207 lines
13 KiB
TeX
207 lines
13 KiB
TeX
%!TEX root=./main.tex
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Missing details from Section~\ref{sec:background}}\label{sec:proofs-background}
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\subsection{$\semK$-relations and $\semNX$-PDBs}\label{subsec:supp-mat-background}\label{subsec:supp-mat-krelations}
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\input{app_notation-background}
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\section{Missing details from Section~\ref{sec:hard}}
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\label{app:single-mult-p}
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\input{app_hardness-results}
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\section{Missing Details from Section~\ref{sec:algo}}\label{sec:proofs-approx-alg}
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\input{app_approx-alg-defs-and-examples}
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\input{app_approx-alg-analysis}
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\input{app_one-pass-analysis}
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\input{app_samp-monom-analysis}
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\subsection{Experimental Results}\label{app:subsec:experiment}
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\input{experiments}
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\section{Circuits}\label{app:sec-cicuits}
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\subsection{Representing Polynomials with Circuits}\label{app:subsec-rep-poly-lin-circ}
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\subsubsection{Circuits for query plans}
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\label{sec:circuits-formal}
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We now formalize circuits and the construction of circuits for SPJU queries.
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As mentioned earlier, we represent lineage polynomials as arithmetic circuits over $\mathbb N$-valued variables with $+$, $\times$.
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A circuit for query $Q$ and $\semNX$-PDB $\pxdb$ is a directed acyclic graph $\tuple{V_{Q,\pxdb}, E_{Q,\pxdb}, \phi_{Q,\pxdb}, \ell_{Q,\pxdb}}$ with vertices $V_{Q,\pxdb}$ and directed edges $E_{Q,\pxdb} \subset {V_{Q,\pxdb}}^2$.
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The sink function $\phi_{Q,\pxdb} : \udom^n \rightarrow V_{Q,\pxdb}$ is a partial function that maps the tuples of the $n$-ary relation $Q(\pxdb)$ to vertices.
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We require that $\phi_{Q,\pxdb}$'s range be limited to sink vertices (i.e., vertices with out-degree 0).
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%We call a sink vertex not in the range of $\phi_R$ a \emph{dead sink}.
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A function $\ell_{Q,\pxdb} : V_{Q,\pxdb} \rightarrow \{\;+,\times\;\}\cup \mathbb N \cup \vct X$ assigns a label to each node: Source nodes (i.e., vertices with in-degree 0) are labeled with constants or variables (i.e., $\mathbb N \cup \vct X$), while the remaining nodes are labeled with the symbol $+$ or $\times$.
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We require that vertices have an in-degree of at most two.
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%For the specifics on how to construct a circuit to encode the polynomials of all result tuples for a query and $\semNX$-PDB see \Cref{app:subsec-rep-poly-lin-circ}.
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Note that we can construct circuits for \bis in time linear in the time required for deterministic query processing over a possible world of the \bi under the aforementioned assumption that $\abs{\pxdb} \leq c \cdot \abs{\db}$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsubsection{Circuit size vs. runtime}
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\label{sec:circuit-runtime}
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\newcommand{\bagdbof}{\textsc{bag}(\pxdb)}
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We now connect the size of a circuit (where the size of a circuit is the number of vertices in the corresponding DAG) %\footnote{since each node has indegree at most two, this also is the same up to constants to counting the number of edges in the DAG.})
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for a given SPJU query $Q$ and $\semNX$-PDB $\pxdb$ to
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the runtime $\qruntime{Q,\dbbase}$ of the PDB's \dbbaseName $\dbbase$.
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We do this formally by showing that the size of the circuit is asymptotically no worse than the corresponding runtime of a large class of deterministic query processing algorithms.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\getpoly}[1]{\textbf{lin}\inparen{#1}}
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Each vertex $v \in V_{Q,\pxdb}$ in the arithmetic circuit for
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\[\tuple{V_{Q,\pxdb}, E_{Q,\pxdb}, \phi_{Q,\pxdb}, \ell_{Q,\pxdb}}\]
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encodes a polynomial, realized as
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\[\getpoly{v} = \begin{cases}
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\sum_{v' : (v',v) \in E_{Q,\pxdb}} \getpoly{v'} & \textbf{if } \ell(v) = +\\
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\prod_{v' : (v',v) \in E_{Q,\pxdb}} \getpoly{v'} & \textbf{if } \ell(v) = \times\\
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\ell(v) & \textbf{otherwise}
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\end{cases}\]
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We define the circuit for a select-union-project-join $Q$ recursively by cases as follows. In each case, let $\tuple{V_{Q_i,\pxdb}, E_{Q_i,\pxdb}, \phi_{Q_{i},\pxdb}, \ell_{Q_i,\pxdb}}$ denote the circuit for subquery $Q_i$.
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\caseheading{Base Relation}
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Let $Q$ be a base relation $R$. We define one node for each tuple. Formally, let $V_{Q,\pxdb} = \comprehension{v_t}{t\in R}$, let $\phi_{Q,\pxdb}(t) = v_t$, let $\ell_{Q,\pxdb}(v_t) = R(t)$, and let $E_{Q,\pxdb} = \emptyset$.
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This circuit has $|D_\Omega.R|$ vertices.
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\caseheading{Selection}
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Let $Q = \sigma_\theta \inparen{Q_1}$.
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We re-use the circuit for $Q_1$. %, but define a new distinguished node $v_0$ with label $0$ and make it the sink node for all tuples that fail the selection predicate.
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Formally, let $V_{Q,\pxdb} = V_{Q_1,\pxdb}$, let $\ell_{Q,\pxdb}(v_0) = 0$, and let $\ell_{Q,\pxdb}(v) = \ell_{Q_1,\pxdb}(v)$ for any $v \in V_{Q_1,\pxdb}$. Let $E_{Q,\pxdb} = E_{Q_1,\pxdb}$, and define
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$$\phi_{Q,\pxdb}(t) =
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\phi_{Q_{1}, \pxdb}(t) \text{ for } t \text{ s.t.}\; \theta(t).$$
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Dead sinks are iteratively removed, and so
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%\AH{While not explicit, I assume a reviewer would know that the notation above discards tuples/vertices not satisfying the selection predicate.}
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%v_0 & \textbf{otherwise}
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%\end{cases}$$
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this circuit has at most $|V_{Q_1,\pxdb}|$ vertices.
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\caseheading{Projection}
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Let $Q = \pi_{\vct A} {Q_1}$.
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We extend the circuit for ${Q_1}$ with a new set of sum vertices (i.e., vertices with label $+$) for each tuple in $Q$, and connect them to the corresponding sink nodes of the circuit for ${Q_1}$.
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Naively, let $V_{Q,\pxdb} = V_{Q_1,\pxdb} \cup \comprehension{v_t}{t \in \pi_{\vct A} {Q_1}}$, let $\phi_{Q,\pxdb}(t) = v_t$, and let $\ell_{Q,\pxdb}(v_t) = +$. Finally let
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$$E_{Q,\pxdb} = E_{Q_1,\pxdb} \cup \comprehension{(\phi_{Q_{1}, \pxdb}(t'), v_t)}{t = \pi_{\vct A} t', t' \in {Q_1}, t \in \pi_{\vct A} {Q_1}}$$
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This formulation will produce vertices with an in-degree greater than two, a problem that we correct by replacing every vertex with an in-degree over two by an equivalent fan-in tree. The resulting structure has at most $|{Q_1}|-1$ new vertices.
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% \AH{Is the rightmost operator \emph{supposed} to be a $-$? In the beginning we add $|\pi_{\vct A}{Q_1}|$ vertices.}
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The corrected circuit thus has at most $|V_{Q_1,\pxdb}|+|{Q_1}|$ vertices.
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\caseheading{Union}
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Let $Q = {Q_1} \cup {Q_2}$.
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We merge graphs and produce a sum vertex for all tuples in both sides of the union.
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Formally, let $V_{Q,\pxdb} = V_{Q_1,\pxdb} \cup V_{Q_2,\pxdb} \cup \comprehension{v_t}{t \in {Q_1} \cap {Q_2}}$, let $\ell_{Q,\pxdb}(v_t) = +$, and let
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\[E_{Q,\pxdb} = E_{Q_1,\pxdb} \cup E_{Q_2,\pxdb} \cup \comprehension{(\phi_{Q_{1}, \pxdb}(t), v_t), (\phi_{Q_{2}, \pxdb}(t), v_t)}{t \in {Q_1} \cap {Q_2}}\]
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\[
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\phi_{Q,\pxdb}(t) = \begin{cases}
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v_t & \textbf{if } t \in {Q_1} \cap {Q_1}\\
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\phi_{Q_{1}, \pxdb}(t) & \textbf{if } t \not \in {Q_2}\\
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\phi_{Q_{2}, \pxdb}(t) & \textbf{if } t \not \in {Q_1}\\
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\end{cases}\]
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This circuit has $|V_{Q_1,\pxdb}|+|V_{Q_2,\pxdb}|+|{Q_1} \cap {Q_2}|$ vertices.
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\caseheading{$k$-ary Join}
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Let $Q = {Q_1} \bowtie \ldots \bowtie {Q_k}$.
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We merge graphs and produce a multiplication vertex for all tuples resulting from the join
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Naively, let $V_{Q,\pxdb} = V_{Q_1,\pxdb} \cup \ldots \cup V_{Q_k,\pxdb} \cup \comprehension{v_t}{t \in {Q_1} \bowtie \ldots \bowtie {Q_k}}$, let
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{\small
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\begin{multline*}
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E_{Q,\pxdb} = E_{Q_1,\pxdb} \cup \ldots \cup E_{Q_k,\pxdb} \cup
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\left\{\;
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(\phi_{Q_{1}, \pxdb}(\pi_{\sch({Q_1})}t), v_t), \right.\\
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\ldots, (\phi_{Q_k,\pxdb}(\pi_{\sch({Q_k})}t), v_t)
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\;\left|\;t \in {Q_1} \bowtie \ldots \bowtie {Q_k}\;\right\}
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\end{multline*}
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}
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Let $\ell_{Q,\pxdb}(v_t) = \times$, and let $\phi_{Q,\pxdb}(t) = v_t$
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As in projection, newly created vertices will have an in-degree of $k$, and a fan-in tree is required.
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There are $|{Q_1} \bowtie \ldots \bowtie {Q_k}|$ such vertices, so the corrected circuit has $|V_{Q_1,\pxdb}|+\ldots+|V_{Q_k,\pxdb}|+(k-1)|{Q_1} \bowtie \ldots \bowtie {Q_k}|$ vertices.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Lemma}\label{lem:circ-model-runtime}
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\label{lem:circuits-model-runtime}
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Given a $\semNX$-PDB $\pxdb$ with \dbbaseName $\dbbase$, and query plan $Q$, the runtime of $Q$ over $\dbbase$ has the same or greater complexity as the size of the lineage of $Q(\pxdb)$. That is, we have $\abs{V_{Q,\pxdb}} \leq (k-1)\qruntime{Q, \dbbase}$, where $k$ is the maximal degree of any polynomial in $Q(\pxdb)$.
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\end{Lemma}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\noindent The proof is shown in \Cref{app:subsec-lem-lin-vs-qplan}.
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%\subsection{Proof for \Cref{lem:circuits-model-runtime}}\label{app:subsec-lem-lin-vs-qplan}
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\begin{proof}
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Proof by induction. The base case is a base relation: $Q = R$ and is trivially true since $|V_{R,\pxdb}| = |D_\Omega.R|$.
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For the inductive step, we assume that we have circuits for subplans $Q_1, \ldots, Q_n$ such that $|V_{Q_i,\pxdb}| \leq (k_i-1)\qruntime{Q_i,\dbbase}$ where $k_i$ is the degree of $Q_i$.
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\caseheading{Selection}
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Assume that $Q = \sigma_\theta(Q_1)$.
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In the circuit for $Q$, $|V_{Q,\pxdb}| = |V_{Q_1,\dbbase}|$ vertices, so from the inductive assumption and $\qruntime{Q,\dbbase} = \qruntime{Q_1,\dbbase}$ by definition, we have $|V_{Q,\pxdb}| \leq (k-1) \qruntime{Q,\dbbase} $.
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% \AH{Technically, $\kElem$ is the degree of $\poly_1$, but I guess this is a moot point since one can argue that $\kElem$ is also the degree of $\poly$.}
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% OK: Correct
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\caseheading{Projection}
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Assume that $Q = \pi_{\vct A}(Q_1)$.
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The circuit for $Q$ has at most $|V_{Q_1,\pxdb}|+|{Q_1}|$ vertices.
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% \AH{The combination of terms above doesn't follow the details for projection above.}
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\begin{align*}
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|V_{Q,\pxdb}| & \leq |V_{Q_1,\pxdb}| + |Q_1|\\
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%\intertext{By \Cref{prop:queries-need-to-output-tuples} $\qruntime{Q_1,\dbbase} \geq |Q_1|$}
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%& \leq |V_{Q_1,\pxdb}| + 2 \qruntime{Q_1,\pxdb}\\
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\intertext{(From the inductive assumption)}
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& \leq (k-1)\qruntime{Q_1,\dbbase} + \abs{Q_1}\\
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\intertext{(By definition of $\qruntime{Q,\dbbase}$)}
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& \le (k-1)\qruntime{Q,\dbbase}.
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\end{align*}
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\caseheading{Union}
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Assume that $Q = Q_1 \cup Q_2$.
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The circuit for $Q$ has $|V_{Q_1,\pxdb}|+|V_{Q_2,\pxdb}|+|{Q_1} \cap {Q_2}|$ vertices.
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\begin{align*}
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|V_{Q,\pxdb}| & \leq |V_{Q_1,\pxdb}|+|V_{Q_2,\pxdb}|+|{Q_1}|+|{Q_2}|\\
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%\intertext{By \Cref{prop:queries-need-to-output-tuples} $\qruntime{Q_1,\dbbase} \geq |Q_1|$}
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%& \leq |V_{Q_1,\pxdb}|+|V_{Q_2,\pxdb}|+\qruntime{Q_1,\pxdb}+\qruntime{Q_2,\dbbase}|\\
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\intertext{(From the inductive assumption)}
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& \leq (k-1)(\qruntime{Q_1,\dbbase} + \qruntime{Q_2,\dbbase}) + (b_1 + b_2)
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\intertext{(By definition of $\qruntime{Q,\dbbase}$)}
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& \leq (k-1)(\qruntime{Q,\dbbase}).
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\end{align*}
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\caseheading{$k$-ary Join}
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Assume that $Q = Q_1 \bowtie \ldots \bowtie Q_k$.
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The circuit for $Q$ has $|V_{Q_1,\pxdb}|+\ldots+|V_{Q_k,\pxdb}|+(k-1)|{Q_1} \bowtie \ldots \bowtie {Q_k}|$ vertices.
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\begin{align*}
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|V_{Q,\pxdb}| & = |V_{Q_1,\pxdb}|+\ldots+|V_{Q_k,\pxdb}|+(k-1)|{Q_1} \bowtie \ldots \bowtie {Q_k}|\\
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\intertext{From the inductive assumption and noting $\forall i: k_i \leq k-1$}
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& \leq (k-1)\qruntime{Q_1,\dbbase}+\ldots+(k-1)\qruntime{Q_k,\dbbase}+\\
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&\;\;\; (k-1)|{Q_1} \bowtie \ldots \bowtie {Q_k}|\\
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& \leq (k-1)(\qruntime{Q_1,\dbbase}+\ldots+\qruntime{Q_k,\dbbase}+\\
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&\;\;\;|{Q_1} \bowtie \ldots \bowtie {Q_k}|)\\
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\intertext{(By definition of $\qruntime{Q,\dbbase}$)}
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& = (k-1)\qruntime{Q,\dbbase}.
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\end{align*}
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The property holds for all recursive queries, and the proof holds.
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\qed
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\end{proof}
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With \cref{lem:circ-model-runtime} and our upper bound results on \approxq, we now have all the pieces to argue that using our approximation algorithm, the expected multiplicities of an $\raPlus$ query can be computed in essentially the same runtime as deterministic query processing for the same query, proving claim (iv) of the Introduction.
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\section{Proof of \Cref{cor:cost-model}}
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\begin{proof}
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This follows from \Cref{lem:circuits-model-runtime} (\Cref{sec:circuit-runtime}) and \Cref{cor:approx-algo-const-p} (where the latter is used with $\delta$ being substituted\footnote{Recall that \Cref{cor:approx-algo-const-p} is stated for a single output tuple so to get the required guarantee for all (at most $n^k$) output tuples of $Q$ we get at most $\frac \delta{n^k}$ probability of failure for each output tuple and then just a union bound over all output tuples. } with $\frac \delta{n^k}$).
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\qed
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\end{proof}
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\mypar{Higher Moments}
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%\label{sec:momemts}
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%
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We make a simple observation to conclude the presentation of our results.
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So far we have only focused on the expectation of $\poly$.
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In addition, we could e.g. prove bounds of the probability of a tuple's multiplicity being at least $1$.
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Progress can be made on this as follows:
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For any positive integer $m$ we can compute the $m$-th moment of the multiplicities, allowing us to e.g. use the Chebyschev inequality or other high moment based probability bounds on the events we might be interested in.
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We leave further investigations for future work.
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "main"
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%%% End:
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