approx algo desc.

app_approx-alg-analysis.tex

@@ -5,7 +5,7 @@
 The following results assume input circuit \circuit computed from an arbitrary $\raPlus$ query $\query$ and arbitrary \abbrBIDB $\pdb$.  We refer to \circuit as a \abbrBIDB circuit.
 \AH{Verify that the proof for \Cref{lem:approx-alg} doesn't rely on properties of $\raPlus$ or \abbrBIDB.}
 \begin{Theorem}\label{lem:approx-alg}
-Let \circuit be an arbitrary \abbrBIDB circuit %for a UCQ over \bi 
+Let \circuit be an arbitrary \abbrBIDB circuit %for a UCQ over \bi
 and define $\poly(\vct{X})=\polyf(\circuit)$ and let $k=\degree(\circuit)$.
 Then an estimate $\mathcal{E}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ can be computed in time
 {\small
@@ -22,7 +22,7 @@ The slight abuse of notation seen in $\abs{\circuit}\inparen{1,\ldots,1}$ is exp
 
 \subsection{Proof of Theorem \ref{lem:approx-alg}}\label{sec:proof-lem-approx-alg}
 \input{app_approx-alg-pseudo-code}
-In order to prove \Cref{lem:approx-alg}, we will need to argue the correctness of \approxq, which relies on the correctness of auxiliary algorithms \onepass and \sampmon.
+We prove \Cref{lem:approx-alg} constructively by presenting an algorithm \approxq (\Cref{alg:mon-sam}) which has the desired runtime and computes an approximation with the desired approximation guarantee. Algorithm \approxq uses Algorithm \onepass to compute weights on the edges of a circuits. These weights are then used to sample a set of monomials of $\poly(\circuit)$ from the circuit $\circuit$ by traversing the circuit using the weights to ensure that monomials are sampled with an appropriate probability. The correctness of \approxq relies on the correctness (and runtime behavior) of auxiliary algorithms \onepass and \sampmon that we state in the following lemmas (and prove later in this part of the appendix).
 
 \begin{Lemma}\label{lem:one-pass}
 The $\onepass$ function completes in time:
@@ -39,7 +39,7 @@ $$O(\log{k} \cdot k \cdot \depth(\circuit)\cdot\multc{\log\left(\abs{\circuit}(1
 
 With the above two lemmas, we are ready to argue the following result: % (proof in \Cref{sec:proofs-approx-alg}):
 \begin{Theorem}\label{lem:mon-samp}
-For any $\circuit$ with 
+For any $\circuit$ with
 \AH{Pretty sure this is $\degree\inparen{\abs{\circuit}}$.  Have to read on to be sure.}
 $\degree(poly(|\circuit|)) = k$, algorithm \ref{alg:mon-sam} outputs an estimate $\vari{acc}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ such that
 \[\probOf\left(\left|\vari{acc} - \rpoly(\prob_1,\ldots, \prob_\numvar)\right|> \error \cdot \abs{\circuit}(1,\ldots, 1)\right) \leq \conf,\]
@@ -47,12 +47,12 @@ $\degree(poly(|\circuit|)) = k$, algorithm \ref{alg:mon-sam} outputs an estimate
 \end{Theorem}
 
 
-Before proving \Cref{lem:mon-samp}, we use it to argue the claimed runtime of our main result, \Cref{lem:approx-alg}.  
+Before proving \Cref{lem:mon-samp}, we use it to argue the claimed runtime of our main result, \Cref{lem:approx-alg}.
 
 \begin{proof}[Proof of \Cref{lem:approx-alg}]
 Set $\mathcal{E}=\approxq({\circuit}, (\prob_1,\dots,\prob_\numvar),$ $\conf, \error')$, where
 \[\error' = \error \cdot \frac{\rpoly(\prob_1,\ldots, \prob_\numvar)}{\abs{{\circuit}}(1,\ldots, 1)},\]
- which achieves the claimed error bound on $\mathcal{E}$ (\vari{acc}) trivially due to the assignment to $\error'$ and \cref{lem:mon-samp}, since $\error' \cdot \abs{\circuit}(1,\ldots, 1) = \error\cdot\frac{\rpoly(1,\ldots, 1)}{\abs{\circuit}(1,\ldots, 1)} \cdot \abs{\circuit}(1,\ldots, 1) = \error\cdot\rpoly(1,\ldots, 1)$. 
+ which achieves the claimed error bound on $\mathcal{E}$ (\vari{acc}) trivially due to the assignment to $\error'$ and \cref{lem:mon-samp}, since $\error' \cdot \abs{\circuit}(1,\ldots, 1) = \error\cdot\frac{\rpoly(1,\ldots, 1)}{\abs{\circuit}(1,\ldots, 1)} \cdot \abs{\circuit}(1,\ldots, 1) = \error\cdot\rpoly(1,\ldots, 1)$.
 
 The claim on the runtime follows from \Cref{lem:mon-samp} since
 
@@ -78,7 +78,7 @@ where in the first equality we use the fact that $\vari{sgn}_{\vari{i}}\cdot \ab
 
 Let $\empmean = \frac{1}{\samplesize}\sum_{i = 1}^{\samplesize}\randvar_\vari{i}$.  It is also true that
 
-\[\expct\pbox{\empmean}  
+\[\expct\pbox{\empmean}
 = \frac{1}{\samplesize}\sum_{i = 1}^{\samplesize}\expct\pbox{\randvar_\vari{i}}
 = \frac{\rpoly(\prob_1,\ldots, \prob_\numvar)}{\abs{{\circuit}}(1,\ldots, 1)}.\]
 
@@ -134,7 +134,7 @@ For the base case, we have that \depth(\circuit) $= 0$, and there can only be on
 
 Assume for $\ell > 0$ an arbitrary circuit \circuit of $\depth(\circuit) \leq \ell$ that it is true that $\abs{\circuit}(1,\ldots, 1) \leq N^{k+1 }$.% for $k \geq 1$ when \depth(C) $\geq 1$.
 
-For the inductive step we consider a circuit \circuit such that $\depth(\circuit) = \ell + 1$.  The sink can only be either a $\circmult$ or $\circplus$ gate.  Let $k_\linput, k_\rinput$ denote \degree($\circuit_\linput$) and \degree($\circuit_\rinput$) respectively.  Consider when sink node is $\circmult$.  
+For the inductive step we consider a circuit \circuit such that $\depth(\circuit) = \ell + 1$.  The sink can only be either a $\circmult$ or $\circplus$ gate.  Let $k_\linput, k_\rinput$ denote \degree($\circuit_\linput$) and \degree($\circuit_\rinput$) respectively.  Consider when sink node is $\circmult$.
  Then note that
 \begin{align}
 \abs{\circuit}(1,\ldots, 1) &= \abs{\circuit_\linput}(1,\ldots, 1)\cdot \abs{\circuit_\rinput}(1,\ldots, 1) \nonumber\\
@@ -160,7 +160,7 @@ In the above, the first inequality follows from the inductive hypothes and \cref
 The upper bound in \Cref{lem:val-ub} for the general case is a simple variant of the above proof (but we present a proof sketch of the bound below for completeness):
 \begin{Lemma}
 \label{lem:C-ub-gen}
-Let $\circuit$ be a (general) circuit. % tree (i.e. the sub-circuits corresponding to two children of a node in $\circuit$ are completely disjoint). 
+Let $\circuit$ be a (general) circuit. % tree (i.e. the sub-circuits corresponding to two children of a node in $\circuit$ are completely disjoint).
 Then we have
 \[\abs{\circuit}(1,\dots,1)\le 2^{2^{\degree(\circuit)}\cdot \depth(\circuit)}.\]
 \end{Lemma}
@@ -193,4 +193,9 @@ In the above the first inequality follows from the inductive hypothesis while th
 
 %Finally, we consider the case when $\circuit$ encodes the run of the algorithm from~\cite{DBLP:conf/pods/KhamisNR16} on an FAQ query. We cannot handle the full generality of an FAQ query but we can handle an FAQ query that has a ``core'' join query on $k$ relations and then a subset of the $k$ attributes are ``summed'' out (e.g. the sum could be because of projecting out a subset of attributes from the join query). While the algorithm~\cite{DBLP:conf/pods/KhamisNR16} essentially figures out when to `push in' the sums, in our case since we only care about $\abs{\circuit}(1,\dots,1)$.  We will consider the obvious circuit that computes the ``inner join'' using a worst-case optimal join (WCOJ) algorithm like~\cite{NPRR} and then adding in the addition gates. The basic idea is very simple: we will argue that the there are at most $\size(\circuit)^k$ tuples in the join output (each with having a value of $1$ in $\abs{\circuit}(1,\dots,1)$). Then the largest value we can see in $\abs{\circuit}(1,\dots,1)$ is by summing up these at most $\size(\circuit)^k$ values of $1$. Note that this immediately implies the claimed bound in \Cref{lem:val-ub}.
 
-%We now sketch the argument for the claim about the join query above. First, we note that the computation of a WCOJ algorithm like~\cite{NPRR} can be expressed as a circuit with {\em multiple} sinks (one for each output tuple). Note that annotation corresponding to $\mathbf{t}$ in $\circuit$ is the polynomial $\prod_{e\in E} R(\pi_e(\mathbf{t}))$ (where $E$ indexes the set of relations). It is easy to see that in this case the value of  $\mathbf{t}$ in $\abs{\circuit}(1,\dots,1)$ will be $1$ (by multiplying $1$ $k$ times). The claim on the number of output tuples follow from the trivial bound of multiplying the input size bound (each relation has at most $n\le \size(\circuit)$ tuples and hence we get an overall bound of $n^k\le\size(\circuit)^k$. Note that we did not really use anything about the WCOJ algorithm except for the fact that $\circuit$ for the join part only is built only of multiplication gates. In fact, we do not need the better WCOJ join size bounds either (since we used the trivial $n^k$ bound). As a final remark, we note that we can build the circuit for the join part by running say the algorithm from~\cite{DBLP:conf/pods/KhamisNR16} on an FAQ query that just has the join query but each tuple is annotated with the corresponding variable $X_i$ (i.e. the semi-ring for the FAQ query is $\mathbb{N}[\mathbf{X}]$). 
+%We now sketch the argument for the claim about the join query above. First, we note that the computation of a WCOJ algorithm like~\cite{NPRR} can be expressed as a circuit with {\em multiple} sinks (one for each output tuple). Note that annotation corresponding to $\mathbf{t}$ in $\circuit$ is the polynomial $\prod_{e\in E} R(\pi_e(\mathbf{t}))$ (where $E$ indexes the set of relations). It is easy to see that in this case the value of  $\mathbf{t}$ in $\abs{\circuit}(1,\dots,1)$ will be $1$ (by multiplying $1$ $k$ times). The claim on the number of output tuples follow from the trivial bound of multiplying the input size bound (each relation has at most $n\le \size(\circuit)$ tuples and hence we get an overall bound of $n^k\le\size(\circuit)^k$. Note that we did not really use anything about the WCOJ algorithm except for the fact that $\circuit$ for the join part only is built only of multiplication gates. In fact, we do not need the better WCOJ join size bounds either (since we used the trivial $n^k$ bound). As a final remark, we note that we can build the circuit for the join part by running say the algorithm from~\cite{DBLP:conf/pods/KhamisNR16} on an FAQ query that just has the join query but each tuple is annotated with the corresponding variable $X_i$ (i.e. the semi-ring for the FAQ query is $\mathbb{N}[\mathbf{X}]$).
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