bounding depth

appendix.tex

@@ -37,8 +37,8 @@ We require that vertices have an in-degree of at most two.
 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 $\abs{\pxdb} \leq c \cdot \abs{\db}$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsubsection{Circuit size vs. runtime}
-\label{sec:circuit-runtime}
+
+\subsection{Modeling Circuit Construction}
 
 \newcommand{\bagdbof}{\textsc{bag}(\pxdb)}
 
@@ -184,6 +184,32 @@ As in projection, newly created vertices will have an in-degree of $k$, and a fa
 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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsubsection{Bounding circuit depth}
+\label{sec:circuit-depth}
+
+We first show that the depth of the circuit (\depth; \Cref{def:size-depth}) is bounded by the size of the query.  Denote by $|\query|$ the number of relational operators in query $\query$.
+
+\begin{Proposition}[Circuit depth is bounded]
+\label{prop:circuit-depth}
+Let $\query$ be a relational query and $\dbbase$ be a \dbbaseName.  There exists a (lineage) circuit $\circuit$ encoding the lineage of all tuples $\tup \in \query(\dbbase)$ for which 
+$\depth(\circuit) \leq O_k(|\query|\log(n))$
+\end{Proposition}
+
+\begin{proof}
+We show that the bound of \Cref{prop:circuit-depth} holds for the circuit constructed by \Cref{alg:lc}.
+First, observe that \Cref{alg:lc} is invoked exactly once for every relational operator or base relation in $\query$; It thus suffices to show that an invocation \Cref{alg:lc} adds at most $O_k(\log(n))$ to the depth of any circuit produced by a recursive invocation.
+Second, observe that modulo the logarithmic fan-in of the projection and join cases, the depth of the output is at most one greater than the depth of any input.
+For the join case, the number of in-edges can be no greater than the join width, which itself is bounded by $k$.  The depth thus increases by at most a constant factor of $\lceil \log(k) \rceil = O_k(1)$.  
+For the projection case, observe that the fan-in is bounded by $|\query'(\dbbase)|$, which is in turn bounded by $n^k$.  The depth increase for any projection node is thus at most $\lceil \log(n^k)\rceil = O(k\log(n)) = O_k(\log(n))$.  
+\qed
+\end{proof}
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsubsection{Circuit size vs. runtime}
+\label{sec:circuit-runtime}
+
 \begin{Lemma}\label{lem:circ-model-runtime}
 \label{lem:circuits-model-runtime}
 Given a \abbrNXPDB $\pxdb$ 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 $\abs{V_{Q,\pxdb}} \leq (k-1)\qruntime{Q, \dbbase}+1$, where $k$ is the maximal degree of  any  polynomial in $Q(\pxdb)$.
@@ -248,6 +274,9 @@ The property holds for all recursive queries, and the proof holds.
 \qed
 \end{proof}
 
+\subsubsection{Runtime of \abbrStepOne}
+\label{sec:lc-runtime}
+
 We next need to show that we can construct the circuit in time linear in the deterministic runtime.  
 \begin{lemma}\label{lem:tlc-is-the-same-as-det}
 Given a query $\query$ over a \dbbaseName $\dbbase$, the runtime $\timeOf{\abbrStepOne}(\query,\dbbase,\circuit) \le O(\qruntime{\query, \dbbase})$