Done with my pass
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%\input{app_approx-alg-pseudo-code}
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The following results assume input circuit \circuit computed from an arbitrary $\raPlus$ query $\query$ and arbitrary \abbrBIDB $\pdb$. We refer to \circuit as a \abbrBIDB circuit.
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\AH{Verify that the proof for \Cref{lem:approx-alg} doesn't rely on properties of $\raPlus$ or \abbrBIDB.}
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\begin{Theorem}\label{lem:approx-alg}
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Let \circuit be an arbitrary \abbrBIDB circuit %for a UCQ over \bi
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and define $\poly(\vct{X})=\polyf(\circuit)$ and let $k=\degree(\circuit)$.
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Then an estimate $\mathcal{E}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ can be computed in time
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{\small
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\[O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot \abs{\circuit}^2(1,\ldots, 1)\cdot k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error}^2\cdot\rpoly^2(\prob_1,\ldots, \prob_\numvar)}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)\]
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}
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such that
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\begin{equation}
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\label{eq:approx-algo-bound}
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\probOf\left(\left|\mathcal{E} - \rpoly(\prob_1,\dots,\prob_\numvar)\right|> \error \cdot \rpoly(\prob_1,\dots,\prob_\numvar)\right) \leq \conf.
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\end{equation}
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\end{Theorem}
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\AR{\textbf{Aaron:}Just copied over from S4. The above text might need smoothening into the appendix.}
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\subsection{Proof of Theorem \ref{lem:approx-alg}}\label{sec:proof-lem-approx-alg}
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\input{app_approx-alg-pseudo-code}
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In order to prove \Cref{lem:approx-alg}, we will need to argue the correctness of \approxq, which relies on the correctness of auxiliary algorithms \onepass and \sampmon.
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132
approx_alg.tex
132
approx_alg.tex
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@ -4,13 +4,13 @@
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\section{$1 \pm \epsilon$ Approximation Algorithm}\label{sec:algo}
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In \Cref{sec:hard}, we showed that the answer to \Cref{prob:intro-stmt} is no.
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With this result, we now design an approximation algorithm for our problem that runs in $\bigO{\abs{\circuit}}$.\footnote{For a very broad class of circuits: please see the discussion after \Cref{lem:val-ub} for more.}
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The folowing approximation algorithm applies to \abbrBIDB lineage polynomials (over $\raPlus$ queries), though our bounds are more meaningful for a non-trivial subclass of queries over \bis that contains all queries on \tis, as well as the queries of the PDBench benchmark~\cite{pdbench}. As before, all proofs and pseudocode can be found in \Cref{sec:proofs-approx-alg}.
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With this result, we now design an approximation algorithm for our problem that runs in $\bigO{\abs{\circuit}}$ for a very broad class of circuits (see the discussion after \Cref{lem:val-ub} for more).
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The folowing approximation algorithm applies to \abbrBIDB lineage polynomials (over $\raPlus$ queries), though our bounds are more meaningful for a non-trivial subclass of queries over \bis that contains all queries on \tis, as well as the queries of the PDBench benchmark~\cite{pdbench}. All proofs and pseudocode can be found in \Cref{sec:proofs-approx-alg}.
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%it is then desirable to have an algorithm to approximate the multiplicity in linear time, which is what we describe next.
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\subsection{Preliminaries and some more notation}
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We now introduce useful definitions and notation related to circuits and polynomials.
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We now introduce definitions and notation related to circuits and polynomials that we will need to state our upper bound results.
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\begin{Definition}[$\expansion{\circuit}$]\label{def:expand-circuit}
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For a circuit $\circuit$, we define $\expansion{\circuit}$ as a list of tuples $(\monom, \coef)$, where $\monom$ is a set of variables and $\coef \in \domN$.
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@ -49,29 +49,12 @@ $\degree(\circuit)$ is defined recursively as follows:
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\]
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\end{Definition}
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Finally, we use the following notation for the complexity of multiplying integers:
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Next, we use the following notation for the complexity of multiplying integers:
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\begin{Definition}[$\multc{\cdot}{\cdot}$]\footnote{We note that when doing arithmetic operations on the RAM model for input of size $N$, we have that $\multc{O(\log{N})}{O(\log{N})}=O(1)$. More generally we have $\multc{N}{O(\log{N})}=O(N\log{N}\log\log{N})$.}
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In a RAM model of word size of $W$-bits, $\multc{M}{W}$ denotes the complexity of multiplying two integers represented with $M$-bits. (We will assume that for input of size $N$, $W=O(\log{N})$.
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In a RAM model of word size of $W$-bits, $\multc{M}{W}$ denotes the complexity of multiplying two integers represented with $M$-bits. (We will assume that for input of size $N$, $W=O(\log{N})$.)
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\end{Definition}
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\subsection{Our main result}\label{sec:algo:sub:main-result}
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The following results assume input circuit \circuit computed from an arbitrary $\raPlus$ query $\query$ and arbitrary \abbrBIDB $\pdb$. We refer to \circuit as a \abbrBIDB circuit.
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\AH{Verify that the proof for \Cref{lem:approx-alg} doesn't rely on properties of $\raPlus$ or \abbrBIDB.}
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\begin{Theorem}\label{lem:approx-alg}
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Let \circuit be an arbitrary \abbrBIDB circuit %for a UCQ over \bi
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and define $\poly(\vct{X})=\polyf(\circuit)$ and let $k=\degree(\circuit)$.
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Then an estimate $\mathcal{E}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ can be computed in time
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{\small
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\[O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot \abs{\circuit}^2(1,\ldots, 1)\cdot k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error}^2\cdot\rpoly^2(\prob_1,\ldots, \prob_\numvar)}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)\]
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}
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such that
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\begin{equation}
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\label{eq:approx-algo-bound}
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\probOf\left(\left|\mathcal{E} - \rpoly(\prob_1,\dots,\prob_\numvar)\right|> \error \cdot \rpoly(\prob_1,\dots,\prob_\numvar)\right) \leq \conf.
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\end{equation}
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\end{Theorem}
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To get linear runtime results from \Cref{lem:approx-alg}, we will need to define another parameter modeling the (weighted) number of monomials in %$\poly\inparen{\vct{X}}$
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Finally, to get linear runtime results from \Cref{lem:approx-alg}, we will need to define another parameter modeling the (weighted) number of monomials in %$\poly\inparen{\vct{X}}$
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$\expansion{\circuit}$
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that need to be `canceled' when monomials with dependent variables are removed (\Cref{def:reduced-bi-poly}). %def:hen it is modded with $\mathcal{B}$ (\Cref{def:mod-set-polys}).
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Let $\isInd{\cdot}$ be a boolean function returning true if monomial $\encMon$ is composed of independent variables and false otherwise; further, let $\indicator{\theta}$ also be a boolean function returning true if $\theta$ evaluates to true.
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@ -82,36 +65,14 @@ Given a \abbrBIDB circuit $\circuit$ define
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\[\gamma(\circuit)=\frac{\sum_{(\monom, \coef)\in \expansion{\circuit}} \abs{\coef}\cdot \indicator{\neg\isInd{\encMon}} }%\encMon\mod{\mathcal{B}}\equiv 0}}
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{\abs{\circuit}(1,\ldots, 1)}.\]
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\end{Definition}
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\noindent We next present a few corollaries of \Cref{lem:approx-alg}.
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\begin{Corollary}
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\label{cor:approx-algo-const-p}
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Let $\poly(\vct{X})$ be as in \Cref{lem:approx-alg} and let $\gamma=\gamma(\circuit)$ for \abbrBIDB circuit \circuit. Further let it be the case that $\prob_i\ge \prob_0$ for all $i\in[\numvar]$. Then an estimate $\mathcal{E}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ satisfying \Cref{eq:approx-algo-bound} can be computed in time
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\[O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)\]
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In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$.
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\end{Corollary}
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The restriction on $\gamma$ is satisfied by any \ti (where $\gamma=0$) as well as for all three queries of the PDBench \bi benchmark (see \Cref{app:subsec:experiment} for experimental results).
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Finally, we address the $\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}$ term in the runtime. %In \Cref{susec:proof-val-up}, we show the following:
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\begin{Lemma}
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\label{lem:val-ub}
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For any \abbrBIDB circuit $\circuit$ with $\degree(\circuit)=k$, we have
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$\abs{\circuit}(1,\ldots, 1)\le 2^{2^k\cdot \size(\circuit)}.$
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Further, under either of the following conditions:
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\begin{enumerate}
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\item $\circuit$ is a tree,
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\item $\circuit$ encodes the run of the algorithm on a FAQ~\cite{DBLP:conf/pods/KhamisNR16}\AH{AJAR citation.} query,
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\end{enumerate}
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we have $\abs{\circuit}(1,\ldots, 1)\le \size(\circuit)^{O(k)}.$
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\end{Lemma}
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Note that the above implies that with the assumption $\prob_0>0$ and $\gamma<1$ are absolute constants from \Cref{cor:approx-algo-const-p}, then the runtime there simplies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)^2\cdot \log{\frac{1}{\conf}}\right)$ for general circuits $\circuit$ and to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$ for the case when $\circuit$ satisfies the specific conditions in \Cref{lem:val-ub}. In \Cref{app:proof-lem-val-ub} we argue that these conditions are very general and encompass many interesting scenarios, including query evaluation under \raPlus or FAQ.
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\AH{AJAR reference.}
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\subsection{Our main result}\label{sec:algo:sub:main-result}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Approximating $\rpoly$}
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We prove \Cref{lem:approx-alg} by developing an approximation algorithm (\approxq pseudo code in \Cref{sec:proof-lem-approx-alg}) with the desired runtime. This algorithm is based on the following observation.
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\mypar{Algorithm Idea}
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%We prove \Cref{lem:approx-alg} by developing an
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Our approximation algorithm (\approxq pseudo code in \Cref{sec:proof-lem-approx-alg})
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%with the desired runtime. This algorithm
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is based on the following observation.
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% The algorithm (\approxq detailed in \Cref{alg:mon-sam}) to prove \Cref{lem:approx-alg} follows from the following observation.
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Given a lineage polynomial $\poly(\vct{X})=\polyf(\circuit)$ for circuit \circuit over $\bi$, we have: % can exactly represent $\rpoly(\vct{X})$ as follows:
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@ -126,9 +87,76 @@ Given a lineage polynomial $\poly(\vct{X})=\polyf(\circuit)$ for circuit \circui
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Given the above, the algorithm is a sampling based algorithm for the above sum: we sample (via \sampmon) $(\monom,\coef)\in \expansion{\circuit}$ with probability proportional
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to $\abs{\coef}$ and compute $\vari{Y}=\indicator{\isInd{\encMon}}
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\cdot \prod_{X_i\in \monom} p_i$. Taking $\ceil{\frac{2 \log{\frac{2}{\conf}}}{\error^2}}$ samples and computing the average of $\vari{Y}$ gives us our final estimate. \onepass is used to compute the sampling probabilities needed in \sampmon (details are in \Cref{sec:proofs-approx-alg}).
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\cdot \prod_{X_i\in \monom} p_i$. %Taking $\ceil{\frac{2 \log{\frac{2}{\conf}}}{\error^2}}$ samples
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Repeating the sampling appropriate number of times
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and computing the average of $\vari{Y}$ gives us our final estimate. \onepass is used to compute the sampling probabilities needed in \sampmon (details are in \Cref{sec:proofs-approx-alg}).
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%%%%%%%%%%%%%%%%%%%%%%%
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%The following results assume input circuit \circuit computed from an arbitrary $\raPlus$ query $\query$ and arbitrary \abbrBIDB $\pdb$. We refer to \circuit as a \abbrBIDB circuit.
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%\AH{Verify that the proof for \Cref{lem:approx-alg} doesn't rely on properties of $\raPlus$ or \abbrBIDB.}
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%\begin{Theorem}\label{lem:approx-alg}
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%Let \circuit be an arbitrary \abbrBIDB circuit %for a UCQ over \bi
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%and define $\poly(\vct{X})=\polyf(\circuit)$ and let $k=\degree(\circuit)$.
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%Then an estimate $\mathcal{E}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ can be computed in time
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%{\small
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%\[O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot \abs{\circuit}^2(1,\ldots, 1)\cdot k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error}^2\cdot\rpoly^2(\prob_1,\ldots, \prob_\numvar)}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)\]
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%}
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%such that
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%\begin{equation}
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%\label{eq:approx-algo-bound}
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%\probOf\left(\left|\mathcal{E} - \rpoly(\prob_1,\dots,\prob_\numvar)\right|> \error \cdot \rpoly(\prob_1,\dots,\prob_\numvar)\right) \leq \conf.
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%\end{equation}
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%\end{Theorem}
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\mypar{Runtime analysis} We can argue the following runtime for the algorithm outlined above:
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% We next present a few corollaries of \Cref{lem:approx-alg}.
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\begin{Theorem}
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\label{cor:approx-algo-const-p}
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Let \circuit be an arbitrary \abbrBIDB circuit %for a UCQ over \bi
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and define $\poly(\vct{X})=\polyf(\circuit)$ and let $k=\degree(\circuit)$.
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%Let $\poly(\vct{X})$ be as in \Cref{lem:approx-alg} and
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Let $\gamma=\gamma(\circuit)$ for \abbrBIDB circuit \circuit. Further let it be the case that $\prob_i\ge \prob_0$ for all $i\in[\numvar]$. Then an estimate $\mathcal{E}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$
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satisfying
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\begin{equation}
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\label{eq:approx-algo-bound-main}
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\probOf\left(\left|\mathcal{E} - \rpoly(\prob_1,\dots,\prob_\numvar)\right|> \error' \cdot \rpoly(\prob_1,\dots,\prob_\numvar)\right) \leq \conf
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\end{equation}
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can be computed in time
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\begin{equation}
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\label{eq:approx-algo-runtime}
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O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right).
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\end{equation}
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In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$.
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\end{Theorem}
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The restriction on $\gamma$ is satisfied by any \ti (where $\gamma=0$) as well as for all three queries of the PDBench \bi benchmark (see \Cref{app:subsec:experiment} for experimental results).
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We briefly connect the runtime in \Cref{eq:approx-algo-runtime} to the algorithm outline earlier (where we ignore the dependence on $\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}$). The $\size(\circuit)$ comes from the time take to run \onepass once (\onepass essentially computes $\abs{\circuit}(1,\ldots, 1)$ using the natural circuit evaluation algorithm on $\circuit$). We make $\frac{\log{\frac{1}{\conf}}}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}$ many calls to \sampmon (each of which essentially traces $O(k)$ random sink to source paths in $\circuit$ all of which by definition have length at most $\depth(\circuit)$.
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Finally, we address the $\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}$ term in the runtime. %In \Cref{susec:proof-val-up}, we show the following:
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\begin{Lemma}
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\label{lem:val-ub}
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For any \abbrBIDB circuit $\circuit$ with $\degree(\circuit)=k$, we have
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$\abs{\circuit}(1,\ldots, 1)\le 2^{2^k\cdot \size(\circuit)}.$
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Further, under either of the following conditions:
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\begin{enumerate}
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\item $\circuit$ is a tree,
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\item $\circuit$ encodes the run of the algorithm on a FAQ~\cite{DBLP:conf/pods/KhamisNR16}/AJAR~\cite{ajar} query,
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\end{enumerate}
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we have $\abs{\circuit}(1,\ldots, 1)\le \size(\circuit)^{O(k)}.$
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\end{Lemma}
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Note that the above implies that with the assumption $\prob_0>0$ and $\gamma<1$ are absolute constants from \Cref{cor:approx-algo-const-p}, then the runtime there simplies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)^2\cdot \log{\frac{1}{\conf}}\right)$ for general circuits $\circuit$ and
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\begin{Corollary}
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Let $\poly(\vct{X})$ be a \abbrBIDB-lineage polynomial correspoding to an \abbrBIDB circuit $\circuit$ that satisfies the specific conditions in \Cref{lem:val-ub}. Then one can compute an approximation satisfying \Cref{eq:approx-algo-bound-main} in time
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$O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$. % for the case when $\circuit$ satisfies the specific conditions in \Cref{lem:val-ub}.
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\end{Corollary}
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\AR{The above Corollary needs to be improved/generalized. This is a place-holder for now.}
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In \Cref{app:proof-lem-val-ub} we argue that these conditions are very general and encompass many interesting scenarios, including query evaluation under FAQ/AJAR setup.
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%\AH{AJAR reference.}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "main"
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@ -1,17 +1,18 @@
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%!TEX root=./main.tex
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\section{Conclusions and Future Work}\label{sec:concl-future-work}
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We have studied the problem of calculating the expectation of lineage polynomials over BIDBs. %random integer variables.
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This problem has a practical application in probabilistic databases over multisets, where it corresponds to calculating the expected multiplicity of a query result tuple.
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We have studied the problem of calculating the expected multiplicity of a query result tuple, %expectation of lineage polynomials over BIDBs. %random integer variables.
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a problem that has a practical application in probabilistic databases over multisets. %, where it corresponds to calculating the expected multiplicity of a query result tuple.
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% It has been studied extensively for sets (lineage formulas), but the bag settings has not received much attention.
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While the expectation of a polynomial can be calculated in linear time for % in the size of
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polynomials % that are
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in SOP form, the problem is \sharpwonehard for factorized polynomials (proven through a reduction from the problem of counting k-matchings).
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%While the expectation of a polynomial can be calculated in linear time for % in the size of
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% polynomials % that are
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%in SOP form, the problem is \sharpwonehard for factorized polynomials (proven through a reduction from the problem of counting k-matchings).
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%We have proven this claim through a reduction from the problem of counting k-matchings.
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We show that under various parameterized complexity hardness results/conjectures computing the expected multiplicities exactly is not possible in time linear in the corresponding deterministic query processing time.
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We prove that it is possible to approximate the expectation of a lineage polynomial in linear time
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% When only considering polynomials for result tuples of
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for UCQs over TIDBs and BIDBs (assuming that there are few cancellations).
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Interesting directions for future work include development of a dichotomy for bag \abbrPDB\xplural. While we handle higher moments in \Cref{app:sec-cicuits}, more general approximations are an interesting area for exploration, including those for more general data models. % beyond what we consider in this paper.
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in the deterministic query processing over TIDBs and BIDBs (assuming that there are few cancellations).
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Interesting directions for future work include development of a dichotomy for bag \abbrPDB\xplural. While we can handle higher moments (this follows fairly easily from our existing results-- see \Cref{app:sec-cicuits}), more general approximations are an interesting area for exploration, including those for more general data models. % beyond what we consider in this paper.
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% Furthermore, it would be interesting to see whether our approximation algorithm can be extended to support queries with negations, perhaps using circuits with monus as a representation system.
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% \BG{I am not sure what interesting future work is here. Some wild guesses, if anybody agrees I'll try to flesh them out:
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@ -17,14 +17,14 @@ Dalvi et al.~\cite{DS12} and Olteanu et al.~\cite{FO16} proved dichotomies for U
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% Olteanu et al.~\cite{FO16} presented dichotomies for two classes of queries with negation.
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% R\'e et al~\cite{RS09b} present a trichotomy for HAVING queries.
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Amarilli et al. investigated tractable classes of databases for more complex queries~\cite{AB15}. %,AB15c
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Another line of work, studies which structural properties of lineage formulas lead to tractable cases~\cite{kenig-13-nclexpdc,roy-11-f,sen-10-ronfqevpd}.
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Another line of work studies which structural properties of lineage formulas lead to tractable cases~\cite{kenig-13-nclexpdc,roy-11-f,sen-10-ronfqevpd}.
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In this paper we focus on intensional query evaluation with polynomials.
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Many data models have been proposed for encoding PDBs more compactly than as sets of possible worlds.
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These include tuple-independent databases~\cite{VS17} (\tis), block-independent databases (\bis)~\cite{RS07}, and \emph{PC-tables}~\cite{GT06}.
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%
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Fink et al.~\cite{FH12} study aggregate queries over a probabilistic version of the extension of K-relations for aggregate queries proposed in~\cite{AD11d} (\emph{pvc-tables}) that supports bags, and has runtime complexity linear in the size of the lineage.
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However, this lineage is encoded as a tree; the size (and thus the runtime) are still superlinear in $\qruntime(\query, \dbbase)$.
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However, this lineage is encoded as a tree; the size (and thus the runtime) are still superlinear in $\qruntime{\query, \dbbase}$.
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The runtime bound is also limited to a specific class of (hierarchical) queries, suggesting the possibility of a generalization of \cite{DS12}'s dichotomy result to \abbrBPDB\xplural.
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% Probabilities are computed using a decomposition approach~\cite{DBLP:conf/icde/OlteanuHK10}.
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% over the symbolic expressions that are used as tuple annotations and values in pvc-tables.
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Reference in New Issue