Changes to Algorithm OnePass and Lemma 4.10 proof.
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approx_alg.tex
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approx_alg.tex
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@ -24,74 +24,53 @@ We now introduce useful definitions and notation related to polynomials. We use
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%\begin{Definition}[Expression Tree]\label{def:express-tree}
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%An expression tree $\etree$ is a binary %an ADT logically viewed as an n-ary
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%tree, whose internal nodes are from the set $\{+, \times\}$, with leaf nodes being either from the set $\mathbb{R}$ $(\tnum)$ or from the set of monomials $(\var)$. The members of $\etree$ are \type, \val, \vari{partial}, \vari{children}, and \vari{weight}, where \type is the type of value stored in the node $\etree$ (i.e. one of $\{+, \times, \var, \tnum\}$, \val is the value stored, and \vari{children} is the list of $\etree$'s children where $\etree_\lchild$ is the left child and $\etree_\rchild$ the right child. Remaining fields hold values whose semantics we will fix later. When $\etree$ is used as input of ~\Cref{alg:mon-sam} and ~\Cref{alg:one-pass}, the values of \vari{partial} and \vari{weight} will not be set. %SEMANTICS FOR \etree: \vari{partial} is the sum of $\etree$'s coefficients , n, and \vari{weight} is the probability of $\etree$ being sampled.
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%An expression tree $\circuit$ is a binary %an ADT logically viewed as an n-ary
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%tree, whose internal nodes are from the set $\{+, \times\}$, with leaf nodes being either from the set $\mathbb{R}$ $(\tnum)$ or from the set of monomials $(\var)$. The members of $\circuit$ are \type, \val, \vari{partial}, \vari{children}, and \vari{weight}, where \type is the type of value stored in the node $\circuit$ (i.e. one of $\{+, \times, \var, \tnum\}$, \val is the value stored, and \vari{children} is the list of $\circuit$'s children where $\circuit_\lchild$ is the left child and $\circuit_\rchild$ the right child. Remaining fields hold values whose semantics we will fix later. When $\circuit$ is used as input of ~\Cref{alg:mon-sam} and ~\Cref{alg:one-pass}, the values of \vari{partial} and \vari{weight} will not be set. %SEMANTICS FOR \circuit: \vari{partial} is the sum of $\circuit$'s coefficients , n, and \vari{weight} is the probability of $\circuit$ being sampled.
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%\end{Definition}
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%Note that $\etree$ need not encode an expression in the standard monomial basis. For instance, $\etree$ could represent a compressed form of the polynomial in~\Cref{eq:poly-eg}, such as $(x + 2y)(2x - y)$.
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%Note that $\circuit$ need not encode an expression in the standard monomial basis. For instance, $\circuit$ could represent a compressed form of the polynomial in~\Cref{eq:poly-eg}, such as $(x + 2y)(2x - y)$.
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\begin{Definition}[$\polyf(\cdot)$]\label{def:poly-func}
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Denote $\polyf(\etree)$ to be the function from expression tree $\etree$ to its corresponding polynomial. $poly(\cdot)$ is recursively defined on $\etree$ as follows, where $\etree_\lchild$ and $\etree_\rchild$ denote the left and right child of $\etree$ respectively.
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With addition and multiplication following the standard interpretation for polynomials:
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%
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% \begin{align*}
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% &\etree.\type = +\mapsto&& \polyf(\etree_\lchild) + \polyf(\etree_\rchild)\\
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% &\etree.\type = \times\mapsto&& \polyf(\etree_\lchild) \cdot \polyf(\etree_\rchild)\\
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% &\etree.\type = \var \text{ OR } \tnum\mapsto&& \etree.\val
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% \end{align*}
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%
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Denote \revision{$\polyf(\circuit)$}~ to be the function from circuit \revision{$\circuit$}~ to its corresponding polynomial. $\polyf(\cdot)$ is recursively defined on \revision{$\circuit$}~ as follows, with addition and multiplication following the standard interpretation for polynomials:
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\begin{equation*}
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\polyf(\etree) = \begin{cases}
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\polyf(\etree_\lchild) + \polyf(\etree_\rchild) &\text{ if \etree.\type } = +\\
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\polyf(\etree_\lchild) \cdot \polyf(\etree_\rchild) &\text{ if \etree.\type } = \times\\
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\etree.\val &\text{ if \etree.\type } = \var \text{ OR } \tnum.
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\polyf(\revision{\circuit}) = \begin{cases}
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\polyf(\revision{\circuit_\lchild}) + \polyf(\revision{\circuit_\rchild}) &\text{ if \revision{\circuit}.\type } = \revision{OR-gate}\\
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\polyf(\revision{\circuit_\lchild}) \cdot \polyf(\revision{\circuit_\rchild}) &\text{ if \revision{\circuit}.\type } = \revision{AND-gate}\\
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\revision{\circuit.\val} &\text{ if \revision{\circuit}.\type } = \var \text{ OR } \tnum.
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\end{cases}
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\end{equation*}
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\end{Definition}
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%Specifically, when adding two monomials whose variables and respective exponents agree, the coefficients corresponding to the monomials are added and their sum is multiplied to the monomial. Multiplication here is denoted by concatenation of the monomial and coefficient. When two monomials are multiplied, the product of each corresponding coefficient is computed, and the variables in each monomial are multiplied, i.e., the exponents of like variables are added. Again we notate this by the direct product of coefficient product and all disitinct variables in the two monomials, with newly computed exponents.
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%\begin{Definition}[Expression Tree Set]\label{def:express-tree-set}$\etreeset{\smb}$ is the set of all possible expression trees $\etree$, such that $poly(\etree) = \poly(\vct{X})$.
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%\end{Definition}
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%
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%For the polynomial in~\Cref{eq:poly-eg}, $\etreeset{\smb}$ would include the following (represented as their corresponding expression trees): $2x^2 + 3xy - 2y^2, (x + 2y)(2x - y), x(2x - y) + 2y(2x - y), 2x(x + 2y) - y(x + 2y)$. Note that \Cref{def:express-tree-set} implies that for any expression tree $\etree$, we have $\etree \in \etreeset{poly(\etree)}$.
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\begin{Definition}[Expanded T]\label{def:expand-tree}
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$\expandtree{\etree}$ is the reduced sum of products expansion of $\etree$.
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The logical view of \expandtree{\etree} is a list of tuples $(\monom, \coef)$, where $\monom$ is a set of variables and $\coef$ is in $\mathbb{R}$.
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\expandtree{\etree} has the following recursive definition ($\circ$ is list concatenation).
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%
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% recursively defined as
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% \begin{align*}
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% &\etree.\type = + \mapsto&& \elist{\expandtree{\etree_\lchild}, \expandtree{\etree_\rchild}}\\
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% &\etree.\type = \times \mapsto&& \elist{\expandtree{\etree_\lchild} \otimes \expandtree{\etree_\rchild}}\\
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% &\etree.\type = \tnum \mapsto&& \elist{(\emptyset, \etree.\val)}\\
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% &\etree.\type = \var \mapsto&& \elist{(\etree.\val, 1)}
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% \end{align*}
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\begin{Definition}[Expanded \revision{\circuit}]\label{def:expand-circuit}
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\revision{$\expansion{\circuit}$} is the reduced sum of products expansion of $\revision{\circuit}$.
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The logical view of \revision{\expansion{\circuit}} is a list of tuples $(\monom, \coef)$, where $\monom$ is a set of variables and $\coef$ is in $\reals$.
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\revision{\expansion{\circuit}} has the following recursive definition ($\circ$ is list concatenation).
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{\small
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\begin{multline*}
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\expandtree{\etree} =
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\expansion{\circuit} =
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\begin{cases}
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\expandtree{\etree_\lchild} \circ \expandtree{\etree_\rchild} &\textbf{ if }\etree.\type = +\\
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\left\{(\monom_\lchild \cup \monom_\rchild, \coef_\lchild \cdot \coef_\rchild) ~|~\right.&\\ \quad \left.(\monom_\lchild, \coef_\lchild) \in \expandtree{\etree_\lchild}, (\monom_\rchild, \coef_\rchild) \in \expandtree{\etree_\rchild}\right\} &\textbf{ if }\etree.\type = \times\\
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\elist{(\emptyset, \etree.\val)} &\textbf{ if }\etree.\type = \tnum\\
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\elist{(\{\etree.\val\}, 1)} &\textbf{ if }\etree.\type = \var.\\
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\revision{\expansion{\circuit_\lchild} \circ \expansion{\circuit_\rchild}} &\textbf{ if }\revision{\circuit.\type = OR-gate}\\
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\left\{(\monom_\lchild \cup \monom_\rchild, \coef_\lchild \cdot \coef_\rchild) ~|~\right.&\\ \quad \left.(\monom_\lchild, \coef_\lchild) \in \revision{\expansion{\circuit_\lchild}}, (\monom_\rchild, \coef_\rchild) \in \revision{\expansion{\circuit_\rchild}}\right\} &\textbf{ if }\revision{\circuit.\type = AND-gate}\\
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\elist{(\emptyset, \revision{\circuit.\val})} &\textbf{ if }\revision{\circuit}.\type = \tnum\\
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\elist{(\{\revision{\circuit}.\val\}, 1)} &\textbf{ if }\revision{\circuit}.\type = \var.\\
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\end{cases}
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\end{multline*}
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}
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\end{Definition}
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%where that the multiplication of two tuples %is the standard multiplication over monomials and the standard multiplication over coefficients to produce the product tuple, as in
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%is their direct product $(\monom_1, \coef_1) \cdot (\monom_2, \coef_2) = (\monom_1 \cdot \monom_2, \coef_1 \times \coef_2)$ such that monomials $\monom_1$ and $\monom_2$ are concatenated in a product operation, while the standard product operation over reals applies to $\coef_1 \times \coef_2$. The product of $\expandtree{\etree_\lchild} \cdot \expandtree{\etree'_\rchild}$ is then the cross product of the multiplication of all such tuples returned to both $\expandtree{\etree_\lchild}$ and $\expandtree{\etree_\rchild}$. %The operator $\otimes$ is defined as the cross-product tuple multiplication of all such tuples returned by both $\expandtree{\etree_\lchild}$ and $\expandtree{\etree_\rchild}$.
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%is their direct product $(\monom_1, \coef_1) \cdot (\monom_2, \coef_2) = (\monom_1 \cdot \monom_2, \coef_1 \times \coef_2)$ such that monomials $\monom_1$ and $\monom_2$ are concatenated in a product operation, while the standard product operation over reals applies to $\coef_1 \times \coef_2$. The product of $\expansion{\circuit_\lchild} \cdot \expansion{\circuit'_\rchild}$ is then the cross product of the multiplication of all such tuples returned to both $\expansion{\circuit_\lchild}$ and $\expansion{\circuit_\rchild}$. %The operator $\otimes$ is defined as the cross-product tuple multiplication of all such tuples returned by both $\expansion{\circuit_\lchild}$ and $\expansion{\circuit_\rchild}$.
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In the following, we abuse notation and write $\monom$ to denote the monomial obtained as the products of the variables in the set.
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\begin{Example}\label{example:expr-tree-T}
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Consider the factorized representation $(X+ 2Y)(2X - Y)$ of the polynomial in~\Cref{eq:poly-eg}.
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Its expression tree $\etree$ is illustrated in Figure ~\ref{fig:expr-tree-T}.
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The pure expansion of the product is $2X^2 - XY + 4XY - 2Y^2$ and the $\expandtree{\etree}$ is $[(X, 2), (XY, -1), (XY, 4), (Y, -2)]$.
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Its circuit $\etree$ is illustrated in Figure ~\ref{fig:expr-tree-T}.
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The pure expansion of the product is $2X^2 - XY + 4XY - 2Y^2$ and the $\expansion{\circuit}$ is $[(X, 2), (XY, -1), (XY, 4), (Y, -2)]$.
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\end{Example}
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$\expandtree{\etree}$ encodes the \emph{reduced} form of $\polyf\inparen{\etree}$, decoupling each monomial into a set of variables $\monom$ and a real coefficient $\coef$.
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Note, however, that unlike $\rpoly$, $\expandtree{\etree}$ does not need to be in SOP form.
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$\expansion{\circuit}$ encodes the \emph{reduced} form of $\polyf\inparen{\circuit}$, decoupling each monomial into a set of variables $\monom$ and a real coefficient $\coef$.
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Note, however, that unlike $\rpoly$, $\expansion{\circuit}$ does not need to be in SOP form.
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\begin{figure}[t]
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@ -118,44 +97,50 @@ Note, however, that unlike $\rpoly$, $\expandtree{\etree}$ does not need to be i
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child{node[tree_node] (neg-leaf) {-1}}
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child{node[tree_node]{y}}
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}
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%child[sibling distance= 0cm, grow=north east, red]{node[tree_node]{$\etree_\rchild$}}
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%child[sibling distance= 0cm, grow=north east, red]{node[tree_node]{$\circuit_\rchild$}}
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};
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% \node[below=2pt of neg-leaf, inner sep=1pt, blue] (neg-comment) {\textbf{Negation pushed to leaf nodes}};
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% \draw[<-|, blue] (neg-leaf) -- (neg-comment);
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\node[above right=0.7cm of TR, highlight_color, inner sep=0pt, font=\bfseries] (tr-label) {$\etree_\rchild$};
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\node[above right=0.7cm of root, highlight_color, inner sep=0pt, font=\bfseries] (t-label) {$\etree$};
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\node[above right=0.7cm of TR, highlight_color, inner sep=0pt, font=\bfseries] (tr-label) {$\circuit_\rchild$};
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\node[above right=0.7cm of root, highlight_color, inner sep=0pt, font=\bfseries] (t-label) {$\circuit$};
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\draw[<-|, highlight_color] (TR) -- (tr-label);
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\draw[<-|, highlight_color] (root) -- (t-label);
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\end{tikzpicture}
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}
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\vspace*{-2mm}
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\caption{Expression tree $\etree$ for the product $\boldsymbol{(x + 2y)(2x - y)}$.}
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\caption{Expression tree $\circuit$ for the product $\boldsymbol{(x + 2y)(2x - y)}$.}
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\label{fig:expr-tree-T}
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\trimfigurespacing
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\end{figure}
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\begin{Definition}[Positive T]\label{def:positive-tree}
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For any expression tree $\etree$, the corresponding
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{\em positive tree}, denoted $\abs{\etree}$ obtained from $\etree$ as follows. For each leaf node $\ell$ of $\etree$ where $\ell.\type$ is $\tnum$, update $\ell.\vari{value}$ to $|\ell.\vari{value}|$. %value $\coef$ of each coefficient leaf node in $\etree$ is set to %$\coef_i$ in $\etree$ is exchanged with its absolute value$|\coef|$.
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\revision{
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\begin{Definition}[Positive \circuit]\label{def:positive-circuit}
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For any circuit $\circuit$, the corresponding
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{\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 $|\ell.\vari{value}|$.
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\end{Definition}
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Using the same factorization from ~\Cref{example:expr-tree-T}, $\polyf(\abs{\etree}) = (X + 2Y)(2X + Y) = 2X^2 +XY +4XY + 2Y^2 = 2X^2 + 5XY + 2Y^2$. Note that this \textit{is not} the same as the polynomial from~\Cref{eq:poly-eg}.
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Using the same factorization from ~\Cref{example:expr-tree-T}, $\polyf(\abs{\circuit}) = (X + 2Y)(2X + Y) = 2X^2 +XY +4XY + 2Y^2 = 2X^2 + 5XY + 2Y^2$. Note that this \textit{is not} the same as the polynomial from~\Cref{eq:poly-eg}.
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\begin{Definition}[Evaluation]\label{def:exp-poly-eval}
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Given an expression tree $\etree$ and $\vct{v} \in \mathbb{R}^\numvar$, we define the evaluation of $\etree$ on $\vct{v}$ as $\etree(\vct{v}) = \polyf(\etree)(\vct{v})$.
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Given an expression tree $\circuit$ and $\vct{v} \in \mathbb{R}^\numvar$, we define the evaluation of $\circuit$ on $\vct{v}$ as $\circuit(\vct{v}) = \polyf(\circuit)(\vct{v})$.
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\end{Definition}
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}
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\subsection{Our main result}
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In the subsequent subsections we will prove the following theorem.
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\begin{Theorem}\label{lem:approx-alg}
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Let $\etree$ be an expression tree for a UCQ over \bi and define $\poly(\vct{X})=\polyf(\etree)$ and let $k=\degree(\poly)$.
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Let $\circuit$ be an expression tree for a UCQ over \bi and define $\poly(\vct{X})=\polyf(\circuit)$ and let $k=\degree(\poly)$.
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%Let $\poly(\vct{X})$ be a query polynomial corresponding to the output of a UCQ in a \bi.
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Then an estimate $\mathcal{E}$ %=\approxq(\etree, P_1,\dots,p_\numvar), \conf, \error')$
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Then an estimate $\mathcal{E}$ %=\approxq(\circuit, P_1,\dots,p_\numvar), \conf, \error')$
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of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ can be computed in time
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\[O\left(\treesize(\etree) + \frac{\log{\frac{1}{\conf}}\cdot \abs{\etree}^2(1,\ldots, 1)\cdot k\cdot \log{k} \cdot depth(\etree))}{\inparen{\error'}^2\cdot\rpoly^2(\prob_1,\ldots, \prob_\numvar)}\right)\]
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\[O\left(\treesize(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot \abs{\circuit}^2(1,\ldots, 1)\cdot k\cdot \log{k} \cdot depth(\circuit))}{\inparen{\error'}^2\cdot\rpoly^2(\prob_1,\ldots, \prob_\numvar)}\right)\]
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such that
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\begin{equation}
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\label{eq:approx-algo-bound}
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@ -166,10 +151,10 @@ such that
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\noindent The proof of~\Cref{lem:approx-alg} can be found in~\Cref{sec:proofs-approx-alg}.
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To get linear runtime results from~\Cref{lem:approx-alg}, we will need to define another parameter modeling the (weighted) number of monomials in $\expandtree{\etree}$ to be `canceled' when it is modded with $\mathcal{B}$:
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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 $\mathcal{B}$:
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\begin{Definition}[Parameter $\gamma$]\label{def:param-gamma}
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Given an expression tree $\etree$, define
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\[\gamma(\etree)=\frac{\sum_{(\monom, \coef)\in \expandtree{\etree}} \abs{\coef}\cdot \indicator{\monom\mod{\mathcal{B}}\equiv 0}}{\abs{\etree}(1,\ldots, 1)}\]
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Given an expression tree $\circuit$, define
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\[\gamma(\circuit)=\frac{\sum_{(\monom, \coef)\in \expansion{\circuit}} \abs{\coef}\cdot \indicator{\monom\mod{\mathcal{B}}\equiv 0}}{\abs{\circuit}(1,\ldots, 1)}\]
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\end{Definition}
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%\AH{This....combined with \Cref{def:mod-set-polys} is \emph{really} nice notation!}
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%\AR{Need to make sure use of indicator variable $\onesymbol$ above is consistent with the rest of the paper.}
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@ -178,9 +163,9 @@ Given an expression tree $\etree$, define
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\noindent We next present couple of corollaries of~\Cref{lem:approx-alg}.
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\begin{Corollary}
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\label{cor:approx-algo-const-p}
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Let $\poly(\vct{X})$ be as in~\Cref{lem:approx-alg} and let $\gamma=\gamma(\etree)$. Further let it be the case that $\prob_i\ge \prob_0$ for all $i\in[\numvar]$. Then an estimate $\mathcal{E}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ satisfying~\Cref{eq:approx-algo-bound} can be computed in time
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\[O\left(\treesize(\etree) + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot depth(\etree))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\]
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In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\treesize(\etree)\cdot \log{\frac{1}{\conf}}\right)$.
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Let $\poly(\vct{X})$ be as in~\Cref{lem:approx-alg} and let $\gamma=\gamma(\circuit)$. Further let it be the case that $\prob_i\ge \prob_0$ for all $i\in[\numvar]$. Then an estimate $\mathcal{E}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ satisfying~\Cref{eq:approx-algo-bound} can be computed in time
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\[O\left(\treesize(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot depth(\circuit))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\]
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In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\treesize(\circuit)\cdot \log{\frac{1}{\conf}}\right)$.
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\end{Corollary}
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The proof for~\Cref{cor:approx-algo-const-p} can be seen in~\Cref{sec:proofs-approx-alg}.
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@ -195,12 +180,12 @@ Thus, we expect the corollary to hold in general.
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\subsection{Approximating $\rpoly$}
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The algorithm to prove~\Cref{lem:approx-alg} follows from the following observation. Given a query polynomial $\poly(\vct{X})=\polyf(\etree)$ for expression tree $\etree$ over $\bi$, we can exactly represent $\rpoly(\vct{X})$ as follows:
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The algorithm to prove~\Cref{lem:approx-alg} follows from the following observation. Given a query polynomial $\poly(\vct{X})=\polyf(\circuit)$ for expression tree $\circuit$ over $\bi$, we can exactly represent $\rpoly(\vct{X})$ as follows:
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\begin{equation}
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\label{eq:tilde-Q-bi}
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\rpoly\inparen{X_1,\dots,X_\numvar}=\hspace*{-1mm}\sum_{(\monom,\coef)\in \expandtree{\etree}} \hspace*{-2mm} \indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \coef\cdot\hspace*{-2mm}\prod_{X_i\in \var\inparen{\monom}}\hspace*{-2mm} X_i
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\rpoly\inparen{X_1,\dots,X_\numvar}=\hspace*{-1mm}\sum_{(\monom,\coef)\in \expansion{\circuit}} \hspace*{-2mm} \indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \coef\cdot\hspace*{-2mm}\prod_{X_i\in \var\inparen{\monom}}\hspace*{-2mm} X_i
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\end{equation}
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Given the above, the algorithm is a sampling based algorithm for the above sum: we sample $(\monom,\coef)\in \expandtree{\etree}$ with probability proportional\footnote{We could have also uniformly sampled from $\expandtree{\etree}$ but this gives better parameters.}
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Given the above, the algorithm is a sampling based algorithm for the above sum: we sample $(\monom,\coef)\in \expansion{\circuit}$ with probability proportional\footnote{We could have also uniformly sampled from $\expansion{\circuit}$ but this gives better parameters.}
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%\AH{Regarding the footnote, is there really a difference? I \emph{suppose} technically, but in this case they are \emph{effectively} the same. Just wondering.}
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%\AR{Yes, there is! If we used uniform distribution then in our bounds we will have a parameter that depends on the largest $\abs{coef}$, which e.g. could be dependent on $n$. But with the weighted probability distribution, we avoid paying this price. Though I guess perhaps we can say for the kinds of queries we consider thhese coefficients are all constants?}
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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.
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@ -213,7 +198,7 @@ The number of samples is computed by (see \Cref{app:subsec-th-mon-samp}):
|
|||
|
||||
%We state the approximation algorithm in terms of a $\bi$.
|
||||
%\subsubsection{Description}
|
||||
%Algorithm ~\ref{alg:mon-sam} approximates $\rpoly$ using the following steps. First, a call to $\onepass$ on its input $\etree$ produces a non-biased weight distribution over the monomials of $\expandtree{\etree}$ and a correct count of $|\etree|(1,\ldots, 1)$, i.e., the number of monomials in $\expandtree{\etree}$. Next, ~\Cref{alg:mon-sam} calls $\sampmon$ to sample one monomial and its sign from $\expandtree{\etree}$. The sampling is repeated $\ceil{\frac{2\log{\frac{2}{\delta}}}{\epsilon^2}}$ times, where each of the samples are evaluated with input $\vct{p}$, multiplied by $1 \times sign$, and summed. The final result is scaled accordingly returning an estimate of $\rpoly$ with the claimed $(\error, \conf)$-bound of ~\Cref{lem:mon-samp}.
|
||||
%Algorithm ~\ref{alg:mon-sam} approximates $\rpoly$ using the following steps. First, a call to $\onepass$ on its input $\circuit$ produces a non-biased weight distribution over the monomials of $\expansion{\circuit}$ and a correct count of $|\circuit|(1,\ldots, 1)$, i.e., the number of monomials in $\expansion{\circuit}$. Next, ~\Cref{alg:mon-sam} calls $\sampmon$ to sample one monomial and its sign from $\expansion{\circuit}$. The sampling is repeated $\ceil{\frac{2\log{\frac{2}{\delta}}}{\epsilon^2}}$ times, where each of the samples are evaluated with input $\vct{p}$, multiplied by $1 \times sign$, and summed. The final result is scaled accordingly returning an estimate of $\rpoly$ with the claimed $(\error, \conf)$-bound of ~\Cref{lem:mon-samp}.
|
||||
|
||||
%\AR{Seems like the notation below belongs to the notation section (if we decide to state this explicitly at all)?}
|
||||
%\AH{Yes, I only included this per your request a few months ago. Based on @lordpretzel removing my definition of monomial, perhaps we can assume that the reader understands the notation below. I \emph{think} this should be a reasonable assumption.}
|
||||
|
|
@ -222,19 +207,19 @@ The number of samples is computed by (see \Cref{app:subsec-th-mon-samp}):
|
|||
|
||||
%Original \ti Algorithm
|
||||
%\begin{algorithm}[H]
|
||||
% \caption{$\approxq$($\etree$, $\vct{p}$, $\conf$, $\error$)}
|
||||
% \caption{$\approxq$($\circuit$, $\vct{p}$, $\conf$, $\error$)}
|
||||
% \label{alg:mon-sam}
|
||||
% \begin{algorithmic}[1]
|
||||
% \Require \etree: Binary Expression Tree
|
||||
% \Require \circuit: Binary Expression Tree
|
||||
% \Require $\vct{p} = (\prob_1,\ldots, \prob_\numvar)$ $\in [0, 1]^N$
|
||||
% \Require $\conf$ $\in [0, 1]$
|
||||
% \Require $\error$ $\in [0, 1]$
|
||||
% \Ensure \vari{acc} $\in \mathbb{R}$
|
||||
% \State $\accum \gets 0$\label{alg:mon-sam-global1}
|
||||
% \State $\numsamp \gets \ceil{\frac{2 \log{\frac{2}{\conf}}}{\error^2}}$\label{alg:mon-sam-global2}
|
||||
% \State $(\vari{\etree}_\vari{mod}, \vari{size}) \gets $ \onepass($\etree$)\label{alg:mon-sam-onepass}\Comment{$\onepass$ is ~\Cref{alg:one-pass} \;and \sampmon \; is ~\Cref{alg:sample}}\newline
|
||||
% \State $(\vari{\circuit}_\vari{mod}, \vari{size}) \gets $ \onepass($\circuit$)\label{alg:mon-sam-onepass}\Comment{$\onepass$ is ~\Cref{alg:one-pass} \;and \sampmon \; is ~\Cref{alg:sample}}\newline
|
||||
% \For{\vari{i} \text{ in } $1\text{ to }\numsamp$}\Comment{Perform the required number of samples}
|
||||
% \State $(\vari{M}_\vari{i}, \vari{sgn}_\vari{i}) \gets $ \sampmon($\etree_\vari{mod}$)\label{alg:mon-sam-sample}
|
||||
% \State $(\vari{M}_\vari{i}, \vari{sgn}_\vari{i}) \gets $ \sampmon($\circuit_\vari{mod}$)\label{alg:mon-sam-sample}
|
||||
% \State $\vari{Y}_\vari{i} \gets 1$\label{alg:mon-sam-assign1}
|
||||
% \For{$\vari{x}_{\vari{j}}$ \text{ in } $\vari{M}_{\vari{i}}$}
|
||||
% \State $\vari{Y}_\vari{i} \gets \vari{Y}_\vari{i} \times \; \vari{\prob}_\vari{j}$\label{alg:mon-sam-product2} \Comment{$\vari{p}_\vari{j}$ is the assignment to $\vari{x}_\vari{j}$ from input $\vct{p}$}
|
||||
|
|
@ -252,10 +237,10 @@ The number of samples is computed by (see \Cref{app:subsec-th-mon-samp}):
|
|||
|
||||
|
||||
\begin{algorithm}[t]
|
||||
\caption{$\approxq(\etree, \vct{p}, \conf, \error)$}
|
||||
\caption{$\approxq(\circuit, \vct{p}, \conf, \error)$}
|
||||
\label{alg:mon-sam}
|
||||
\begin{algorithmic}[1]
|
||||
\Require \etree: Binary Expression Tree
|
||||
\Require \circuit: Binary Expression Tree
|
||||
\Require $\vct{p} = (\prob_1,\ldots, \prob_\numvar)$ $\in [0, 1]^N$
|
||||
\Require $\conf$ $\in [0, 1]$
|
||||
\Require $\error$ $\in [0, 1]$
|
||||
|
|
@ -265,12 +250,12 @@ The number of samples is computed by (see \Cref{app:subsec-th-mon-samp}):
|
|||
%\State $\vari{sample}_\vari{next} \gets 0$
|
||||
\State $\accum \gets 0$\label{alg:mon-sam-global1}
|
||||
\State $\numsamp \gets \ceil{\frac{2 \log{\frac{2}{\conf}}}{\error^2}}$\label{alg:mon-sam-global2}
|
||||
\State $(\vari{\etree}_\vari{mod}, \vari{size}) \gets $ \onepass($\etree$)\label{alg:mon-sam-onepass}\Comment{$\onepass$ is ~\Cref{alg:one-pass}}
|
||||
\State $(\vari{\circuit}_\vari{mod}, \vari{size}) \gets $ \onepass($\circuit$)\label{alg:mon-sam-onepass}\Comment{$\onepass$ is ~\Cref{alg:one-pass}}
|
||||
%\newline
|
||||
%\State $\vari{i} \gets 1$
|
||||
\For{$\vari{i} \in 1 \text{ to }\numsamp$}\label{alg:sampling-loop}\Comment{Perform the required number of samples}
|
||||
%\State $\bivec \gets [0]^{\abs{\block}}$\Comment{$\bivec$ is an array whose size is the number of blocks, used to check for cross-terms}\newline
|
||||
\State $(\vari{M}, \vari{sgn}_\vari{i}) \gets $ \sampmon($\etree_\vari{mod}$)\label{alg:mon-sam-sample}\Comment{\sampmon \; is ~\Cref{alg:sample}}
|
||||
\State $(\vari{M}, \vari{sgn}_\vari{i}) \gets $ \sampmon($\circuit_\vari{mod}$)\label{alg:mon-sam-sample}\Comment{\sampmon \; is ~\Cref{alg:sample}}
|
||||
%\For{$\vari{x}_\vari{\block,i}$ \text{ in } $\vari{M}$}
|
||||
% \If{$\bivec[\block] = 1$}\label{alg:mon-sam-check}\Comment{If we have already had a variable from this block, $\rpoly$ drops the sample.}
|
||||
% \newline
|
||||
|
|
@ -301,17 +286,17 @@ The number of samples is computed by (see \Cref{app:subsec-th-mon-samp}):
|
|||
\EndIf
|
||||
\EndFor
|
||||
|
||||
%\State $\gamma \gets $ $\algname{Estimate}$ $\gamma(\etree, \numsamp, \abs{\block})$
|
||||
%\State $\gamma \gets $ $\algname{Estimate}$ $\gamma(\circuit, \numsamp, \abs{\block})$
|
||||
\State $\vari{acc} \gets \vari{acc} \times \frac{\vari{size}}{\numsamp}$\label{alg:mon-sam-global3}
|
||||
\State \Return \vari{acc}
|
||||
\end{algorithmic}
|
||||
\end{algorithm}
|
||||
|
||||
%\begin{algorithm}[H]
|
||||
% \caption{$\algname{Estimate}$ $\gamma(\etree, \numsamp, \abs{\block})$}
|
||||
% \caption{$\algname{Estimate}$ $\gamma(\circuit, \numsamp, \abs{\block})$}
|
||||
% \label{alg:est-gamma}
|
||||
% \begin{algorithmic}[1]
|
||||
% \Require \etree: Binary Expression Tree
|
||||
% \Require \circuit: Binary Expression Tree
|
||||
% \Require $\numsamp \in \mathbb{N}$
|
||||
% \Require $\abs{\block} \in \mathbb{N}$
|
||||
% \Ensure \vari{cTerms} $]in \mathbb{R}$
|
||||
|
|
@ -320,7 +305,7 @@ The number of samples is computed by (see \Cref{app:subsec-th-mon-samp}):
|
|||
% \State $\vari{isCross} \gets 0$
|
||||
% \For{$\vari{i} \text{ in } 1 \text{ to } \numsamp$}
|
||||
% \State $\bivec \gets [0]^{\abs{\block}}$
|
||||
% \State $(\vari{M}, \vari{sgn}) \gets $ \sampmon($\etree_\vari{mod}$)
|
||||
% \State $(\vari{M}, \vari{sgn}) \gets $ \sampmon($\circuit_\vari{mod}$)
|
||||
% \For{$\vari{x}_{\vari{b}, \vari{j}} \text{ in } \vari{M}$}
|
||||
% \If{$\bivec[b] = 1$}
|
||||
% \State $\vari{isCross} \gets 1$
|
||||
|
|
@ -346,24 +331,24 @@ we first state the lemmas that summarize the relevant properties of $\onepass$ a
|
|||
|
||||
|
||||
\begin{Lemma}\label{lem:one-pass}
|
||||
The $\onepass$ function completes in $O(size(\etree))$ time. $\onepass$ guarantees two post conditions: First, for each subtree $\vari{S}$ of $\etree$, we have that $\vari{S}.\vari{partial}$ is set to $\abs{\vari{S}}(1,\ldots, 1)$. Second, when $\vari{S}.\type = +$, each $\vari{child}$ of $\vari{S}$, $\vari{child}.\vari{weight}$ is set to $\frac{\abs{\vari{S}_{\vari{child}}}(1,\ldots, 1)}{\abs{\vari{S}}(1,\ldots, 1)}$. % is correctly computed for each child of $\vari{S}.$
|
||||
The $\onepass$ function completes in $O(size(\circuit))$ time. $\onepass$ guarantees two post conditions: First, for each subtree $\vari{S}$ of $\circuit$, we have that $\vari{S}.\vari{partial}$ is set to $\abs{\vari{S}}(1,\ldots, 1)$. Second, when $\vari{S}.\type = +$, each $\vari{child}$ of $\vari{S}$, $\vari{child}.\vari{weight}$ is set to $\frac{\abs{\vari{S}_{\vari{child}}}(1,\ldots, 1)}{\abs{\vari{S}}(1,\ldots, 1)}$. % is correctly computed for each child of $\vari{S}.$
|
||||
\end{Lemma}
|
||||
To prove correctness of~\Cref{alg:mon-sam}, we only use the following fact that follows from the above lemma: $\etree_{\vari{mod}}.\vari{partial}=\abs{\etree}(1,\dots,1)$.
|
||||
%\AH{I'm wondering if there is a better notation to use here. I myself got confused by my own notation of $\etree_{\vari{mod}}$. \emph{But}, we need to to be referencing the modified $\etree$ returned by $\onepass$ in the algorithm, so maybe this is the best we can do?}
|
||||
To prove correctness of~\Cref{alg:mon-sam}, we only use the following fact that follows from the above lemma: $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.
|
||||
%\AH{I'm wondering if there is a better notation to use here. I myself got confused by my own notation of $\circuit_{\vari{mod}}$. \emph{But}, we need to to be referencing the modified $\circuit$ returned by $\onepass$ in the algorithm, so maybe this is the best we can do?}
|
||||
%\AR{yeah, I think this is fine.}
|
||||
%At the conclusion of $\onepass$, $\etree.\vari{partial}$ will hold the sum of all coefficients in $\expandtree{\abs{\etree}}$, i.e., $\sum\limits_{(\monom, \coef) \in \expandtree{\abs{\etree}}}\coef$. $\etree.\vari{weight}$ will hold the weighted probability that $\etree$ is sampled from from its parent $+$ node.
|
||||
%At the conclusion of $\onepass$, $\circuit.\vari{partial}$ will hold the sum of all coefficients in $\expansion{\abs{\circuit}}$, i.e., $\sum\limits_{(\monom, \coef) \in \expansion{\abs{\circuit}}}\coef$. $\circuit.\vari{weight}$ will hold the weighted probability that $\circuit$ is sampled from from its parent $+$ node.
|
||||
|
||||
\begin{Lemma}\label{lem:sample}
|
||||
The function $\sampmon$ completes in $O(\log{k} \cdot k \cdot depth(\etree))$ time, where $k = \degree(poly(\abs{\etree})$. Upon completion, every $\left(\monom, sign(\coef)\right)\in \expandtree{\abs{\etree}}$ is returned with probability $\frac{|\coef|}{\abs{\etree}(1,\ldots, 1)}$. %, $\sampmon$ returns the sampled term $\left(\monom, sign(\coef)\right)$ from $\expandtree{\abs{\etree}}$.
|
||||
The function $\sampmon$ completes in $O(\log{k} \cdot k \cdot depth(\circuit))$ time, where $k = \degree(poly(\abs{\circuit})$. Upon completion, every $\left(\monom, sign(\coef)\right)\in \expansion{\abs{\circuit}}$ is returned with probability $\frac{|\coef|}{\abs{\circuit}(1,\ldots, 1)}$. %, $\sampmon$ returns the sampled term $\left(\monom, sign(\coef)\right)$ from $\expansion{\abs{\circuit}}$.
|
||||
\end{Lemma}
|
||||
|
||||
Armed 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}
|
||||
%If the contracts for $\onepass$ and $\sampmon$ hold, then
|
||||
For any $\etree$ with $\degree(poly(|\etree|)) = k$, algorithm \ref{alg:mon-sam} outputs an estimate $\vari{acc}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ such that %$\expct\pbox{\empmean} = \frac{\rpoly(\prob_1,\ldots, \prob_\numvar)\cdot(1 - \gamma)}{\abs{\etree}(1,\ldots, 1)}$. %within an additive $\error \cdot \abs{\etree}(1,\ldots, 1)$ error with
|
||||
For any $\circuit$ with $\degree(poly(|\circuit|)) = k$, algorithm \ref{alg:mon-sam} outputs an estimate $\vari{acc}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ such that %$\expct\pbox{\empmean} = \frac{\rpoly(\prob_1,\ldots, \prob_\numvar)\cdot(1 - \gamma)}{\abs{\circuit}(1,\ldots, 1)}$. %within an additive $\error \cdot \abs{\circuit}(1,\ldots, 1)$ error with
|
||||
%$\empmean$ has bounds
|
||||
\[\probOf\left(\left|\vari{acc} - \rpoly(\prob_1,\ldots, \prob_\numvar)\right|> \error \cdot \abs{\etree}(1,\ldots, 1)\right) \leq \conf,\]
|
||||
in $O\left(\treesize(\etree)\right.$ $+$ $\left.\left(\frac{\log{\frac{1}{\conf}}}{\error^2} \cdot k \cdot\log{k} \cdot depth(\etree)\right)\right)$ time.
|
||||
\[\probOf\left(\left|\vari{acc} - \rpoly(\prob_1,\ldots, \prob_\numvar)\right|> \error \cdot \abs{\circuit}(1,\ldots, 1)\right) \leq \conf,\]
|
||||
in $O\left(\treesize(\circuit)\right.$ $+$ $\left.\left(\frac{\log{\frac{1}{\conf}}}{\error^2} \cdot k \cdot\log{k} \cdot depth(\circuit)\right)\right)$ time.
|
||||
\end{Theorem}
|
||||
|
||||
|
||||
|
|
@ -373,38 +358,38 @@ For any $\etree$ with $\degree(poly(|\etree|)) = k$, algorithm \ref{alg:mon-sam}
|
|||
%\subsubsection{Description}
|
||||
%Algorithm ~\ref{alg:one-pass} satisfies the requirements of lemma ~\ref{lem:one-pass}.
|
||||
|
||||
The evaluation of $\abs{\etree}(1,\ldots, 1)$ can be defined recursively, as follows (where $\etree_\lchild$ and $\etree_\rchild$ are the `left' and `right' children of $\etree$ if they exist):
|
||||
The evaluation of $\abs{\circuit}(1,\ldots, 1)$ can be defined recursively, as follows (where $\circuit_\lchild$ and $\circuit_\rchild$ are the `left' and `right' children of $\circuit$ if they exist):
|
||||
|
||||
{\small
|
||||
\begin{align}
|
||||
\label{eq:T-all-ones}
|
||||
\abs{\etree}(1,\ldots, 1) = \begin{cases}
|
||||
\abs{\etree_\lchild}(1,\ldots, 1) \cdot \abs{\etree_\rchild}(1,\ldots, 1) &\textbf{if }\etree.\type = \times\\
|
||||
\abs{\etree_\lchild}(1,\ldots, 1) + \abs{\etree_\rchild}(1,\ldots, 1) &\textbf{if }\etree.\type = + \\
|
||||
|\etree.\val| &\textbf{if }\etree.\type = \tnum\\
|
||||
1 &\textbf{if }\etree.\type = \var.
|
||||
\abs{\circuit}(1,\ldots, 1) = \begin{cases}
|
||||
\abs{\circuit_\lchild}(1,\ldots, 1) \cdot \abs{\circuit_\rchild}(1,\ldots, 1) &\textbf{if }\circuit.\type = \revision{AND}\\
|
||||
\abs{\circuit_\lchild}(1,\ldots, 1) + \abs{\circuit_\rchild}(1,\ldots, 1) &\textbf{if }\circuit.\type = \revision{OR} \\
|
||||
|\circuit.\val| &\textbf{if }\circuit.\type = \tnum\\
|
||||
1 &\textbf{if }\circuit.\type = \var.
|
||||
\end{cases}
|
||||
\end{align}
|
||||
}
|
||||
|
||||
%\begin{align*}
|
||||
%&\eval{\etree ~|~ \etree.\type = +}_{\abs{\etree}} =&& \eval{\etree_\lchild}_{\abs{\etree}} + \eval{\etree_\rchild}_{\abs{\etree}}\\
|
||||
%&\eval{\etree ~|~ \etree.\type = \times}_{\abs{\etree}} = && \eval{\etree_\lchild}_{\abs{\etree}} \cdot \eval{\etree_\rchild}_{\abs{\etree}}\\
|
||||
%&\eval{\etree ~|~ \etree.\type = \tnum}_{\abs{\etree}} = && \etree.\val\\
|
||||
%&\eval{\etree ~|~ \etree.\val = \var}_{\abs{\etree}} = && 1
|
||||
%&\eval{\circuit ~|~ \circuit.\type = +}_{\abs{\circuit}} =&& \eval{\circuit_\lchild}_{\abs{\circuit}} + \eval{\circuit_\rchild}_{\abs{\circuit}}\\
|
||||
%&\eval{\circuit ~|~ \circuit.\type = \times}_{\abs{\circuit}} = && \eval{\circuit_\lchild}_{\abs{\circuit}} \cdot \eval{\circuit_\rchild}_{\abs{\circuit}}\\
|
||||
%&\eval{\circuit ~|~ \circuit.\type = \tnum}_{\abs{\circuit}} = && \circuit.\val\\
|
||||
%&\eval{\circuit ~|~ \circuit.\val = \var}_{\abs{\circuit}} = && 1
|
||||
%\end{align*}
|
||||
|
||||
%In the same fashion the weighted distribution can be described as above with the following modification for the case when $\etree.\type = +$:
|
||||
It turns out that for proof of~\Cref{lem:sample}, we need to argue that when $\etree.\type = +$, we indeed have
|
||||
%In the same fashion the weighted distribution can be described as above with the following modification for the case when $\circuit.\type = +$:
|
||||
It turns out that for proof of~\Cref{lem:sample}, we need to argue that when $\circuit.\type = +$, we indeed have
|
||||
\begin{align}
|
||||
\label{eq:T-weights}
|
||||
%&\abs{\etree_\lchild}(1,\ldots, 1) + \abs{\etree_\rchild}(1,\ldots, 1); &\textbf{if }\etree.\type = + \\
|
||||
\etree_\lchild.\vari{weight} &\gets \frac{\abs{\etree_\lchild}(1,\ldots, 1)}{\abs{\etree_\lchild}(1,\ldots, 1) + \abs{\etree_\rchild}(1,\ldots, 1)};\\
|
||||
\etree_\rchild.\vari{weight} &\gets \frac{\abs{\etree_\rchild}(1,\ldots, 1)}{\abs{\etree_\lchild}(1,\ldots, 1)+ \abs{\etree_\rchild}(1,\ldots, 1)}
|
||||
%&\abs{\circuit_\lchild}(1,\ldots, 1) + \abs{\circuit_\rchild}(1,\ldots, 1); &\textbf{if }\circuit.\type = + \\
|
||||
\circuit_\lchild.\vari{weight} &\gets \frac{\abs{\circuit_\lchild}(1,\ldots, 1)}{\abs{\circuit_\lchild}(1,\ldots, 1) + \abs{\circuit_\rchild}(1,\ldots, 1)};\\
|
||||
\circuit_\rchild.\vari{weight} &\gets \frac{\abs{\circuit_\rchild}(1,\ldots, 1)}{\abs{\circuit_\lchild}(1,\ldots, 1)+ \abs{\circuit_\rchild}(1,\ldots, 1)}
|
||||
\end{align}
|
||||
|
||||
%\begin{align*}
|
||||
%&\eval{\etree~|~\etree.\type = +}_{\wght} =&&\eval{\etree_\lchild}_{\abs{\etree}} + \eval{\etree_\rchild}_{\abs{\etree}}; \etree_\lchild.\wght = \frac{\eval{\etree_\lchild}_{\abs{\etree}}}{\eval{\etree_\lchild}_{\abs{\etree}} + \eval{\etree_\rchild}_{\abs{\etree}}}; \etree_\rchild.\wght = \frac{\eval{\etree_\rchild}_{\abs{\etree}}}{\eval{\etree_\lchild}_{\abs{\etree}} + \eval{\etree_\rchild}_{\abs{\etree}}}
|
||||
%&\eval{\circuit~|~\circuit.\type = +}_{\wght} =&&\eval{\circuit_\lchild}_{\abs{\circuit}} + \eval{\circuit_\rchild}_{\abs{\circuit}}; \circuit_\lchild.\wght = \frac{\eval{\circuit_\lchild}_{\abs{\circuit}}}{\eval{\circuit_\lchild}_{\abs{\circuit}} + \eval{\circuit_\rchild}_{\abs{\circuit}}}; \circuit_\rchild.\wght = \frac{\eval{\circuit_\rchild}_{\abs{\circuit}}}{\eval{\circuit_\lchild}_{\abs{\circuit}} + \eval{\circuit_\rchild}_{\abs{\circuit}}}
|
||||
%\end{align*}
|
||||
\noindent \onepass\ (Algorithm ~\ref{alg:one-pass} in \Cref{sec:proofs-approx-alg}) essentially populates the \wght and \vpartial variables on each node with the definitions above. Lemma~\ref{lem:one-pass} is also proved in~\Cref{sec:proofs-approx-alg}.
|
||||
|
||||
|
|
@ -417,12 +402,12 @@ It turns out that for proof of~\Cref{lem:sample}, we need to argue that when $\e
|
|||
\subsection{\sampmon\ Algorithm}
|
||||
\label{sec:samplemonomial}
|
||||
|
||||
%Algorithm ~\ref{alg:sample} takes $\etree$ as input, samples an arbitrary $(\monom, \coef)$ from $\expandtree{\etree}$ with probabilities $\stree_\lchild.\wght$ and $\stree_\rchild.\wght$ for each subtree $\stree$ with $\stree.\type = +$, outputting the tuple $(\monom, \sign(\coef))$. While one cannot compute $\expandtree{\etree}$ in time better than $O(N^k)$, the algorithm, similar to \textsc{OnePass}, uses a technique on $\etree$ which produces a sample from $\expandtree{\etree}$ without ever materializing $\expandtree{\etree}$.
|
||||
%Algorithm ~\ref{alg:sample} takes $\circuit$ as input, samples an arbitrary $(\monom, \coef)$ from $\expansion{\circuit}$ with probabilities $\stree_\lchild.\wght$ and $\stree_\rchild.\wght$ for each subtree $\stree$ with $\stree.\type = +$, outputting the tuple $(\monom, \sign(\coef))$. While one cannot compute $\expansion{\circuit}$ in time better than $O(N^k)$, the algorithm, similar to \textsc{OnePass}, uses a technique on $\circuit$ which produces a sample from $\expansion{\circuit}$ without ever materializing $\expansion{\circuit}$.
|
||||
|
||||
A naive (slow) implementation of \sampmon\ would first compute $\expandtree{\etree}$ and then sample from it.
|
||||
A naive (slow) implementation of \sampmon\ would first compute $\expansion{\circuit}$ and then sample from it.
|
||||
% However, this would be too time consuming.
|
||||
%
|
||||
Instead, \Cref{alg:sample} selects a monomial from $\expandtree{\etree}$ by top-down traversal.
|
||||
Instead, \Cref{alg:sample} selects a monomial from $\expansion{\circuit}$ by top-down traversal.
|
||||
For a parent $+$ node, the child to be visited is sampled from the weighted distribution precomputed by \onepass.
|
||||
When a parent $\times$ node is visited, both children are visited.
|
||||
The algorithm computes two properties: the set of all variable leaf nodes visited, and the product of signs of visited coefficient leaf nodes.
|
||||
|
|
@ -436,34 +421,34 @@ We will assume the TreeSet data structure to maintain sets with logarithmic time
|
|||
$\sampmon$ is given in \Cref{alg:sample}, and a proof of its correctness (via \Cref{lem:sample}) is provided in \Cref{sec:proofs-approx-alg}.
|
||||
|
||||
\begin{algorithm}[t]
|
||||
\caption{\sampmon(\etree)}
|
||||
\caption{\sampmon(\circuit)}
|
||||
\label{alg:sample}
|
||||
\begin{algorithmic}[1]
|
||||
\Require \etree: Binary Expression Tree
|
||||
\Require \circuit: Binary Expression Tree
|
||||
\Ensure \vari{vars}: TreeSet
|
||||
\Ensure \vari{sgn} $\in \{-1, 1\}$
|
||||
\Comment{\Cref{alg:one-pass} should have been run before this one} % algorithm ~\ref{alg:sample}}
|
||||
\State $\vari{vars} \gets \emptyset$ \label{alg:sample-global1}
|
||||
\If{$\etree.\type = +$}\Comment{Sample at every $+$ node}
|
||||
\State $\etree_{\vari{samp}} \gets$ Sample from left subtree ($\etree_{\lchild}$) and right subtree ($\etree_{\rchild}$) w.p. $\etree_\lchild.\wght$ and $\etree_\rchild.\wght$. \label{alg:sample-plus-bsamp}
|
||||
\State $(\vari{v}, \vari{s}) \gets \sampmon(\etree_{\vari{samp}})$\label{alg:sample-plus-traversal}
|
||||
\If{$\circuit.\type = +$}\Comment{Sample at every $+$ node}
|
||||
\State $\circuit_{\vari{samp}} \gets$ Sample from left subtree ($\circuit_{\lchild}$) and right subtree ($\circuit_{\rchild}$) w.p. $\circuit_\lchild.\wght$ and $\circuit_\rchild.\wght$. \label{alg:sample-plus-bsamp}
|
||||
\State $(\vari{v}, \vari{s}) \gets \sampmon(\circuit_{\vari{samp}})$\label{alg:sample-plus-traversal}
|
||||
% \State $\vari{vars} \gets \vari{vars} \;\cup \;\{\vari{v}\}$\label{alg:sample-plus-union}
|
||||
% \State $\vari{sgn} \gets \vari{sgn} \times \vari{s}$\label{alg:sample-plus-product}
|
||||
\State $\Return ~(\vari{v}, \vari{s})$
|
||||
\ElsIf{$\etree.\type = \times$}\Comment{Multiply the sampled values of all subtree children}
|
||||
\ElsIf{$\circuit.\type = \times$}\Comment{Multiply the sampled values of all subtree children}
|
||||
\State $\vari{sgn} \gets 1$\label{alg:sample-global2}
|
||||
\For {$child$ in $\etree.\vari{children}$}
|
||||
\For {$child$ in $\circuit.\vari{children}$}
|
||||
\State $(\vari{v}, \vari{s}) \gets \sampmon(child)$
|
||||
\State $\vari{vars} \gets \vari{vars} \cup \{\vari{v}\}$\label{alg:sample-times-union}
|
||||
\State $\vari{sgn} \gets \vari{sgn} \times \vari{s}$\label{alg:sample-times-product}
|
||||
\EndFor
|
||||
\State $\Return ~(\vari{vars}, \vari{sgn})$
|
||||
\ElsIf{$\etree.\type = numeric$}\Comment{The leaf is a coefficient}
|
||||
%\State $\vari{sgn} \gets \vari{sgn} \times sign(\etree.\val)$
|
||||
\State $\Return ~\left(\{\}, sign(\etree.\val)\right)$\label{alg:sample-num-return}
|
||||
\ElsIf{$\etree.\type = var$}
|
||||
%\State $\vari{vars} \gets \vari{vars} \; \cup \; \{\;\etree.\val\;\}\label{alg:sample-var-union}$\Comment{Add the variable to the set}
|
||||
\State $\Return~\left(\{\etree.\val\}, 1\right) $\label{alg:sample-var-return}
|
||||
\ElsIf{$\circuit.\type = numeric$}\Comment{The leaf is a coefficient}
|
||||
%\State $\vari{sgn} \gets \vari{sgn} \times sign(\circuit.\val)$
|
||||
\State $\Return ~\left(\{\}, sign(\circuit.\val)\right)$\label{alg:sample-num-return}
|
||||
\ElsIf{$\circuit.\type = var$}
|
||||
%\State $\vari{vars} \gets \vari{vars} \; \cup \; \{\;\circuit.\val\;\}\label{alg:sample-var-union}$\Comment{Add the variable to the set}
|
||||
\State $\Return~\left(\{\circuit.\val\}, 1\right) $\label{alg:sample-var-return}
|
||||
\EndIf
|
||||
\end{algorithmic}
|
||||
\end{algorithm}
|
||||
|
|
|
|||
|
|
@ -443,39 +443,39 @@ Applying this bound in the runtime bound in~\Cref{lem:approx-alg} gives the firs
|
|||
|
||||
\subsection{$\onepass$ Pseudocode}
|
||||
\begin{algorithm}[h!]
|
||||
\caption{\onepass$(\etree)$}
|
||||
\caption{\onepass$(\revision{\circuit})$}
|
||||
\label{alg:one-pass}
|
||||
\begin{algorithmic}[1]
|
||||
\Require \etree: Binary Expression Tree
|
||||
\Ensure \etree: Binary Expression Tree
|
||||
\Ensure \vari{sum} $\in \mathbb{R}$
|
||||
\If{$\etree.\type = +$}\label{alg:one-pass-equality1}
|
||||
\Require \revision{\circuit}: \revision{Circuit}
|
||||
\Ensure \revision{\circuit}: \revision{Circuit}
|
||||
\Ensure \vari{sum} $\in \reals$
|
||||
\If{$\revision{\circuit}.\type = \revision{\andgate}$}\label{alg:one-pass-equality1}
|
||||
\State $\accum \gets 0$\label{alg:one-pass-plus-assign1}
|
||||
\For{$child$ in $\etree.\vari{children}$}\Comment{Sum up all children coefficients}
|
||||
\For{$child$ in $\revision{\circuit}.\vari{children}$}\Comment{Sum up all children coefficients}
|
||||
\State $(child, \vari{s}) \gets \onepass(child)$
|
||||
\State $\accum \gets \accum + \vari{s}$\label{alg:one-pass-plus-add}
|
||||
\EndFor
|
||||
\State $\etree.\vari{partial} \gets \accum$\label{alg:one-pass-plus-assign2}
|
||||
\For{$child$ in $\etree.\vari{children}$}\Comment{Record distributions for each child}
|
||||
\For{$child$ in $\revision{\circuit}.\vari{children}$}\Comment{Record distributions for each child}
|
||||
\State $child.\vari{weight} \gets \frac{child.\vari{partial}}{\etree.\vari{partial}}$\label{alg:one-pass-plus-prob}
|
||||
\EndFor
|
||||
%\State $\vari{sum} \gets \etree.\vari{partial}$\label{alg:one-pass-plus-assign3}
|
||||
\State \Return (\etree, \etree.\vari{partial})
|
||||
\ElsIf{$\etree.\type = \times$}\label{alg:one-pass-equality2}
|
||||
\State \Return (\revision{\circuit, \circuit}.\vari{partial})
|
||||
\ElsIf{$\revision{\circuit}.\type = \revision{\orgate}$}\label{alg:one-pass-equality2}
|
||||
\State $\accum \gets 1$\label{alg:one-pass-times-assign1}
|
||||
\For{$child \text{ in } \etree.\vari{children}$}\Comment{Compute the product of all children coefficients}
|
||||
\For{$child \text{ in } \revision{\circuit}.\vari{children}$}\Comment{Compute the product of all children coefficients}
|
||||
\State $(child, \vari{s}) \gets \onepass(child)$
|
||||
\State $\accum \gets \accum \times \vari{s}$\label{alg:one-pass-times-product}
|
||||
\EndFor
|
||||
\State $\etree.\vari{partial}\gets \accum$\label{alg:one-pass-times-assign2}
|
||||
\State $\revision{\circuit}.\vari{partial}\gets \accum$\label{alg:one-pass-times-assign2}
|
||||
%\State $\vari{sum} \gets \etree.\vari{partial}$\label{alg:one-pass-times-assign3}
|
||||
\State \Return (\etree, \etree.\vari{partial})
|
||||
\ElsIf{$\etree.\type = numeric$}\Comment{Base case}\label{alg:one-pass-equality3}
|
||||
\State $\vari{sum} \gets |\etree.\val|$\label{alg:one-pass-leaf-assign1}\Comment{This step effectively converts $\etree$ into $\abs{\etree}$}
|
||||
\State \Return (\etree, \vari{sum})
|
||||
\Else\Comment{$\etree.\type = \var$}\label{alg:one-pass-equality4}
|
||||
\State \Return (\revision{\circuit, \circuit}.\vari{partial})
|
||||
\ElsIf{$\revision{\circuit}.\type = numeric$}\Comment{Base case}\label{alg:one-pass-equality3}
|
||||
\State $\vari{sum} \gets |\revision{\circuit}.\val|$\label{alg:one-pass-leaf-assign1}\Comment{This step effectively converts $\etree$ into $\abs{\revision{\circuit}}$}
|
||||
\State \Return (\revision{\circuit}, \vari{sum})
|
||||
\Else\Comment{$\revision{\circuit}.\type = \var$}\label{alg:one-pass-equality4}
|
||||
%\State $\vari{sum} \gets 1$\label{alg:one-pass-global-assign}
|
||||
\State \Return (\etree,$1$) % \vari{sum})
|
||||
\State \Return (\revision{\circuit},$1$) % \vari{sum})
|
||||
\EndIf
|
||||
\end{algorithmic}
|
||||
\end{algorithm}
|
||||
|
|
@ -546,17 +546,17 @@ level 2/.style={sibling distance=0.7cm},
|
|||
|
||||
\subsection{Proof of~\Cref{lem:one-pass}}
|
||||
|
||||
We prove the first part of lemma ~\ref{lem:one-pass}, i.e., correctness, by structural induction over the depth $d$ of the binary tree $\etree$.
|
||||
We prove the first part of lemma ~\ref{lem:one-pass}, i.e., correctness, by structural induction over the depth $d$ of the \revision{circuit \circuit}.
|
||||
|
||||
For the base case, $d = 0$, it is the case that the node is a leaf and therefore by definition ~\ref{def:express-tree} must be a variable or coefficient. When it is a variable, \textsc{OnePass} returns $1$, and we have in this case that $\polyf(\etree) = X_i = \polyf(\abs{\etree})$ for some $i$ in $[\numvar]$, and this evaluated at all $1$'s indeed gives $1$, verifying the correctness of the returned value of $\abs{\etree}(1,\ldots, 1) = 1$. When the root is a coefficient, the absolute value of the coefficient is returned, which is indeed $\abs{\etree}(1,\ldots, 1)$. This proves the base case.
|
||||
For the base case, $d = 0$, it is the case that the node is a leaf and therefore by definition ~\ref{def:express-tree} must be a variable or coefficient. When it is a variable, \textsc{OnePass} returns $1$, and we have in this case that $\polyf(\revision{\circuit}) = X_i = \polyf(\abs{\revision{\circuit}})$ for some $i$ in $[\numvar]$, and this evaluated at all $1$'s indeed gives $1$, verifying the correctness of the returned value of $\abs{\revision{\circuit}}(1,\ldots, 1) = 1$. When the root is a coefficient, the absolute value of the coefficient is returned, which is indeed $\abs{\revision{\circuit}}(1,\ldots, 1)$. This proves the base case.
|
||||
%\AH{The inductive step assumes $k \geq 0$ rather than $k \geq 1$, correct?}
|
||||
%\AR{yep!}
|
||||
For the inductive hypothesis, assume that for $d \leq k$ for some $k \geq 0$,~\Cref{lem:one-pass} is true for~\Cref{alg:one-pass}.
|
||||
|
||||
Now prove that lemma ~\ref{lem:one-pass} holds for $k + 1$. Notice that $\etree$ has at most two children, $\etree_\lchild$ and $\etree_\rchild$. Note also, that for each child, it is the case that $d \leq k$. Then, by inductive hypothesis, lemma ~\ref{lem:one-pass} holds for each existing child, and we are left with two possibilities for $\etree$. The first case is when $\etree$ is a $+$ node. When this happens,~\Cref{alg:one-pass} computes $|T_\lchild|(1,\ldots, 1) + |T_\rchild|(1,\ldots, 1)$ on line ~\ref{alg:one-pass-plus-add} which by definition is $\abs{\etree}(1,\ldots, 1)$ and hence the inductive hypothesis holds in this case. For the weight computation of the children of $+$, by lines ~\ref{alg:one-pass-plus-add}, ~\ref{alg:one-pass-plus-assign2}, and ~\ref{alg:one-pass-plus-prob} algorithm ~\ref{alg:one-pass} computes $\etree_i.\wght = \frac{|T_i|(1,\ldots, 1)}{|T_\lchild|(1,\ldots, 1) + |T_\rchild|(1,\ldots, 1)}$ which is indeed as claimed. The second case is when the $\etree.\val = \times$. By inductive hypothesis, it is the case that both $\abs{\etree_\lchild}\polyinput{1}{1}$ and $\abs{\etree_\rchild}\polyinput{1}{1}$ have been correctly computed. On line~\ref{alg:one-pass-times-product} algorithm ~\ref{alg:one-pass} then computes the product of the subtree partial values, $|T_\lchild|(1,\ldots, 1) \cdot |T_\rchild|(1,\ldots, 1)$ which by definition is $\abs{\etree}(1,\ldots, 1)$.
|
||||
Now prove that lemma ~\ref{lem:one-pass} holds for $k + 1$. Notice that $\revision{\circuit}$ has at most two children, $\revision{\circuit}_\lchild$ and $\revision{\circuit}_\rchild$. Note also, that for each child, it is the case that $d \leq k$. Then, by inductive hypothesis, lemma ~\ref{lem:one-pass} holds for each existing child, and we are left with two possibilities for $\revision{\circuit}$. The first case is when $\revision{\circuit}$ is an $\orgate$ node. When this happens,~\Cref{alg:one-pass} computes $|\revision{\circuit}_\lchild|(1,\ldots, 1) + |\revision{\circuit}_\rchild|(1,\ldots, 1)$ on line ~\ref{alg:one-pass-plus-add} which by definition is $\abs{\revision{\circuit}}(1,\ldots, 1)$ and hence the inductive hypothesis holds in this case. For the weight computation of the children of $\orgate$, by lines ~\ref{alg:one-pass-plus-add}, ~\ref{alg:one-pass-plus-assign2}, and ~\ref{alg:one-pass-plus-prob} algorithm ~\ref{alg:one-pass} computes $\revision{\circuit}_i.\wght = \frac{|\revision{\circuit}_i|(1,\ldots, 1)}{|\revision{\circuit}_\lchild|(1,\ldots, 1) + |\revision{\circuit}_\rchild|(1,\ldots, 1)}$ which is indeed as claimed. The second case is when the $\revision{\circuit}.\val = \times$. By inductive hypothesis, it is the case that both $\abs{\etree_\lchild}\polyinput{1}{1}$ and $\abs{\etree_\rchild}\polyinput{1}{1}$ have been correctly computed. On line~\ref{alg:one-pass-times-product} algorithm ~\ref{alg:one-pass} then computes the product of the children's partial values, $|\revision{\circuit}_\lchild|(1,\ldots, 1) \cdot |\revision{\circuit}_\rchild|(1,\ldots, 1)$ which by definition is $\abs{\revision{\circuit}}(1,\ldots, 1)$.
|
||||
|
||||
\paragraph{Run-time Analysis}
|
||||
The runtime for \textsc{OnePass} is fairly straight forward. Note first that each node is visited at most one time. Second, for each type of node visited, it can be trivially verified that there are only a constant number of operations. This concludes then with a $O\left(\treesize(\etree)\right)$ runtime.
|
||||
The runtime for \textsc{OnePass} is fairly straight forward. Note first that each node is visited at most one time. Second, for each type of node visited, it can be trivially verified that there are only a constant number of operations. This concludes then with a $O\left(\treesize(\revision{\circuit})\right)$ runtime.
|
||||
|
||||
|
||||
\subsection{Proof of~\Cref{lem:sample}}
|
||||
|
|
|
|||
10
macros.tex
10
macros.tex
|
|
@ -98,7 +98,9 @@
|
|||
\newcommand{\smb}{\poly\left(\vct{X}\right)}%smb for standard monomial basis
|
||||
\newcommand{\smbOf}[1]{\textsc{SMB}(#1)}
|
||||
\newcommand{\etreeset}[1]{\vari{ET}\left(#1\right)}
|
||||
%\expandtree is a placeholder until I change other files with the new macro name \expansion
|
||||
\newcommand{\expandtree}[1]{\vari{E}(#1)}
|
||||
\newcommand{\expansion}[1]{\vari{E}(#1)}
|
||||
\newcommand{\elist}[1]{\vari{List}\pbox{#1}}
|
||||
|
||||
%expandtree tuple elements:
|
||||
|
|
@ -271,15 +273,17 @@
|
|||
%God Loves You Aaron Huber! <3
|
||||
%Revision Parameters
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\revision}[1]{\color{blue}{#1}\color{black}}
|
||||
\newcommand{\oldstuff}[1]{\color{gray}{#1}\color{black}}
|
||||
\newcommand{\revision}[1]{\color{blue}{#1}\color{black}\xspace}
|
||||
\newcommand{\oldstuff}[1]{\color{gray}{#1}\color{black}\xspace}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%God Loves You Aaron Huber! <3
|
||||
%Circuit Notation
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\circuit}{\vari{C}}
|
||||
\newcommand{\circuit}{\vari{C}\xspace}
|
||||
\newcommand{\circuitset}[1]{\vari{CSet}(#1)}
|
||||
\newcommand{\andgate}{AND}
|
||||
\newcommand{\orgate}{OR}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Sets
|
||||
|
|
|
|||
Loading…
Reference in New Issue