153 lines
8.2 KiB
TeX
153 lines
8.2 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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Supplementary Material for~\Cref{prop:expection-of-polynom}}\label{subsec:supp-mat-background}
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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-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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\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 $|R|$ 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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\subsection{Proof for~\Cref{lem:circuits-model-runtime}}\label{app:subsec-lem-lin-vs-qplan}
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Proof by induction. The base case is a base relation: $Q = R$ and is trivially true since $|V_{R,\pxdb}| = |R|$.
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For the inductive step, we assume that we have circuits for subplans $Q_1, \ldots, Q_n$ such that $|V_{Q_i,\pxdb}| \leq (k_i-1)\qruntime{Q_i,\pxdb}$ 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 $Q$, $|V_{Q,\pxdb}| = |V_{Q_1,\pxdb}|$ vertices, so from the inductive assumption and $\qruntime{Q,\pxdb} = \qruntime{Q_1,\pxdb}$ by definition, we have $|V_{Q,\pxdb}| \leq (k-1) \qruntime{Q,\pxdb} $.
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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} $\qruntime{Q_1,\pxdb} \geq |Q_1|$}
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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,\pxdb} + \abs{Q_1}\\
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\intertext{(By definition of $\qruntime{Q,\pxdb}$)}
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& \le (k-1)\qruntime{Q,\pxdb}.
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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} $\qruntime{Q_1,\pxdb} \geq |Q_1|$}
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%& \leq |V_{Q_1,\pxdb}|+|V_{Q_2,\pxdb}|+\qruntime{Q_1,\pxdb}+\qruntime{Q_2,\pxdb}|\\
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\intertext{(From the inductive assumption)}
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& \leq (k-1)(\qruntime{Q_1,\pxdb} + \qruntime{Q_2,\pxdb}) + (b_1 + b_2)
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\intertext{(By definition of $\qruntime{Q,\pxdb}$)}
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& \leq (k-1)(\qruntime{Q,\pxdb}).
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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,\pxdb}+\ldots+(k-1)\qruntime{Q_k,\pxdb}+\\
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&\;\;\; (k-1)|{Q_1} \bowtie \ldots \bowtie {Q_k}|\\
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& \leq (k-1)(\qruntime{Q_1,\pxdb}+\ldots+\qruntime{Q_k,\pxdb}+\\
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&\;\;\;|{Q_1} \bowtie \ldots \bowtie {Q_k}|)\\
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\intertext{(By definition of $\qruntime{Q,\pxdb}$)}
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& = (k-1)\qruntime{Q,\pxdb}.
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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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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "main"
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%%% End:
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