Finished pass on S.4.

app_approx-alg-pseudo-code.tex

@@ -16,7 +16,7 @@
 		\For{$\vari{i} \in 1 \text{ to }\numsamp$}\label{alg:sampling-loop}\Comment{Perform the required number of samples}
 			\State $(\vari{M}, \vari{sgn}_\vari{i}) \gets  $ \sampmon($\circuit_\vari{mod}$)\label{alg:mon-sam-sample}\Comment{\sampmon is \Cref{alg:sample}. Note that $\vari{sgn}_\vari{i}$ is the \emph{sign} of the monomial's coefficient and \emph{not} the coefficient itself}
 			\If{$\vari{M}$ has at most one variable from each block}\label{alg:check-duplicate-block}
-				\State $\vari{Y}_\vari{i} \gets \prod_{X_j\in\var\inparen{\vari{M}}}p_j$\label{alg:mon-sam-assign1}
+				\State $\vari{Y}_\vari{i} \gets \prod_{X_j\in\vari{M}}p_j$\label{alg:mon-sam-assign1}\Comment{\vari{M} is the sampled monomial's set of variables (cref. \cref{subsec:sampmon-remarks})}
 				\State $\vari{Y}_\vari{i} \gets \vari{Y}_\vari{i} \times\; \vari{sgn}_\vari{i}$\label{alg:mon-sam-product}
 			\State $\accum \gets \accum + \vari{Y}_\vari{i}$\Comment{Store the sum over all samples}\label{alg:mon-sam-add}
 			\EndIf

approx_alg.tex

@@ -6,19 +6,20 @@
 In \Cref{sec:hard}, we showed that computing the expected multiplicity of a compressed lineage polynomial for \ti (even just based on project-join queries), and by extension \bi (or more general \abbrPDB models) %any $\semNX$-PDB) 
 is unlikely to be possible in linear time (\Cref{thm:mult-p-hard-result}), even if all tuples have the same probability  (\Cref{th:single-p-hard}).
 Given this, we now design an approximation algorithm for our problem that runs in {\em linear time}.\footnote{For a very broad class of circuits: please see the discussion after \Cref{lem:val-ub} for more.}
-The folowing approximation algorithm applies to \bi, though our bounds are more meaningful for a non-trivial subclass of \bis that contains both \tis, as well as the PDBench benchmark~\cite{pdbench}.
+The folowing approximation algorithm applies to \bi, though our bounds are more meaningful for a non-trivial subclass of \bis that contains both \tis, as well as the PDBench benchmark~\cite{pdbench}.  As before, all proofs and pseudocode can be found in \Cref{sec:proofs-approx-alg}.
 %it is then desirable to have an algorithm to approximate the multiplicity in linear time, which is what we describe next.
 
 \subsection{Preliminaries and some more notation}
 
-We now introduce useful definitions and notation related to circuits and polynomials.  All proofs and missing pseudocode can be found in \Cref{sec:proofs-approx-alg}.
+We now introduce useful definitions and notation related to circuits and polynomials.  
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %\begin{Definition}[Variables in a monomial]\label{def:vars}
 % Given a monomial $v$, we use $\var(v)$ to denote the set of variables in $v$.
 %\end{Definition}
+%\noindent For example the monomial $XY$ has $\var(XY)=\inset{X,Y}$.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\noindent For example the monomial $XY$ has $\var(XY)=\inset{X,Y}$.
+
 
 
 \begin{Definition}[$\expansion{\circuit}$]\label{def:expand-circuit}
@@ -34,33 +35,42 @@ $\expansion{\circuit} =
 \end{cases}
 $
 \end{Definition}
-For further explanation, please refer to \Cref{example:expr-tree-T}.
+Consider $\circuit$ illustrated in \Cref{fig:circuit}.  $\expansion{\circuit}$ is then $[(X, 2), (XY, -1), (XY, 4), (Y, -2)]$.
 
 \begin{Definition}[$\abs{\circuit}(\vct{X})$]\label{def:positive-circuit}
 For any circuit $\circuit$, the corresponding
 {\em positive circuit}, denoted $\abs{\circuit}$, is obtained from $\circuit$ as follows. For each leaf node $\ell$ of $\circuit$ where $\ell.\type$ is $\tnum$, update $\ell.\vari{value}$ to $|\ell.\vari{value}|$.
 \end{Definition}
-Please see \Cref{ex:def-pos-circ} for an illustration.
+Conveniently, $\abs{\circuit}\inparen{1,\ldots,1}$ gives us the number of terms represented in $\expansion{\circuit}$, i.e. $\sum\limits_{\inparen{\monom, \coef} \in \expansion{\circuit}}\abs{\coef}$.
 
-\begin{Definition}[\size($\cdot$)]\label{def:size}
-The function \size~ takes a circuit $\circuit$ as input and outputs the number of gates (nodes) in \circuit.
+\begin{Definition}[\size($\cdot$), \depth$\inparen{\cdot}$]\label{def:size-depth}
+The functions \size and \depth output the number of gates and levels respectively for input \circuit.
 \end{Definition}
 
-\begin{Definition}[\depth($\cdot$)]
-The function \depth~ has circuit $\circuit$ as input and outputs the number of levels in \circuit.
-\end{Definition}
+%\begin{Definition}[\depth($\cdot$)]
+%The function \depth has circuit $\circuit$ as input and outputs the number of levels in \circuit.
+%\end{Definition}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%NEEDS to be moved to appendix
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%\begin{Definition}[$\degree(\cdot)$]\label{def:degree}\footnote{Note that the degree of $\polyf(\abs{\circuit})$ is always upper bounded by $\degree(\circuit)$ and the latter can be strictly larger (e.g. consider the case when $\circuit$ multiplies two copies of the constant $1$-- here we have $\deg(\circuit)=1$ but degree of $\polyf(\abs{\circuit})$ is $0$).}
+%$\degree(\circuit)$ is defined recursively as follows:
+%\[\degree(\circuit)=
+%\begin{cases}
+%\max(\degree(\circuit_\linput),\degree(\circuit_\rinput)) & \text{ if }\circuit.\type=+\\
+%\degree(\circuit_\linput) + \degree(\circuit_\rinput)+1 &\text{ if }\circuit.\type=\times\\
+%1 & \text{ if }\circuit.\type = \var\\
+%0 & \text{otherwise}.
+%\end{cases}
+%\]
+%\end{Definition}
+%%%%%%%%%%%%%%%%%%%%%%%%%%
+%END move to appendix
+%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\begin{Definition}[$\degree(\cdot)$]\label{def:degree}\footnote{Note that the degree of $\polyf(\abs{\circuit})$ is always upper bounded by $\degree(\circuit)$ and the latter can be strictly larger (e.g. consider the case when $\circuit$ multiplies two copies of the constant $1$-- here we have $\deg(\circuit)=1$ but degree of $\polyf(\abs{\circuit})$ is $0$).}
-$\degree(\circuit)$ is defined recursively as follows:
-\[\degree(\circuit)=
-\begin{cases}
-\max(\degree(\circuit_\linput),\degree(\circuit_\rinput)) & \text{ if }\circuit.\type=+\\
-\degree(\circuit_\linput) + \degree(\circuit_\rinput)+1 &\text{ if }\circuit.\type=\times\\
-1 & \text{ if }\circuit.\type = \var\\
-0 & \text{otherwise}.
-\end{cases}
-\]
-\end{Definition}
 Finally, we will need the following notation for the complexity of multiplying large integers:
 \begin{Definition}[$\multc{\cdot}{\cdot}$]\footnote{We note that when doing arithmetic operations on the RAM model for input of size $N$, we have that $\multc{O(\log{N})}{O(\log{N})}=O(1)$. More generally we have $\multc{N}{O(\log{N})}=O(N\log{N}\log\log{N})$.}
 In a RAM model of word size of $W$-bits, $\multc{M}{W}$ denotes the complexity of multiplying two integers represented with $M$-bits. (We will assume that for input of size $N$, $W=O(\log{N})$.
@@ -82,12 +92,16 @@ such that
 \end{equation}
 \end{Theorem}
 
-To get linear runtime results from \Cref{lem:approx-alg}, we will need to define another parameter modeling the (weighted) number of monomials in $\expansion{\circuit}$ to be `canceled' when it is modded with $\mathcal{B}$ (\Cref{def:mod-set-polys}).
+To get linear runtime results from \Cref{lem:approx-alg}, we will need to define another parameter modeling the (weighted) number of monomials in %$\poly\inparen{\vct{X}}$ 
+$\expansion{\circuit}$ 
+to be `canceled' monomials with dependent variables are removed (\cref{def:reduced-bi-poly}).  %def:hen it is modded with $\mathcal{B}$ (\Cref{def:mod-set-polys}).
+Let $\isInd{\cdot}$ be a boolean function returning true if monomial $\encMon$ is composed of independent variables and false otherwise.
 \begin{Definition}[Parameter $\gamma$]\label{def:param-gamma}
 Given an expression tree $\circuit$, define
-\AH{Technically, $\monom$ is a set of variables rather than a monomial.  Perhaps we don't need the $\var(\cdot)$ function and can replace is with a function that returns the monomial represented by a set of variables.}
+\AH{Technically, $\monom$ is a set of variables rather than a monomial.  Perhaps we don't need the $\var(\cdot)$ function and can replace is with a function that returns the monomial represented by a set of variables.  FIXED: need to propogate this to the appendix ($\encMon$)}
 \AH{To add, this is an issue on line 1073, 1117 of app C.}
-\[\gamma(\circuit)=\frac{\sum_{(\monom, \coef)\in \expansion{\circuit}} \abs{\coef}\cdot \indicator{\encMon\mod{\mathcal{B}}\equiv 0}}{\abs{\circuit}(1,\ldots, 1)}\]
+\[\gamma(\circuit)=\frac{\sum_{(\monom, \coef)\in \expansion{\circuit}} \abs{\coef}\cdot \indicator{\neg\isInd{\encMon}} }%\encMon\mod{\mathcal{B}}\equiv 0}}
+{\abs{\circuit}(1,\ldots, 1)}.\]
 \end{Definition}
 
 \noindent We next present a few corollaries of \Cref{lem:approx-alg}.
@@ -108,7 +122,7 @@ $\abs{\circuit}(1,\ldots, 1)\le 2^{2^k\cdot \size(\circuit)}.$
 Further, under either of the following conditions:
 \begin{enumerate}
 \item $\circuit$ is a tree,
-\item $\circuit$ encodes the run of the algorithm in~\cite{DBLP:conf/pods/KhamisNR16} on an FAQ query,
+\item $\circuit$ encodes the run of the algorithm in~\cite{DBLP:conf/pods/KhamisNR16} on an FAQ\AH{citation would help here, as a reviewer complaint on this was ``What is FAQ?'', though we do cite (I think) in the appendix.} query,
 \end{enumerate}
 we have $\abs{\circuit}(1,\ldots, 1)\le  \size(\circuit)^{O(k)}.$
 \end{Lemma}
@@ -124,14 +138,25 @@ Given a query polynomial $\poly(\vct{X})=\polyf(\circuit)$ for circuit \circuit
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{equation}
 \label{eq:tilde-Q-bi}
-\rpoly\inparen{X_1,\dots,X_\numvar}=\hspace*{-1mm}\sum_{(\monom,\coef)\in \expansion{\circuit}} \hspace*{-2mm} \indicator{\encMon\mod{\mathcal{B}}\not\equiv 0}\cdot \coef\cdot\hspace*{-2mm}\prod_{X_i\in \var\inparen{\monom}}\hspace*{-2mm} X_i
+\rpoly\inparen{X_1,\dots,X_\numvar}=\hspace*{-1mm}\sum_{(\monom,\coef)\in \expansion{\circuit}} %\hspace*{-2mm}
+\indicator{\isInd{\encMon}%\mod{\mathcal{B}}\not\equiv 0
+}\cdot \coef\cdot\hspace*{-2mm}\prod_{X_i\in \monom}\hspace*{-2mm} X_i
 \end{equation}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\input{app_approx-alg-pseudo-code}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%NEED to move to appendix
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%\input{app_approx-alg-pseudo-code}
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%END move to appendix
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
 
 Given the above, the algorithm is a sampling based algorithm for the above sum: we sample (via \sampmon) $(\monom,\coef)\in \expansion{\circuit}$ with probability proportional %\footnote{We could have also uniformly sampled from $\expansion{\circuit}$ but this gives better parameters.}
- to $\abs{\coef}$ and compute $Y=\indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \prod_{X_i\in \var\inparen{\monom}} p_i$. Taking $\numsamp$ samples and computing the average of $Y$ gives us our final estimate. \onepass is used to compute the sampling probabilities needed in \sampmon (details are in \Cref{sec:proofs-approx-alg}).
+ to $\abs{\coef}$ and compute $\vari{Y}=\indicator{\isInd{\encMon}}%\monom\mod{\mathcal{B}}\not\equiv 0}
+ \cdot \prod_{X_i\in \monom} p_i$. Taking $\numsamp$ samples and computing the average of $\vari{Y}$ gives us our final estimate. \onepass is used to compute the sampling probabilities needed in \sampmon (details are in \Cref{sec:proofs-approx-alg}).
 %\approxq (\Cref{alg:mon-sam}) modifies \circuit with a call to \onepass.  It then samples from $\circuit_{\vari{mod}}\numsamp$ times and uses that information to approximate $\rpoly$.
 
 

macros.tex

@@ -72,14 +72,14 @@
 %Function Names and Typesetting				         									     %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \newcommand{\domain}{\func{Dom}}
-\newcommand{\func}[1]{\textsc{#1}}
+\newcommand{\func}[1]{\textsc{#1}\xspace}
 \newcommand{\isInd}[1]{\func{isInd}\inparen{#1}}
 \newcommand{\polyf}{\func{poly}}
 \newcommand{\evalmp}{\func{eval}}
 \newcommand{\degree}{\func{deg}}
 \newcommand{\size}{\func{size}}
 \newcommand{\depth}{\func{depth}}
-\newcommand{\topord}{\func{TopOrd}\xspace}
+\newcommand{\topord}{\func{TopOrd}}
 \newcommand{\smbOf}[1]{\func{\abbrSMB}\inparen{#1}}
 %Verify if we need the above...
 %saving \treesize for now to keep latex from breaking
@@ -231,7 +231,7 @@
 \newcommand{\mtrix}[1]{M_{#1}}
 \newcommand{\dtrm}[1]{Det\left(#1\right)}
 \newcommand{\tuple}[1]{\left<#1\right>}
-\newcommand{\indicator}[1]{\onesymbol\inparen{#1}}
+\newcommand{\indicator}[1]{\underset{#1}{\onesymbol}}
 %----------------------------------------------
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%