209 lines
17 KiB
TeX
209 lines
17 KiB
TeX
%root: main.tex
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\setenumerate[1]{label = \Roman*.}
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\setenumerate[2]{label=\Alph*.}
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\setenumerate[3]{label=\roman*.}
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\setenumerate[4]{label=\alph*.}
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\section{Introduction (Outline and Rewrite)}
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%for outline functionality
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\begin{outline}[enumerate]
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\1 Overall Problem
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\2 Hardness of deterministic queries, e.g., counting cliques, multiple joins, etc.
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\2 Assuming $\query$ is easy, how does the probabilistic computation compare in \abbrPDB\xplural?
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\3 Introduce two-step process
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\4 Deterministic Process: compute query, lineage, representation aka circuit
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\4 Probability Computation
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\3 Why the two-step process?
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\4 Semiring provenance nicely follows this model
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\4 Set \abbrPDB\xplural use this process
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\4 Model allows us to separate the deterministic from the probability computation
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\AH{The part below should maybe be moved further down.}
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\3 Assuming a bag-\abbrTIDB, when the probability of all tuples $\prob_i = 1$, the problem of computing the expected count is linear
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\3 However, when $\prob_i < 1$, the problem is not linear (in circuit size)
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\3 An approximation algorithm exists to bring the second step back down to linear time
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\3 For set-\abbrPDB, the problem is \sharpphard with respect to exact computation
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\3 For set-\abbrPDB, the problem is quadratic with respect to approximation
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\2 Note that in all cases, Step 2 is at least as hard as Step 1
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\3 Bag-\abbrPDB\xplural are useful for things like a count query
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\3 This work focuses on Step 2 for bag-\abbrPDB\xplural
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\4 Given a lineage polynomial generated by a query, compute the expected multiplicity
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\3 Why focus on tuple expected multiplicity?
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\4 Note that bag-\abbrPDB query output is a probability distribution over counts; this contrasts the marginal probability paradigm of set-\abbrPDB\xplural
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\4 From a theoretical perspective, bag-\abbrPDB\xplural are not well studied
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\4 There are several statistical measures that can be done
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\4 We focus on expected count since it is a natural and simplistic statistic to consider
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\4 Appendix also considers higher moments
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\2 The setting for our problem assumes set-\abbrPDB inputs
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\3 A simple generalization exists
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\end{outline}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\textwidth]{twostep}
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\caption{Two-Step Computation Model (\abbrPDB\xplural)}
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\label{fig:two-step}
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\end{figure}
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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 $\idb'$. As depicted in \cref{fig:two-step}, 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.
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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.
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A tuple independent database (\abbrTIDB) is a \abbrPDB whose tuples are treated as independent random events. Given a set-\abbrTIDB $\pdb$, $\query(\pdb)$ is essentially limited to computing the \emph{marginal} probability for a member tuple $\tup$. When it is desirable to compute either a probability distribution over the set of possible multiplicities $\tup$ or to compute certain statistical measures over the multiplicity of $\tup$, bag-\abbrPDB\xplural are a natural fit, proving very useful for posing questions such as count queries to the database. While other statistical measures can be computed, we focus primarily on computing the expected multiplicity of $\tup$, a natural interpretation of step two in the bag setting.\footnote{We consider this natural since it is true that computing the marginal probability of $\tup$ in set-\abbrPDB\xplural is essentially computing $\tup$'s expectation.} It is also compelling to consider the expected multiplicity since bag-\abbrPDB\xplural are not well studied from a theoretical perspective, and the expected count is both natural and simplistic to consider as a first building block. We consider higher moments in the appendix.\AH{Pointer here.}
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Traditionally, bag-\abbrPDB\xplural have long been considered to be bottlenecked in step one only, or linear in the size of query. This may partially be due to the prevalence that exists in using a sum of products (\abbrSOP) representation of the lineage polynomial amongst many of the most well-known implementations of set-\abbrPDB\xplural. Such a representation used in the bag-\abbrPDB setting \emph{indeed} allows for step two to be linear in the \emph{size} of the \abbrSOP representation, a result due to linearity of expectation.
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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.} Indeed, if for all $i \in [\numvar]$, $\prob_i = 1$, computation is essentially a deterministic query and dominated by the first step. This changes, however, when $\prob_i < 1$, and for this case our work shows that the problem is not linear in the size of the representation.\AH{Not sure of the wording...this is \emph{true} for all representations, correct? Also, should I be more precise and say ``when $\forall i \in [\numvar]$, $\prob_i < 1$''? I don't think we consider the case of a mixture of probabilities, with some equal to 1 and some less than 1.}
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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.
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%%%%%%%%%%%%%%%%%%%%%%%%%
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%Contributions, Overview, Paper Organization
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%%%%%%%%%%%%%%%%%%%%%%%%%
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Concretely, we make the following contributions:
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(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;
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(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).
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\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 in what follows.%continuing \Cref{ex:intro-tbls}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{figure}[t]
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\begin{subfigure}[b]{0.49\linewidth}
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\centering
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{\small
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\begin{tabular}{ c | c c c}
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$OnTime$ & City$_\ell$ & $\Phi$ & \textbf{p}\\
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\hline
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& Buffalo & $L_a$ & 0.9 \\
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& Chicago & $L_b$ & 0.5\\
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& Bremen & $L_c$ & 0.5\\
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& Zurich & $L_d$ & 1.0\\
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\end{tabular}
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}
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\caption{Relation $OnTime$}
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\label{subfig:ex-shipping-simp-loc}
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\end{subfigure}%
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\begin{subfigure}[b]{0.49\linewidth}
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\centering
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{\small
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\begin{tabular}{ c | c c c c}
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$Route$ & $\text{City}_1$ & $\text{City}_2$ & $\Phi$ & \textbf{p} \\
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\hline
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& Buffalo & Chicago & $R_a$ & 1.0 \\
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& Chicago & Zurich & $R_b$ & 1.0 \\
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%& $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ \\
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& Chicago & Bremen & $R_c$ & 1.0 \\
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\end{tabular}
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}
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\caption{Relation $Route$}
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\label{subfig:ex-shipping-simp-route}
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\end{subfigure}%
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% \begin{subfigure}[b]{0.17\linewidth}
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% \centering
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% \caption{Circuit for $(Chicago)$}
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% \label{subfig:ex-proj-push-circ-q3}
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% \end{subfigure}
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\begin{subfigure}[b]{0.66\linewidth}
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\centering
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{\small
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\begin{tabular}{ c | c c c}
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$\query_1$ & City & $\Phi$ & $\expct_{\idb \sim \probDist}[\query(\db)(t)]$ \\ \hline
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& Buffalo & $L_a \cdot R_a$ & $0.9$ \\
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& Chicago & $L_b \cdot R_b + L_b \cdot R_c$ & $0.5 \cdot 1.0 + 0.5 \cdot 1.0 = 1.0$ \\
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%& $\cdots$ & $\cdots$ & $\cdots$ \\
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\end{tabular}
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}
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\caption{$Q_1$'s Result}
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\label{subfig:ex-shipping-simp-queries}
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\end{subfigure}%
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\begin{subfigure}[b]{0.33\linewidth}
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\centering
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\resizebox{!}{16mm} {
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\begin{tikzpicture}[thick]
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\node[tree_node] (a2) at (0, 0){$R_b$};
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\node[tree_node] (b2) at (1, 0){$L_b$};
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\node[tree_node] (c2) at (2, 0){$R_c$};
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%level 1
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\node[tree_node] (a1) at (0.5, 0.8){$\boldsymbol{\circmult}$};
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\node[tree_node] (b1) at (1.5, 0.8){$\boldsymbol{\circmult}$};
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%level 0
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\node[tree_node] (a0) at (1.0, 1.6){$\boldsymbol{\circplus}$};
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%edges
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\draw[->] (a2) -- (a1);
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\draw[->] (b2) -- (a1);
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\draw[->] (b2) -- (b1);
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\draw[->] (c2) -- (b1);
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\draw[->] (a1) -- (a0);
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\draw[->] (b1) -- (a0);
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\end{tikzpicture}
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}
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\resizebox{!}{16mm} {
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\begin{tikzpicture}[thick]
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\node[tree_node] (a1) at (1, 0){$R_b$};
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\node[tree_node] (b1) at (2, 0){$R_c$};
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%level 1
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\node[tree_node] (a2) at (0.75, 0.8){$L_b$};
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\node[tree_node] (b2) at (1.5, 0.8){$\boldsymbol{\circplus}$};
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%level 0
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\node[tree_node] (a3) at (1.1, 1.6){$\boldsymbol{\circmult}$};
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%edges
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\draw[->] (a1) -- (b2);
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\draw[->] (b1) -- (b2);
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\draw[->] (a2) -- (a3);
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\draw[->] (b2) -- (a3);
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\end{tikzpicture}
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}
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\caption{Two circuits for $Q_1(Chicago)$}
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\label{subfig:ex-proj-push-circ-q4}
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\end{subfigure}%
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\vspace*{-3mm}
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\caption{\ti instance and query results for \cref{ex:overview}}%\Cref{ex:intro-tbls}.}%{$\ti$ relations for $\poly$}
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\label{fig:ex-shipping-simp}
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\trimfigurespacing
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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 $\query^2()\dlImp Q(), Q()$.
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The lineage polynomial for $Q^2$ is given by $\Phi^2$:
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\begin{equation*}
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\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.
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\end{equation*}
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The expectation $\expct\pbox{\Phi^2}$ then is:
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\begin{multline*}
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\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} \\
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+ 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}
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\end{multline*}
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\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:
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\begin{footnotesize}
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\begin{equation*}
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\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}
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\end{equation*}
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\end{footnotesize}
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\noindent This property leads us to consider a structure related to the lineage polynomial.
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\begin{Definition}\label{def:reduced-poly}
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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 $1$.
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\end{Definition}
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With $\Phi^2$ as an example, we have:
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\begin{align*}
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\widetilde{\Phi^2}(L_a, L_b, L_c, L_d)
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=&\; L_aL_b + L_bL_d + L_bL_c + 2L_aL_bL_d + 2L_aL_bL_c + 2L_bL_cL_d
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\end{align*}
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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.
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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 $\widetilde{\Phi}\left(\cdot\right)$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\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}.
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