Main algorithm, some definitions for approx algo.

approx_alg.tex

@@ -27,16 +27,47 @@ Thus, it is the case, that we can approximate $\rpoly(\prob_1,\ldots, \prob_n)$
 \qed
 \AH{{\bf END:} Old Stuff}
 
-Before proceeding to describe the approximation algorithm, let us intrduce notation that will be of use in the following discussion.  First, when we speak of $\smb$, we are speaking of a polynomial $\poly$ of the standard monomial basis, i.e., a polynomial whose monomials are not only in SOP form, but one whose non-distinct monomials have been collapsed into one distinct monomial, with its corresponding coefficient accurately reflecting the number of monomials combined.
+Before proceeding to describe the approximation algorithm, let us intrduce notation that will be of use in the following discussion.  First, unless explicitly stated otherwise, when we speak of a polynomial we assume that it is of the standard monomial basis, i.e., a polynomial whose monomials are not only in SOP form, but one whose non-distinct monomials have been collapsed into one distinct monomial, with its corresponding coefficient accurately reflecting the number of monomials combined.
+
+\begin{Definition}[Expression Tree]\label{def:express-tree}
+An expression tree $\polytree$ is an ADT logically viewed as an n-ary tree, whose internal nodes are from the set $\{+, \times\}$, with leaf nodes being either numerical coefficients or variables, but not both.
+\end{Definition}
+
+Note that $\polytree$ models the input polynomial $\poly$, and is therefore not necessarily of the standard monomial basis.
+
+\begin{Definition}\label{def:express-tree-set}
+Consider input $\polytree$ and its equivalent $\smb$ in the standard monomial basis.  Then $\expresstree{\smb}$ be the set of all possible polynomial expressions equivalent to $\smb$, and $\polytree \subseteq \expresstree{\smb}$.
+\end{Definition}
 
 Let $\expresstree{\smb}$ be the set of all possible polynomial expressions equivalent to $\smb$.  Call the input polynomial $\polytree$, and note that $\polytree \subseteq \expresstree{\smb}$ and need not be of the standard monomial basis.  Refer to the expanded SOP form of $\poly$ as $\expandtree$, which is the SOP form of $\poly$ such that all coefficients $c_i$ are in the set $\{-1, 1\}$, thus relaxing the distinct monomial requirement of the standard monomial basis.  Denote $\abstree$ as the resulting polynomial when all monomial coefficients of $\polytree$ are converted to positive coefficients, and then $\polytree$ itself is converted to the standard monomial basis. 
 
-\subsection{Monomial Sample Algorithm}
-\begin{Lemma}\label{lem:approx-alg}
-For any query polynomial $\poly(\vct{X})$, an approximation of $\rpoly(\prob_1,\ldots, \prob_n)$ can be computed in $O\left(|\poly|\cdot k \frac{\log\frac{1}{\conf}}{\error^2}\right)$, within $1 \pm \error$ multiplicative error with probability $\geq 1 - \conf$, where $k$ denotes the product width of $\poly$.
-\end{Lemma}
+\subsection{Approximating $\rpoly$}
 
-\begin{proof}[Proof of Lemma \ref{lem:approx-alg}]
+\subsubsection{Description}
+The algorithm that approximates $\rpoly$ takes an ADT that we will refer to as an expression tree $\polytree$ for its input.  Using a few helper methods, it samples $\polytree$ $O\left(\frac{\log{\frac{1}{\delta}}}{\epsilon^2}\right)$ times, yielding an estimate of $\rpoly$ within a multiplicative error of $1 \pm \epsilon$ with a probability of $1 - \delta$.
+
+\subsubsection{Psuedo Code}
+
+\begin{algorithm}
+	\caption{\textsc{MonomialSample($\polytree$)}}
+	\label{alg:mon-sam}
+	\begin{algorithmic}
+		\State $acc \gets 0$
+		\State \textsc{OnePass($\polytree$)}
+		\For{$sample$ $in$ $\frac{\log{\frac{1}{\delta}}}{\epsilon^2}$}
+			\State $acc \gets acc + $\textsc{Sample($\polytree$)}
+		\EndFor
+
+		\State Return $acc$
+	\end{algorithmic}
+\end{algorithm}
+
+\subsubsection{Correctness}
+\begin{Theorem}\label{lem:approx-alg}
+For any query polynomial $\poly(\vct{X})$, an approximation of $\rpoly(\prob_1,\ldots, \prob_n)$ can be computed in $O\left(|\poly|\cdot k \frac{\log\frac{1}{\conf}}{\error^2}\right)$, within $1 \pm \error$ multiplicative error with probability $\geq 1 - \conf$, where $k$ denotes the product width of $\poly$.
+\end{Theorem}
+
+\begin{proof}[Proof of Theorem \ref{lem:approx-alg}]
 Consider $\polytree$ in the standard monomial basis and let $c_i$ be the coefficient of the $i^{th}$ monomial and $\distinctvars_i$ be the number of distinct variables appearing in the $i^{th}$ monomial.  Note that sampling each term $t$ in $\polytree$ with probability $\frac{|c_i|}{\abstree(1,\ldots, 1)}$ is the equivalent of sampling uniformly over $\expandtree$.  Now consider $\rpoly$ and note that $\coeffitem{i}$ is the value of the $i^{th}$ monomial term in $\rpoly(\prob_1,\ldots, \prob_n)$.  Let $m$ be the number of terms in $\expandtree$ and  $\coeffset$ to be the set $\{c'_1,\ldots, c'_m\}$ where each $c'_i$ is in $\{-1, 1\}$.  
 
 Consider now a set of $\samplesize$ random variables $\vct{\randvar}$, where $\randvar_i \sim \unidist{\coeffset}$.  Recall that we are estimating for $\rpoly(\prob,\ldots, \prob)$.  Then for random variable $\randvar_i$, it is the case that $\expct\pbox{\randvar_i} = \sum_{i = 1}^{\setsize}\frac{c'_i \cdot \prob^{\distinctvars}}{\setsize} = \frac{\rpoly(\prob,\ldots, \prob)}{\abstree(1,\ldots, 1)}$.  Let $\hoeffest = \frac{1}{\samplesize}\sum_{i = 1}^{\samplesize}\randvar_i$.  It is also true that 
@@ -58,11 +89,6 @@ Solving for the number of samples $\samplesize$ we get
 Equation \cref{eq:hoeff-1} results computing the sum in the denominator of the exponential.  Equation \cref{eq:hoeff-2} is the result of dividing both sides by $2$.  Equation \cref{eq:hoeff-3} follows from taking the reciprocal of both sides, and noting that such an operation flips the inequality sign.  We then derive \cref{eq:hoeff-4} by the taking the base $e$ log of both sides, and \cref{eq:hoeff-5} results from reducing common factors.  We arrive at the final result of \cref{eq:hoeff-6} by simply multiplying both sides by the reciprocal of the RHS fraction without the $\samplesize$ factor.
 
 \end{proof}
-\subsubsection{Description}
-
-\subsubsection{Psuedo Code}
-
-\subsubsection{Correctness}
 
 
 \subsubsection{Run-time Analysis}