address this concern.  The opening paragraph precisely and formally states the query evaluation problem in \abbrBPDB\xplural.  We use a series of problem statements to clearly define the problem we are addressing as it relates to the query evaluation problem.  We have included significant discussion of the standard approach, e.g. see the paragraph \textbf{Relationship to Set-Probabilistic Query Evaluation} on page 4.
+
+\RCOMMENT{Definition 2.6 on reduced BIDB polynomials seem not the right tool for the studied problem.}
+We have chosen to stick with a less formal, ad-hoc definition (please see \Cref{def:reduced-poly} and \Cref{def:reduced-bi-poly}) as suggested by both Reviewer 1 and Reviewer 2.
+
+\RCOMMENT{The paper is very difficult to read. Improvements are needed in particular for the presentation of the approximation results and their proofs. Also for the notation. Missing definitions for used notions need to be added. Ideally use one instead of three query languages (UCQ, RA+, SPJU).}
+\AH{How have we handled the presentation of the approximation results and their proofs?}
+We have chosen one specific query language throughout the paper ($\raPlus$).  We have also made a concerted effort to use clean, defined, non-ambiguous notation.  To the best of our examination, all notation conflicts have been addressed and definitions for used notions are added (see e.g. \Cref{def:Gk} appears before \Cref{lem:3m-G2} and \Cref{lem:lin-sys}.
+
+\subsection{Reviewer 1}
+\RCOMMENT{l.24 "is \#W[1]-hard": parameterized by what?}
+While the above reference does not exist in the revised \Cref{sec:intro} anymore, all theorem statements and claims on \sharpwone runtime have been stated in a way so as to avoid ambiguity in the parameter.  Please see e.g. \Cref{thm:k-match-hard} and \Cref{thm:mult-p-hard-result}
+
+\RCOMMENT{You might want to explain your title somewhere (probably in the  introduction): in the end, what exactly should be considered harmful and why?}
+We have modified the title to better aptly describe our body of work.
+
+\RCOMMENT{l.45 when discussing Dalvi and Suciu's dichotomy, you might want to mention that they consider *data complexity*. Currently the second sentence of your introduction ("take a query Q and a pdb D") suggests that you are considering combined complexity.}
+We have made an explicit mention of data complexity when alluding to Dalvi and Suciu's dichotomy.  We have further rewritten \Cref{sec:intro} in such a way as to explicitly note the type(s) of complexity we are considering.
+
+\RCOMMENT{l.51 "Consider ... and tuples are independent random event": so this is actually a set PDB... You might want to use an example where the input PDB is actually a bag PDB. The last sentence before the example makes the reader *expect* that the example will be of a bag PDB that is not a set PDB}
+Our revision has removed the example referred to above.  While the paper considers predominantly set-\abbrPDB inputs to queries, we have concluded this not to be limiting.  Please see \Cref{footnote:set-not-limit} on \Cpageref{footnote:set-not-limit}.
+

single_p.tex

@@ -1,86 +1,20 @@
 %root: main.tex
 %!TEX root=./main.tex
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Single $\prob$ value}
 \label{sec:single-p}
 
-%In this discussion, let us fix $\kElem = 3$.
 While \Cref{thm:mult-p-hard-result} shows that computing $\rpoly(\prob,\dots,\prob)$ for multiple values of $\prob$ in general is hard it does not rule out the possibility that one can compute this value exactly for a {\em fixed} value of $\prob$. Indeed, it is easy to check that one can compute $\rpoly(\prob,\dots,\prob)$ exactly in linear time for $\prob\in \inset{0,1}$. Next we show that these two are the only possibilities:
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Theorem}\label{th:single-p-hard}
 Fix $\prob\in (0,1)$. Then assuming \Cref{conj:graph} is true, any algorithm that computes $\rpoly_{G}^3(\prob,\dots,\prob)$ for arbitrary $G = (\vset, \edgeSet)$ exactly has to run in time $\Omega\inparen{\abs{\edgeSet}^{1+\eps_0}}$, where $\eps_0$ is as defined in \Cref{conj:graph}.
 \end{Theorem}
-%\begin{proof}[Proof of Corollary ~\ref{th:single-p-gen-k}]
-%Consider $\poly^3_{G}$ and $\poly' = 1$ such that $\poly'' = \poly^3_{G} \cdot \poly'$.  By  \Cref{th:single-p}, query $\poly''$ with $\kElem = 4$ has $\Omega(\numvar^{\frac{4}{3}})$ complexity.
-%\end{proof}
+
 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}.
 The above shows the hardness for a very specific lineage polynomial but it is easy to convert this into a parameterized complexity result as follows.  One can come up with an infinite family of hard query polynomials by `embedding' $\rpoly_{G}^3$ into an infinite family of trivial lineage polynomials.
-%Unlike \Cref{thm:mult-p-hard-result} the above result does not show that computing $\rpoly_{G}^3(\prob,\dots,\prob)$ for a fixed $\prob\in (0,1)$ is \sharpwonehard.
-%However, in \Cref{sec:algo} we show that if we are willing to compute an approximation, then this problem (and indeed solving our problem for a much more general setting) is in linear time, yielding an affirmative answer to \Cref{prob:intro-stmt}.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%
-%NEED to move to appendix
-%%%%%%%%%%%%%%%%%%%%%%%%%
-%%\AH{@atri needs to put in the result for triangles of $\numvar^{\frac{4}{3}}$ runtime.}
-%We will prove the above result by the following reduction:
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%\begin{Theorem}\label{th:single-p}
-%Fix $\prob\in (0,1)$. Let $G$ be a graph on $\numedge$ edges.
-%If we can compute $\rpoly_{G}^3(\prob,\dots,\prob)$ exactly in $T(\numedge)$ time, then we can exactly compute $\numocc{G}{\tri}$ %count the number of triangles, 3-paths, and 3-matchings in $G$
-%in $O\inparen{T(\numedge) + \numedge}$ time.
-%\end{Theorem}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-%
-%%Before we move on to the proof itself, we state the results, lemmas, and defintions that will be useful in the proof.
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-%
-%To prove \Cref{th:single-p}, we use the following notion.
-%\begin{Definition}\label{def:Gk}
-%For $\ell \geq 1$, let graph $\graph{\ell}$ be a graph generated from an arbitrary graph $G$, by replacing every edge $e$ of $G$ with a $\ell$-path, such that all inner vertexes of an $\ell$-path replacement edge are disjoint from all other vertexes.\footnote{Note that $G\equiv \graph{1}$.}% of any other $\ell$-path replacement edge. % in the sense that they only intersect at the original intersection endpoints as seen in $\graph{1}$.
-%\end{Definition}
-%
-%
-%
-%The following result immediately implies \Cref{th:single-p}:
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-%\begin{Lemma}\label{lem:lin-sys}
-%Fix $\prob\in (0,1)$. Given $\rpoly_{\graph{\ell}}^3(\prob,\dots,\prob)$ for $\ell\in [2]$, we can compute in $O(m)$ time a vector $\vct{b}\in\mathbb{R}^3$ such that
-%\[ \begin{pmatrix}
-%1 - 3p                                     &       -(3\prob^2 - \prob^3)\\
-%10(3\prob^2 - \prob^3)		&       10(3\prob^2 - \prob^3)
-%\end{pmatrix}
-%\cdot
-%\begin{pmatrix}
-%\numocc{G}{\tri}]\\
-%\numocc{G}{\threedis}
-%\end{pmatrix}
-%=\vct{b},
-%\]
-%allowing us to compute $\numocc{G}{\tri}$ and $\numocc{G}{\threedis}$ in $O(1)$ time.
-%\end{Lemma}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%END move to appendix
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%The bounds of \Cref{thm:mult-p-hard-result} and \Cref{th:single-p-hard} imply the following corollary.
-%\AH{Corollary needs refinement.}
-%\begin{Corollary}\label{cor:bounds-tlc}
-%The lower bounds of \Cref{thm:mult-p-hard-result} and \Cref{th:single-p-hard} hold with respect to $\timeOf{\abbrStepOne}$.
-%\end{Corollary}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
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+%%% End: