Finished cleaning App C; started App D.

appendix.tex

@@ -13,7 +13,7 @@
 \section{Missing details from Section~\ref{sec:background}}\label{sec:proofs-background}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Background details for proof of~\Cref{prop:expection-of-polynom}}
+\subsection{Background details for proof of~\Cref{prop:expection-of-polynom}}\label{app:subsec:background-nxdbs}
 \subsubsection{$\semK$-relations and \abbrNXPDB\xplural}\label{subsec:supp-mat-background}\label{subsec:supp-mat-krelations}
 \input{app_k-relations}
 \input{app_notation-background}
@@ -39,7 +39,7 @@
 \label{sec:circuits-formal}
 We now formalize circuits and the construction of circuits for $\raPlus$ queries.
 As mentioned earlier, we represent lineage polynomials as arithmetic circuits over $\mathbb N$-valued variables with $+$, $\times$.
-A circuit for query $Q$ and \abbrNXPDB $\pxdb$ 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 $E_{Q,\pxdb} \subset {V_{Q,\pxdb}}^2$.
+A circuit for query $Q$ and \abbrNXPDB $\pxdb$ \footnote{For background on \abbrNXPDB\xplural, see~\Cref{app:subsec:background-nxdbs}} 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 $E_{Q,\pxdb} \subset {V_{Q,\pxdb}}^2$.
 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.
 We require that $\phi_{Q,\pxdb}$'s range be limited to sink vertices (i.e., vertices with out-degree 0).
 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 $\times$.
@@ -54,7 +54,8 @@ Note that we can construct circuits for \bis in time linear in the time required
 
 We now connect the size of a circuit (where the size of a circuit is the number of vertices in the corresponding DAG) 
 for a given $\raPlus$ query $Q$ and \abbrNXPDB $\pxdb$ to
-the runtime $\qruntime{Q,\dbbase}$ of the PDB's \dbbaseName $\dbbase$.
+the runtime $\qruntime{Q,\tupset}$ of the PDB's \dbbaseName $\tupset$.
+\AH{@atri: do we use $\tupset$ or $\gentupset$ here?}
 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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -76,38 +77,38 @@ We define the circuit for a $\raPlus$ query $\query$ recursively by cases as fol
 
 
 \begin{algorithm}
-\caption{\abbrStepOne$(\query, \dbbase, E, V, \ell)$}
+\caption{\abbrStepOne$(\query, \tupset, E, V, \ell)$}
 \label{alg:lc}
   \begin{algorithmic}[1]
   \Require $\query$: query
-  \Require $\dbbase$: a \dbbaseName
+  \Require $\tupset$: a \dbbaseName
   \Require $E, V, \ell$: accumulators for the edge list, vertex list, and vertex label list.
-  \Ensure $\circuit = \tuple{E, V, \phi, \ell}$: a circuit encoding the lineage of each tuple in $\query(\dbbase)$
-  \If{$\query$ is $R$} \Comment{\textbf{Case 1}: $\query$ is a relation atom}
-    \For{$t \in \dbbase.R$}
-      \State $V \leftarrow V \cup \{v_t\}$; $\ell \leftarrow \ell \cup \{(v_t, R(t))\}$  \Comment{Allocate a fresh node $v_t$}
+  \Ensure $\circuit = \tuple{E, V, \phi, \ell}$: a circuit encoding the lineage of each tuple in $\query(\tupset)$
+  \If{$\query$ is $\rel$} \Comment{\textbf{Case 1}: $\query$ is a relation atom}
+    \For{$t \in \tupset.\rel$}
+      \State $V \leftarrow V \cup \{v_t\}$; $\ell \leftarrow \ell \cup \{\inparen{v_t, \rel\inparen{\tup}}\}$  \Comment{Allocate a fresh node $v_t$}
       \State $\phi(t) \gets v_t$
     \EndFor
   \ElsIf{$\query$ is $\sigma_\theta(\query')$} \Comment{\textbf{Case 2}: $\query$ is a Selection}
-    \State $\tuple{V, E, \phi', \ell} \gets \abbrStepOne(\query', \dbbase, V, E, \ell)$
+    \State $\tuple{V, E, \phi', \ell} \gets \abbrStepOne(\query', \tupset, V, E, \ell)$
     \For{$t \in \domain(\phi')$}
       \State \textbf{if }$\theta(t)$
               \textbf{ then } $\phi(t) \gets \phi'(t)$
               \textbf{ else } $\phi(t) \gets v_0$
     \EndFor
   \ElsIf{$\query$ is $\pi_{\vec{A}}(\query')$} \Comment{\textbf{Case 3}: $\query$ is a Projection}
-    \State $\tuple{V, E, \phi', \ell} \gets \abbrStepOne(\query', \dbbase, V, E, \ell)$
-    \For{$t \in \pi_{\vec{A}}(\query'(\dbbase))$}
+    \State $\tuple{V, E, \phi', \ell} \gets \abbrStepOne(\query', \tupset, V, E, \ell)$
+    \For{$t \in \pi_{\vec{A}}(\query'(\tupset))$}
       \State $V \leftarrow V \cup \{v_t\}$; $\ell \leftarrow \ell \cup \{(v_t, +)\}$\Comment{Allocate a fresh node $v_t$}
       \State $\phi(t) \leftarrow v_t$
     \EndFor
-    \For{$t \in \query'(\dbbase)$}
+    \For{$t \in \query'(\tupset)$}
       \State $E \leftarrow E \cup \{(\phi'(t), \phi(\pi_{\vec{A}}t))\}$
     \EndFor
     \State Correct nodes with in-degrees $>2$ by appending an equivalent fan-in two tree instead
   \ElsIf{$\query$ is $\query_1 \cup \query_2$} \Comment{\textbf{Case 4}: $\query$ is a Bag Union}
-    \State $\tuple{V, E, \phi_1, \ell} \gets \abbrStepOne(\query_1, \dbbase, V, E, \ell)$
-    \State $\tuple{V, E, \phi_2, \ell} \gets \abbrStepOne(\query_2, \dbbase, V, E, \ell)$
+    \State $\tuple{V, E, \phi_1, \ell} \gets \abbrStepOne(\query_1, \tupset, V, E, \ell)$
+    \State $\tuple{V, E, \phi_2, \ell} \gets \abbrStepOne(\query_2, \tupset, V, E, \ell)$
     \State $\phi \gets \phi_1 \cup \phi_2$
     \For{$t \in \domain(\phi_1) \cap \domain(\phi_2)$}
       \State $V \leftarrow V \cup \{v_t\}$; $\ell \leftarrow \ell \cup \{(v_t, +)\}$ \Comment{Allocate a fresh node $v_t$}
@@ -116,12 +117,12 @@ We define the circuit for a $\raPlus$ query $\query$ recursively by cases as fol
     \EndFor
   \ElsIf{$\query$ is $\query_1 \bowtie \ldots \bowtie \query_m$} \Comment{\textbf{Case 5}: $\query$ is a $m$-ary Join}
     \For{$i \in [m]$}
-      \State $\tuple{V, E, \phi_i, \ell} \gets \abbrStepOne(\query_i, \dbbase, V, E, \ell)$
+      \State $\tuple{V, E, \phi_i, \ell} \gets \abbrStepOne(\query_i, \tupset, V, E, \ell)$
     \EndFor
     \For{$t \in \domain(\phi_1) \bowtie \ldots \bowtie \domain(\phi_m)$}
       \State $V \leftarrow V \cup \{v_t\}$; $\ell \leftarrow \ell \cup \{(v_t, \times)\}$ \Comment{Allocate a fresh node $v_t$}
       \State $\phi(t) \gets v_t$
-      \State $E \leftarrow E \cup \comprehension{(\phi_i(\pi_{sch(\query_i(\dbbase))}(t)), v_t)}{i \in [n]}$
+      \State $E \leftarrow E \cup \comprehension{(\phi_i(\pi_{sch(\query_i(\tupset))}(t)), v_t)}{i \in [n]}$
     \EndFor
     \State Correct nodes with in-degrees $>2$ by appending an equivalent fan-in two tree instead
 
@@ -134,7 +135,7 @@ We define the circuit for a $\raPlus$ query $\query$ recursively by cases as fol
 \Cref{alg:lc} defines how the circuit for a query result is constructed.  We quickly review the number of vertices emitted in each case.
 
 \caseheading{Base Relation}
-This circuit has $|D_\Omega.R|$ vertices.
+This circuit has $\abs{\tupset.\rel}$ vertices.
 
 \caseheading{Selection}
 If we assume dead sinks are iteratively garbage collected,
@@ -159,7 +160,7 @@ We first show that the depth of the circuit (\depth; \Cref{def:size-depth}) is b
 
 \begin{Proposition}[Circuit depth is bounded]
 \label{prop:circuit-depth}
-Let $\query$ be a relational query and $\dbbase$ be a \dbbaseName with $n$ tuples.  There exists a (lineage) circuit $\circuit^*$ encoding the lineage of all tuples $\tup \in \query(\dbbase)$ for which
+Let $\query$ be a relational query and $\tupset$ be a \dbbaseName with $n$ tuples.  There exists a (lineage) circuit $\circuit^*$ encoding the lineage of all tuples $\tup \in \query(\tupset)$ for which
 $\depth(\circuit^*) \leq O(k|\query|\log(n))$.
 \end{Proposition}
 
@@ -180,18 +181,19 @@ For the projection case, observe that the fan-in is bounded by $|\query'(\dbbase
 
 \begin{Lemma}\label{lem:circ-model-runtime}
 \label{lem:circuits-model-runtime}
-Given a \abbrNXPDB $\pxdb$ with \dbbaseName $\dbbase$, and an $\raPlus$ query  $Q$, 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 $\abs{V_{Q,\pxdb}} \leq k\qruntime{Q, \dbbase}+1$, where $k\ge 1$ is the maximal degree of  any  polynomial in $Q(\pxdb)$.
+Given a \abbrNXPDB $\pxdb$ with \dbbaseName $\tupset$, and an $\raPlus$ query  $Q$, the runtime of $Q$ over $\tupset$ has the same or greater complexity as the size of the lineage of $Q(\pxdb)$.  That is, we have $\abs{V_{Q,\pxdb}} \leq k\qruntime{Q, \tupset}+1$, where $k\ge 1$ is the maximal degree of  any  polynomial in $Q(\pxdb)$.
 \end{Lemma}
+\AH{Why are the number of vertices considered to be the size of the lineage?}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{proof}
-We prove by induction that $\abs{V_{Q,\pxdb} \setminus \{v_0\}} \leq k\qruntime{Q, \dbbase}$.  For clarity, we implicitly exclude $v_0$ in the proof below.
+We prove by induction that $\abs{V_{Q,\pxdb} \setminus \{v_0\}} \leq k\qruntime{Q, \tupset}$.  For clarity, we implicitly exclude $v_0$ in the proof below.
 
-The base case is a base relation: $Q = R$ and is trivially true since $|V_{R,\pxdb}| = |\dbbase.R|=\qruntime{R, \dbbase}$ (note that here the degree $k=1$).
-For the inductive step, we assume that we have circuits for subqueries $Q_1, \ldots, Q_m$ such that $|V_{Q_i,\pxdb}| \leq k_i\qruntime{Q_i,\dbbase}$ where $k_i$ is the degree of $Q_i$.
+The base case is a base relation: $Q = R$ and is trivially true since $|V_{R,\pxdb}| = |\tupset.R|=\qruntime{R, \tupset}$ (note that here the degree $k=1$).
+For the inductive step, we assume that we have circuits for subqueries $Q_1, \ldots, Q_m$ such that $|V_{Q_i,\pxdb}| \leq k_i\qruntime{Q_i,\tupset}$ where $k_i$ is the degree of $Q_i$.
 
 \caseheading{Selection}
 Assume that $Q = \sigma_\theta(Q_1)$.
-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 \qruntime{Q,\dbbase} $.
+In the circuit for $Q$, $|V_{Q,\pxdb}| = |V_{Q_1,\tupset}|$ vertices, so from the inductive assumption and $\qruntime{Q,\tupset} = \qruntime{Q_1,\tupset}$ by definition, we have $|V_{Q,\pxdb}| \leq k \qruntime{Q,\tupset} $.
 
 \caseheading{Projection}
 Assume that $Q = \pi_{\vct A}(Q_1)$.
@@ -199,9 +201,9 @@ The circuit for $Q$ has at most $|V_{Q_1,\pxdb}|+|{Q_1}|$ vertices.
 \begin{align*}
 |V_{Q,\pxdb}| & \leq |V_{Q_1,\pxdb}| + |Q_1|\\
 \intertext{(From the inductive assumption)}
-& \leq k\qruntime{Q_1,\dbbase} + \abs{Q_1}\\
-\intertext{(By definition  of $\qruntime{Q,\dbbase}$)}
-& \le k\qruntime{Q,\dbbase}.
+& \leq k\qruntime{Q_1,\tupset} + \abs{Q_1}\\
+\intertext{(By definition  of $\qruntime{Q,\tupset}$)}
+& \le k\qruntime{Q,\tupset}.
 \end{align*}
 \caseheading{Union}
 Assume that $Q = Q_1 \cup Q_2$.
@@ -209,9 +211,9 @@ The circuit for $Q$ has $|V_{Q_1,\pxdb}|+|V_{Q_2,\pxdb}|+|{Q_1} \cap {Q_2}|$ ver
 \begin{align*}
 |V_{Q,\pxdb}| & \leq |V_{Q_1,\pxdb}|+|V_{Q_2,\pxdb}|+|{Q_1}|+|{Q_2}|\\
 \intertext{(From the inductive assumption)}
-& \leq k(\qruntime{Q_1,\dbbase} + \qruntime{Q_2,\dbbase}) + (|Q_1| + |Q_2|)
-\intertext{(By definition of $\qruntime{Q,\dbbase}$)}
-& \leq k(\qruntime{Q,\dbbase}).
+& \leq k(\qruntime{Q_1,\tupset} + \qruntime{Q_2,\tupset}) + (|Q_1| + |Q_2|)
+\intertext{(By definition of $\qruntime{Q,\tupset}$)}
+& \leq k(\qruntime{Q,\tupset}).
 \end{align*}
 
 \caseheading{$m$-ary Join}
@@ -220,12 +222,12 @@ The circuit for $Q$ has $|V_{Q_1,\pxdb}|+\ldots+|V_{Q_k,\pxdb}|+(m-1)|{Q_1} \bow
 \begin{align*}
 |V_{Q,\pxdb}| & = |V_{Q_1,\pxdb}|+\ldots+|V_{Q_k,\pxdb}|+(m-1)|{Q_1} \bowtie \ldots \bowtie {Q_k}|\\
 \intertext{From the inductive assumption and noting $\forall i: k_i \leq k$ and $m\le k$}
-& \leq k\qruntime{Q_1,\dbbase}+\ldots+k\qruntime{Q_k,\dbbase}+\\
+& \leq k\qruntime{Q_1,\tupset}+\ldots+k\qruntime{Q_k,\tupset}+\\
 &\;\;\; (m-1)|{Q_1} \bowtie \ldots \bowtie {Q_m}|\\
-& \leq k(\qruntime{Q_1,\dbbase}+\ldots+\qruntime{Q_m,\dbbase}+\\
-&\;\;\;|{Q_1} \bowtie \ldots \bowtie {Q_m}|)\\
-\intertext{(By definition of $\qruntime{Q,\dbbase}$ and assumption on $\jointime{\cdot}$)}
-& \le k\qruntime{Q,\dbbase}.
+& \leq k\left(\qruntime{Q_1,\tupset}+\ldots+\qruntime{Q_m,\tupset}+\right.\\
+&\;\;\;\left.|{Q_1} \bowtie \ldots \bowtie {Q_m}|\right)\\
+\intertext{(By definition of $\qruntime{Q,\tupset}$ and assumption on $\jointime{\cdot}$)}
+& \le k\qruntime{Q,\tupset}.
 \end{align*}
 
 The property holds for all recursive queries, and the proof holds.