Finished main parts of first iteration Intro Rewrite

intro-rewrite2.tex

@@ -4,7 +4,7 @@
 		\setenumerate[3]{label=\roman*.}
 		\setenumerate[4]{label=\alph*.}	
 			
-\paragraph{Introduction (Rewrite) Outline}
+\section{Introduction (Outline and Rewrite)}
 		%for outline functionality
 		\begin{outline}[enumerate]
 			\1 Overall Problem
@@ -39,7 +39,7 @@
 		\end{outline}
 
 \AH{Setting}
-A probabilistic database (\abbrPDB) $\pdb$ is a two-tuple ($\idb, \pd$) such that $\idb$ is the set of possible worlds $\db$ represented by $\pdb$, and $\pd$ is the associated probability distribution across each $\db$ in $\idb$.  Given a query $\query$ the output of $\query(\pdb)$ is ($\idb', \pd'$) such that $\idb' = \{\query(\db_i) \suchthat i \in [\numvar]\}$ where $\numvar$ is the number of possible worlds, and $\pd'$ is the resulting probability distribution over $\idb'$.  Computing $\query$ as outlined above can be modeled in two steps, where the first step consists of the deterministic computation of both the query output and result tuple lineage(s) encoded in the respective representation, and the second step consists of computing the probability distributation. This computational model is nicely followed by set-\abbrPDB computation and semiring provenance, and is useful in this work for the purpose separating the deterministic computation from the probability computation.
+A probabilistic database (\abbrPDB) $\pdb$ is a two-tuple ($\idb, \pd$) such that $\idb$ is the set of possible worlds $\db$ represented by $\pdb$, and $\pd$ is the associated probability distribution across each $\db$ in $\idb$.  Given a query $\query$ the output of $\query(\pdb)$ is ($\idb', \pd'$) such that $\idb' = \{\query(\db_i) \suchthat i \in [\numvar]\}$ where $\numvar = \abs{\idb}$, the number of possible worlds in $\pdb$, and $\pd'$ is the resulting probability distribution over $\idb'$.  Computing $\query$ as outlined above can be modeled in two steps, where the first step consists of the deterministic computation of both the query output and result tuple lineage(s) encoded in the respective representation, and the second step consists of computing the probability distributation. This computational model is nicely followed by set-\abbrPDB computation and semiring provenance, and is useful in this work for the purpose separating the deterministic computation from the probability computation.
 
 Much work already exists regarding \abbrPDB\xplural, most of which considers $\pdb$ to be a set, meaning all possible worlds $\db$ are a \emph{set} of tuples.  The problem of computing $\query$ \emph{exactly} over a set-\abbrPDB is known to be \sharpphard in the general case.  The dichotomy of Dalvi and Suicu shows that for set-\abbrPDB\xplural it is the case that $\query(\pdb)$ is either polynomial or \sharpphard.  Further, this dichotomy is \emph{based} on the query structure and in general is independent of the representation of the lineage polynomial.\footnote{We do note that there exist specific cases when given a specific database instance combined with an amenable representation, that a hard $\query$ can become easy, but this is {\emph not} the general case.}  The hardness results for set-\abbrPDB\xplural depend on step two of the computation model.
 
@@ -49,7 +49,51 @@ Traditionally, bag-\abbrPDB\xplural have long been considered to be bottlenecked
 
 However, it is not necessarily satisfying to stop here.  Since typical implementations of \abbrPDB\xplural compute the representation of the lineage polynomial in sync with the particular choice of query plan, it is important that optimizations are allowed if we want to have a true comparison between step one and step two in bag-\abbrPDB queries.  Optimizations like projection push-down produce factorized or non-\abbrSOP representations of the lineage polynomial.  Our work explores whether or not step two in the computation model is \emph{always} linear in the \emph{size} of the representation of the lineage polynomial when step one of $\query(\pdb)$ is easy.\footnote{It is known that, in general, there exist queries that are \emph{not} linear in the size of the data.  Such queries as multiple joins and counting cliques are specific examples of this.  We are considering cases where the query is linear in the size of the data.}
 
-Our work focuses on the following setting for query computation.  Inputs of $\query$ are set-\abbrPDB\xplural, while the output of $\query$ is a bag-\abbrPDB.  This, however, is not limiting as a simple generalization exists, which involves assigning a unique id to each tuple of bag-\abbrPDB inputs.
+Our work focuses on the following setting for query computation.  Inputs of $\query$ are set-\abbrPDB\xplural, while the output of $\query$ is a bag-\abbrPDB.  This setting, however, is not limiting as a simple generalization exists, which involves assigning a unique id to each tuple of bag-\abbrPDB inputs.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%Contributions, Overview, Paper Organization
+%%%%%%%%%%%%%%%%%%%%%%%%%
+Concretely, we make the following contributions:
+(i) We show that the expected result multiplicity problem (\Cref{def:the-expected-multipl}) for conjunctive queries for bag-$\ti$s is \sharpwonehard in the size of a lineage circuit by reduction from counting the number of $k$-matchings over an arbitrary graph;
+(ii) We present an $(1\pm\epsilon)$-\emph{multiplicative} approximation algorithm for bag-$\ti$s and show that for typical database usage patterns (e.g. when the circuit is a tree or is generated by recent worst-case optimal join algorithms or their FAQ followups~\cite{DBLP:conf/pods/KhamisNR16}) its complexity is linear in the size of the compressed lineage encoding (in contrast, known approximation techniques in set-\abbrPDB\xplural are quadratic); (iii) We generalize the approximation algorithm to bag-$\bi$s, a more general model of probabilistic data; (iv) We further prove that for \raPlus queries (an equivalently expressive, but factorizable form of UCQs),  we can approximate the expected output tuple multiplicities with only $O(\log{Z})$ overhead (where $Z$ is the number of output tuples) over the runtime of a broad class of query processing algorithms. We also observe that our results trivially extend to higher moments of the tuple multiplicity (instead of just the expectation).
+
+\mypar{Overview of our Techniques} All of our results rely on working with a {\em reduced} form of the lineage polynomial $\Phi$. In fact, it turns out that for the TIDB (and BIDB) case, computing the expected multiplicity is {\em exactly} the same as evaluating this reduced polynomial over the probabilities that define the TIDB/BIDB. Next, we motivate this reduced polynomial by continuing \Cref{ex:intro-tbls}.
+
+Consider the query $Q()\dlImp$$OnTime(\text{City}), Route(\text{City}, \text{City}'),$ $OnTime(\text{City}')$ over the bag relations of \Cref{fig:ex-shipping-simp}. It can be verified that $\Phi$ for $Q$ is $L_aL_b + L_bL_d + L_bL_c$. Now consider the product query $\query^2()\dlImp Q(), Q()$.
+
+The lineage polynomial for $Q^2$ is given by $\Phi^2$:
+\begin{equation*}
+\left(L_aL_b + L_bL_d + L_bL_c\right)^2=L_a^2L_b^2 + L_b^2L_d^2 + L_b^2L_c^2 + 2L_aL_b^2L_d + 2L_aL_b^2L_c + 2L_b^2L_dL_c.
+\end{equation*}
+The expectation $\expct\pbox{\Phi^2}$ then is:
+\begin{multline*}
+\expct\pbox{L_a}\expct\pbox{L_b^2} + \expct\pbox{L_b^2}\expct\pbox{L_d^2} + \expct\pbox{L_b^2}\expct\pbox{L_c^2} + 2\expct\pbox{L_a}\expct\pbox{L_b^2}\expct\pbox{L_d} \\
++ 2\expct\pbox{L_a}\expct\pbox{L_b^2}\expct\pbox{L_c} + 2\expct\pbox{L_b^2}\expct\pbox{L_d}\expct\pbox{L_c}
+\end{multline*}
+\noindent If the domain of a random variable $W$ is $\{0, 1\}$, then for any $k > 0$, $\expct\pbox{W^k} = \expct\pbox{W}$, which means that $\expct\pbox{\Phi^2}$ simplifies to:
+\begin{footnotesize}
+\begin{equation*}
+\expct\pbox{L_a}\expct\pbox{L_b} + \expct\pbox{L_b}\expct\pbox{L_d} + \expct\pbox{L_b}\expct\pbox{L_c} + 2\expct\pbox{L_a}\expct\pbox{L_b}\expct\pbox{L_d} + 2\expct\pbox{L_a}\expct\pbox{L_b}\expct\pbox{L_c} + 2\expct\pbox{L_b}\expct\pbox{L_d}\expct\pbox{L_c}
+\end{equation*}
+\end{footnotesize}
+\noindent This property leads us to consider a structure related to the lineage polynomial.
+\begin{Definition}\label{def:reduced-poly}
+For any polynomial $\poly(\vct{X})$, define the \emph{reduced polynomial} $\rpoly(\vct{X})$ to be the polynomial obtained by setting all exponents $e > 1$ in the SOP form of $\poly(\vct{X})$ to $1$.
+\end{Definition}
+With $\Phi^2$ as an example, we have:
+\begin{align*}
+\widetilde{\Phi^2}(L_a, L_b, L_c, L_d)
+=&\; L_aL_b + L_bL_d + L_bL_c + 2L_aL_bL_d + 2L_aL_bL_c + 2L_bL_cL_d
+\end{align*}
+It can be verified that the reduced polynomial parameterized with each variable's respective marginal probability is a closed form of the expected count (i.e., $\expct\pbox{\Phi^2} = \widetilde{\Phi^2}(\probOf\pbox{L_a=1},$ $\probOf\pbox{L_b=1}, \probOf\pbox{L_c=1}), \probOf\pbox{L_d=1})$). In fact, we show in \Cref{lem:exp-poly-rpoly} that this equivalence holds for {\em all} UCQs over TIDB/BIDB.
+
+To prove our hardness result we show that for the same $Q$ considered in the running example, the query $Q^k$ is able to encode various hard graph-counting problems.  We do so by analyzing  how the coefficients in the (univariate) polynomial $\widetilde{\Phi}\left(p,\dots,p\right)$ relate to counts of various sub-graphs on $k$ edges in an arbitrary graph $G$ (which is used to define the relations in $Q$). \AH{What is meant by the following sentence?}For the upper bound it is easy to check that if all the probabilties are constant then ${\Phi}\left(\probOf\pbox{X_1=1},\dots, \probOf\pbox{X_n=1}\right)$ (i.e. evaluating the original lineage polynomial over the probability values) is a constant factor approximation. \AH{Why do we say `approximation'?  This is a linear {\emph exact} computation.}  To get an $(1\pm \epsilon)$-multiplicative approximation we sample monomials from $\Phi$ and `adjust' their contribution to $\widetilde{\Phi}\left(\cdot\right)$.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mypar{Paper Organization} We present relevant background and notation in \Cref{sec:background}. We then prove our main hardness results in \Cref{sec:hard} and present our approximation algorithm in \Cref{sec:algo}. We present some (easy) generalizations of our results in \Cref{sec:gen} and also discuss extensions from computing expectations of polynomials to the expected result multiplicity problem (\Cref{def:the-expected-multipl})\AH{Aren't they the same?}. Finally, we discuss related work in \Cref{sec:related-work} and conclude in \Cref{sec:concl-future-work}.
+
+
 	
 
 

main.tex

@@ -111,8 +111,8 @@ sensitive=true
 
 \input{abstract}
 \input{intro-rewrite2}
-\input{outline-intro-new}
-\input{intro-new}
+%\input{outline-intro-new}
+%\input{intro-new}
 % \input{intro}
 \input{ra-to-poly}
 \input{poly-form}