Small changes to poly, smb, degree, etc. defs

approx_alg.tex

@@ -38,7 +38,7 @@ For any circuit $\circuit$, the corresponding
 \end{Definition}
 Please see \Cref{ex:def-pos-circ} for an illustration.
 
-\begin{Definition}[\size($\cdot$)]
+\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.
 \end{Definition}
 

poly-form.tex

@@ -9,41 +9,26 @@ and develop a reduced form (a closed form of the polynomial's expectation) for p
 Note that a polynomial over $\vct{X}=(X_1,\dots,X_n)$ is formally defined as:
 \begin{equation}
   \label{eq:sop-form}
-Q(X_1,\dots,X_n)=\sum_{\vct{i}=(i_1,\dots,i_n)\in \semN^n} c_{\vct{i}}\cdot \prod_{j=1}^n X_j^{i_j}.
+Q(X_1,\dots,X_n)=\sum_{\vct{d}=(d_1,\dots,d_n)\in \semN^n} c_{\vct{d}}\cdot \prod_{i=1}^n X_i^{d_i}.
 \end{equation}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Definition}[Standard Monomial Basis]\label{def:smb}
-%A monomial is a product of variable terms, each raised to a non-negative integer power.
-%  A polynomial in \termSMB (\abbrSMB) has the form: $\sum_{i=1}^n c_i \cdot m_i$ for each of its $n$ terms, where each $c_i \neq 0$ is an integer and each $m_i$ is a monomial and $m_i \neq m_j$ for $i \neq j$. We use $\smbOf{\poly}$ to denote the \abbrSMB of $\poly$.
-The term $\prod_{j=1}^n X_j^{i_j}$ is a {\em monomial}. A polynomial $Q(\vct{X})$ is in standard monomial basis (SMB) if % in the above sum
- terms with $c_{\vct{i}}\ne 0$ are removed from \Cref{eq:sop-form}.
+From above, the term $\prod_{i=1}^n X_i^{d_i}$ is a {\em monomial}. A polynomial $Q(\vct{X})$ is in standard monomial basis (SMB) when we keep only the terms with $c_{\vct{i}}\ne 0$ from \Cref{eq:sop-form}.
 \end{Definition}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-We consider \abbrSMB as the default representation of a polynomial. % When we want to stress the use of the SMB representation,
+We consider \abbrSMB as the default representation of a polynomial. 
 We use $\smbOf{\poly}$ to denote the SMB form of a polynomial $\poly$.
 
-%The \abbrSMB for the running example is $X^2 +2XY + Y^2$.  Note that the example's SOP expansion $X^2 + XY + XY + Y^2$ is is not $\smbOf{(X+Y)^2}$ since $XY$ appears twice.
-
-% \BG{Maybe inline degree?}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Definition}[Degree]\label{def:degree}
-The degree of polynomial $\poly(\vct{X})$ is the largest $\sum_{j=1}^n i_j$ such that $c_{(i_1,\dots,i_n)}\ne 0$. % maximum sum of exponents, over all monomials in $\smbOf{\poly(\vct{X})}$.
+The degree of polynomial $\poly(\vct{X})$ is the largest $\sum_{i=1}^n d_i$ such that $c_{(d_1,\dots,d_n)}\ne 0$. % maximum sum of exponents, over all monomials in $\smbOf{\poly(\vct{X})}$.
 \end{Definition}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 The degree of the polynomial $X^2+2XY+Y^2$ is $2$.
 Product terms in lineage arise only from join operations (\Cref{fig:nxDBSemantics}), so intuitively, the degree of a lineage polynomial is analogous to the largest number of joins in any clause of the UCQ query that created it.
 In this paper we consider only finite degree polynomials.
-%
-% Throughout this paper, we also make the following \textit{assumption}.
-%
-% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% \begin{Assumption}\label{assump:poly-smb}
-% All polynomials considered are in standard monomial basis, i.e., $\poly(\vct{X}) = \sum\limits_{\vct{d} \in \mathbb{N}^\numvar}q_d \cdot \prod\limits_{i = 1, d_i \geq 1}^{\numvar}X_i^{d_i}$, where $q_d$ is the coefficient for the monomial encoded in $\vct{d}$ and $d_i$ is the $i^{th}$ element of $\vct{d}$.
-% \end{Assumption}
-% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
 We call a polynomial $\query(\vct{X})$ a \emph{\bi-lineage polynomial} (resp., \emph{\ti-lineage polynomial}, or simply lineage polynomial), if
 %\AH{Why is it required for the tuple to be n-ary?  I think this slightly confuses me since we have n tuples.}
 % OK: agreed w/ AH, this can be treated as implicit
@@ -184,7 +169,7 @@ to the variables $\vct{X}$. Intuitively, \Cref{lem:exp-poly-rpoly} states that w
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Corollary}\label{cor:expct-sop}
-If $\poly$ is a \bi-lineage polynomial, then the expectation of $\poly$, i.e., $\expct\pbox{\poly} = \rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$ can be computed in $O(\size\inparen{\smbOf{\poly}})$, where $\size\inparen{\poly}$ denotes the total number of multiplication/addition operators in $\poly$.
+If $\poly$ is a \bi-lineage polynomial, then the expectation of $\poly$, i.e., $\expct\pbox{\poly} = \rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$ can be computed in $O(\size\inparen{\smbOf{\poly}})$, where $\size\inparen{\poly}$ (\Cref{def:size}) is proportional to the total number of multiplication/addition operators in $\poly$.
 \end{Corollary}
 %\AH{What if $\poly$ is not in \abbrSMB form?}