Merge branch 'master' of https://git.overleaf.com/61d6263016ff472ac9308dea

approx_alg.tex

@@ -101,7 +101,7 @@ and computing the average of $\vari{Y}$ gives us our final estimate. \onepass is
 \mypar{Runtime analysis} We can argue the following runtime for the algorithm outlined above (which solves \Cref{prob:intro-stmt}):
 \begin{Theorem}
 \label{cor:approx-algo-const-p}
-Let \circuit be an arbitrary \emph{\abbrOneBIDB} circuit, define $\poly(\vct{X})=\polyf(\circuit)$, let $k=\degree(\circuit)$, 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 $\rpoly(\prob_1,\ldots, \prob_\numvar)$
+Let \circuit be an arbitrary \emph{\abbrOneBIDB} circuit, define $\poly(\vct{X})=\polyf(\circuit)$, let $k=\degree(\circuit)$, and let $\gamma=\gamma(\circuit)$. Further let it be the case that $\prob_i\ge \prob_0$ for all $i\in[\numvar]$. Then for all $\epsilon'>0$, an estimate $\mathcal{E}$  of $\rpoly(\prob_1,\ldots, \prob_\numvar)$
 satisfying
 \begin{equation}
 \label{eq:approx-algo-bound-main}

binarybidb.tex

@@ -26,7 +26,7 @@ We call a polynomial $\poly\inparen{\vct{X}}$ a \emph{\abbrCTIDB-lineage polynom
 \subsection{\abbrOneBIDB}\label{subsec:one-bidb}
 \label{subsec:tidbs-and-bidbs}
 
-\noindent A block independent database \abbrBIDB $\pdb'$ models a set of worlds each of which consists of a subset of the possible tuples $\tupset'$, where $\tupset'$ is partitioned into $\numblock$ blocks $\block_i$ and all $\tup\in\block_i$ are independent random events for distinct blocks.  $\pdb'$ further constrains that all $\tup\in\block_i$ for all $i\in\pbox{\numblock}$ of $\tupset'$ be disjoint events.  We refer to any monomial that includes $X_\tup X_{\tup'}$ for $\tup\neq\tup'\in\block_i$ as a \emph{cancellation}.  We define next a specific construction of \abbrBIDB that is useful for our work.
+\noindent A block independent database \abbrBIDB $\pdb'$ models a set of worlds each of which consists of a subset of the possible tuples $\tupset'$, where $\tupset'$ is partitioned into $\numblock$ blocks $\block_i$ and the events $\tup\in\block_i$ and $\tup\in\block_j$ are independent  for $i\ne j$.  $\pdb'$ further constrains that all $\tup\in\block_i$ for all $i\in\pbox{\numblock}$ of $\tupset'$ be disjoint events.  We refer to any monomial that includes $X_\tup X_{\tup'}$ for $\tup\neq\tup'\in\block_i$ as a \emph{cancellation}.  We define next a specific construction of \abbrBIDB that is useful for our work.
 
 \begin{Definition}[\abbrOneBIDB]\label{def:one-bidb}
 Define a \emph{\abbrOneBIDB} to be the pair $\pdb' = \inparen{\bigtimes_{\tup\in\tupset'}\inset{0, \bound_\tup}, \bpd'},$  where $\tupset'$ is the set of possible tuples such that each $\tup \in \tupset'$ has a multiplicity domain of $\inset{0, \bound_\tup}$, with $\bound_\tup\in\mathbb{N}$.  $\tupset'$ is partitioned into $\numblock$ independent blocks $\block_i,$ for $i\in\pbox{\numblock}$, of disjoint tuples.  $\bpd'$ is characterized by the vector $\inparen{\prob_\tup}_{\tup\in\tupset'}$ where for every block $\block_i$, $\sum_{\tup \in \block_i}\prob_\tup \leq 1$.  Given $W\in\onebidbworlds{\tupset'}$ and for $i\in\pbox{\numblock}$, let $\prob_\tup(W) = \begin{cases}