ODIn/paper-BagRelationalPDBsAreHard
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cleaning up a few terms

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Oliver Kennedy 2021-09-18 00:58:03 -04:00
parent e5ca8a4b24
commit 18d105f610
Signed by: okennedy
GPG key ID: 3E5F9B3ABD3FDB60
1 changed files with 11 additions and 11 deletions
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appendix.tex
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@ -71,7 +71,7 @@ We define the circuit for a select-union-project-join $Q$ recursively by cases a
\begin{algorithmic}[1]
\Require $\query$: query
\Require $\dbbase$: a \dbbaseName
\Ensure $\circuit = \tuple{E, V, \ell, \phi}$: a circuit encoding the lineage of each tuple in $\query(\dbbase)$
\Ensure $\circuit = \tuple{E, V, \phi, \ell}$: a circuit encoding the lineage of each tuple in $\query(\dbbase)$
\If{$\query$ is $R$}
\State $V = \comprehension{v_t}{t \in \dbbase.R}$
@ -82,7 +82,7 @@ We define the circuit for a select-union-project-join $Q$ recursively by cases a
\EndFor
\ElsIf{$\query$ is $\sigma_\theta(\query')$}
\State $\tuple{V, E, \phi', \ell} = \abbrStepOne(\query', \dbbase)$
\For{$t \in \dbbase.R$}
\For{$t \in \domain(\phi')$}
\If{$\theta(t)$}
\State $\phi(t) = \phi'(t)$
\Else
@ -91,21 +91,21 @@ We define the circuit for a select-union-project-join $Q$ recursively by cases a
\EndFor
\ElsIf{$\query$ is $\pi_{\vec{A}}(\query')$}
\State $\tuple{V', E', \phi', \ell'} = \abbrStepOne(\query', \dbbase)$
\State $V = V' \cup \comprehension{v_t}{t \in \pi_{\vec{A}}(\query)}$
\State $E = E' \cup \comprehension{(\phi(t'), v_t)}{t \in \pi_{\vec{A}}t', t' \in \query', t \in \pi_{\vec{A}}(\query')}$
\State $V = V' \cup \comprehension{v_t}{t \in \pi_{\vec{A}}(\domain(\phi'))}$
\State $E = E' \cup \comprehension{(\phi(t'), v_t)}{t \in \pi_{\vec{A}}t', t' \in \domain(\phi'), t \in \pi_{\vec{A}}(\domain(\phi'))}$
\Comment{Nodes with in-degrees above 2 are corrected (with logarithmic overhead) with an equivalent fan-in tree.}
\For{$t \in \pi_{\vec{A}}(\query')$}
\For{$t \in \pi_{\vec{A}}(\query'(\dbbase))$}
\State $\phi(t) = v_t$ \Comment{$v_t$ as defined above}
\State $\ell(v_t) = +$
\EndFor
\ElsIf{$\query$ is $\query_1 \cup \query_2$}
\State $\tuple{V_1, E_1, \phi_1, \ell_1} = \abbrStepOne(\query_1, \dbbase)$
\State $\tuple{V_2, E_2, \phi_2, \ell_2} = \abbrStepOne(\query_2, \dbbase)$
\State $V = V_1 \cup V_2 \cup \comprehension{v_t}{t \in \query_1 \cap \query_2}$
\State $E = E_1 \cup E_2 \cup \comprehension{(\phi_1(t), v_t), (\phi_2(t), v_t)}{t \in \query_1 \cap \query_2}$
\State $V = V_1 \cup V_2 \cup \comprehension{v_t}{t \in \domain(\phi_1) \cap \domain(\phi_2)}$
\State $E = E_1 \cup E_2 \cup \comprehension{(\phi_1(t), v_t), (\phi_2(t), v_t)}{t \in \domain(\phi_1) \cap \domain(\phi_2)}$
\State $\phi = \phi_1 \cup \phi_2$
\State $\ell = \ell_1 \cup \ell_2$
\For{$t \in \query_1 \cap \query_2$}
\For{$t \in \domain(\phi_1) \cap \domain(\phi_2)$}
\State $\phi(t) = v_t$ \Comment{$v_t$ as defined above}
\State $\ell(v_t) = +$
\EndFor
@ -113,12 +113,12 @@ We define the circuit for a select-union-project-join $Q$ recursively by cases a
\For{$i \in [1, k]$}
$\tuple{V_i, E_i, \phi_i, \ell_i} = \abbrStepOne(\query_i, \dbbase)$
\EndFor
\State $V = V_1 \cup \ldots \cup V_k \cup \comprehension{v_t}{t \in \query_1 \bowtie \ldots \bowtie \query_k}$
\State $V = V_1 \cup \ldots \cup V_k \cup \comprehension{v_t}{t \in \domain(\phi_1) \bowtie \ldots \bowtie \domain(\phi_k)}$
\State $E = E_1 \cup \ldots \cup E_k \cup \bigcup_{i \in [1,k]}
\comprehension{(\phi_i(\pi_{sch}(\query_i)(t))}{t \in \query_1 \bowtie \ldots \bowtie \query_k}$\Comment{Nodes with in-degrees above 2 are corrected (with $\log_2(k)$ overhead) with an equivalent fan-in tree.}
\comprehension{(\phi_i(\pi_{sch}(\query_i)(t))}{t \in \domain(\phi_1) \bowtie \ldots \bowtie \domain(\phi_k)}$\Comment{Nodes with in-degrees above 2 are corrected (with $\log_2(k)$ overhead) with an equivalent fan-in tree.}
\State $\phi = \phi_1 \cup \ldots \cup \phi_k$
\State $\ell = \ell_1 \cup \ldots \cup \phi_k$
\For{$t \in \query_1 \bowtie \ldots \bowtie \query_k$}
\For{$t \in \domain(\phi_1) \bowtie \ldots \bowtie \domain(\phi_k)$}
\State $\phi(t) = v_t$
\State $\ell(v_t) = \times$
\EndFor