Changes to proof of Lemma 4.8 per @atri 030722 suggestions.

approx_alg.tex

@@ -139,20 +139,24 @@ In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the abo
 
 The restriction on $\gamma$ is satisfied by any
 $1$-\abbrTIDB (where $\gamma=0$ in the equivalent $1$-\abbrBIDB of~\Cref{prop:ctidb-reduct})
-as well as for all three queries of the PDBench \abbrBIDB benchmark (see \Cref{app:subsec:experiment} for experimental results). Further, we can alo argue the following result:
+as well as for all three queries of the PDBench \abbrBIDB benchmark (see \Cref{app:subsec:experiment} for experimental results). Further, we can also argue the following result
+\secrev{
+, recalling from~\Cref{sec:intro} for \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, where $\tupset$ is the set of possible tuples across all possible worlds of $\pdb$.
+} 
 \secrev{
 \begin{Lemma}
 \label{lem:ctidb-gamma}
-Given $\raPlus$ query $\query$ and \abbrCTIDB $\pdb$, let \circuit be the circuit computed by $\query\inparen{\tupset}$.  Then, for the reduced \abbrOneBIDB $\pdb'$ there exists an equivalent circuit \circuit' obtained from $\query\inparen{\tupset'}$, such that $\gamma\inparen{\circuit'}\leq 1 - \inparen{\bound + 1}^{-\inparen{k-1}}$ with $\size\inparen{\circuit'} \leq \size\inparen{\circuit} + \numvar\cdot\inparen{2^{\inparen{\ceil{\log{2\bound}}}+ 1} - 1}$ and $\depth\inparen{\circuit'} = \depth\inparen{\circuit} + \ceil{\log{2\bound}}$.
+Given $\raPlus$ query $\query$ and \abbrCTIDB $\pdb$, let \circuit be the circuit computed by $\query\inparen{\tupset}$.  Then, for the reduced \abbrOneBIDB $\pdb'$ there exists an equivalent circuit \circuit' obtained from $\query\inparen{\tupset'}$, such that $\gamma\inparen{\circuit'}\leq 1 - \inparen{\bound + 1}^{-\inparen{k-1}}$ with $\size\inparen{\circuit'} \leq \size\inparen{\circuit} + \bigO{\numvar\bound}$ %\cdot\inparen{2^{\inparen{\ceil{\log{2\bound}}}+ 1} - 1}$
+ and $\depth\inparen{\circuit'} = \depth\inparen{\circuit} + \bigO{\log{\bound}}$.%\ceil{\log{2\bound}}$.
 \end{Lemma}
 }
 
 \secrev{
 \begin{proof}[Proof of~\Cref{lem:ctidb-gamma}]
 %Let $\pdb' = \inparen{\onebidbworlds{\tupset'}, \pdb'}$ be the reduced \abbrOneBIDB and $\pdb = \inparen{\worlds, \pdb}$ the original \abbrCTIDB.
-The circuit \circuit' is built from \circuit in the following manner.  For each input gate $\gate_i$ with $\gate_i.\val = X_\tup$, replace $\gate_i$ with the circuit \subcircuit encoding the sum $\sum_{j = 1}^\bound j\cdot X_{\tup, j}$.  We argue that \circuit' is a valid circuit by the following facts.  Let $\pdb = \inparen{\worlds, \pdb}$ be the original \abbrCTIDB \circuit was generated from.  Then, by~\Cref{prop:ctidb-reduct} there exists a reduced $\pdb' = \inparen{\onebidbworlds{\tupset'}, \pdb'}$, from which the conversion from \circuit to \circuit' follows.  Both $\polyf\inparen{\circuit}$ and $\polyf\inparen{\circuit'}$ have the same expected multiplicity since (by~\Cref{prop:ctidb-reduct}) the distributions $\bpd$ and $\bpd'$ are equivalent and each $j\cdot\worldvec'_{\tup, j} = \worldvec_\tup$ for $\worldvec'\in\inset{0, 1}^{\bound\numvar}$ and $\worldvec\in\worlds$.  Finally, note that because there exists $\subcircuit'\in\circuitset{\polyf\inparen{\circuit}}$ encoding $\sum_{j = 1}^\bound j\cdot X_{\tup, j}$ that is a \emph{balanced} binary tree, the above conversion implies the claimed size and depth bounds of the lemma.
+The circuit \circuit' is built from \circuit in the following manner.  For each input gate $\gate_i$ with $\gate_i.\val = X_\tup$, replace $\gate_i$ with the circuit \subcircuit encoding the sum $\sum_{j = 1}^\bound j\cdot X_{\tup, j}$.  We argue that \circuit' is a valid circuit by the following facts.  Let $\pdb = \inparen{\worlds, \bpd}$ be the original \abbrCTIDB \circuit was generated from.  Then, by~\Cref{prop:ctidb-reduct} there exists a \abbrOneBIDB $\pdb' = \inparen{\onebidbworlds{\tupset'}, \bpd'}$, with $\tupset' = \inset{\intup{\tup, j}~|~\tup\in\tupset, j\in\pbox{\bound}}$, from which the conversion from \circuit to \circuit' follows.  Both $\polyf\inparen{\circuit}$ and $\polyf\inparen{\circuit'}$ have the same expected multiplicity since (by~\Cref{prop:ctidb-reduct}) the distributions $\bpd$ and $\bpd'$ are equivalent and each $j\cdot\worldvec'_{\tup, j} = \worldvec_\tup$ for $\worldvec'\in\inset{0, 1}^{\bound\numvar}$ and $\worldvec\in\worlds$.  Finally, note that because there exists a (sub) circuit encoding $\sum_{j = 1}^\bound j\cdot X_{\tup, j}$ that is a \emph{balanced} binary tree, the above conversion implies the claimed size and depth bounds of the lemma.
 
-Consider the list of expanded monomials $\expansion{\circuit}$ for \abbrCTIDB circuit \circuit.  Let \monom be an arbitrary monomial such that the set of variables in \monom is $\encMon = X_{\tup, 1}^{d_1},\ldots,X_{\tup, \ell}^{d_\ell}$ with the number of variables $\abs{\encMon} = \ell$.  Then \monom yields the set of monomials $\vari{E}_\monom\inparen{\circuit'}=\inparen{j_1\cdot X_{\tup, j_1}^{d_1},\ldots, j_\ell\cdot X_{\tup, j_\ell}^{d_\ell}}_{j_1,\ldots, j_\ell \in \pbox{0, \bound}}$ in $\expansion{\circuit'}$.  Observe that cancellations can only occur for each $X_{\tup}^{d_\tup}\in \encMon$.  Consider the number of cancellations for $X_{\tup}^{d_\tup}$.  Then $\gamma \leq 1 - \inparen{c + 1}^{d_\tup - 1}$, since for each element in $\inset{\bigtimes_{i\in\pbox{d_\tup}, j_i\in\pbox{0, \bound}}X_{j_i}}$ there are \emph{exactly} $\bound+1$ surviving elements with $j_1=\cdots=j_{d_\tup}$, i.e. $X_j^{d_\tup}$ for each $j\in\pbox{0, \bound}$.  The rest of the $\inparen{\bound + 1}^{d_\tup-1}$ cross terms cancel.  Regarding the whole monomial \monom it is the case that the proportion of non-cancellations across each $X_\tup^{d_\tup}\in\encMon$ multiply as non-cancelling terms for $X_\tup$ can only be joined with non-cancelling terms of $X_{\tup'}^{d_{\tup'}}$.  This then yields the inequality $1 - \prod_{i = 1}^{\ell}\inparen{c +1}^{d_i - 1}\leq \gamma \leq 1 - \inparen{c + 1}^{-\inparen{k - 1}}$ where the inequalities take into account the fact that $\sum_{i = 1}^\ell d_i \leq k$.
+Next we argue the claim on $\gamma\inparen{\circuit'}$.  Consider the list of expanded monomials $\expansion{\circuit}$ for \abbrCTIDB circuit \circuit.  Let \monom be an arbitrary monomial such that the set of variables in \monom is $\encMon = X_{\tup_1}^{d_1},\ldots,X_{\tup_\ell}^{d_\ell}$ with $\ell$ variables.  Then \monom yields the set of monomials $\vari{E}_\monom\inparen{\circuit'}=\inset{j_1^{d_1}\cdot X_{\tup, j_1}^{d_1}\times\cdots\times j_\ell^{d_\ell}\cdot X_{\tup, j_\ell}^{d_\ell}}_{j_1,\ldots, j_\ell \in \pbox{0, \bound}}$ in $\expansion{\circuit'}$.  Recall that a cancellation occurs when we have a monomial \monom such that there exists $\tup\neq\tup'$ in the same block $\block$ where variables $X_\tup, X_{\tup'}$ are in the set of variables $\encMon$ of \monom.  Observe that cancellations can only occur for each $X_{\tup}^{d_\tup}\in \encMon$, where the expansion $\inparen{\sum_{j = 1}^\bound j\cdot X_{\tup, j}}^{d_\tup}$ represents the monomial $X_\tup^{d_\tup}$ in $\tupset'$.  Consider the number of cancellations for $\inparen{\sum_{j = 1}^\bound j\cdot X_{\tup, j}^{d_\tup}}^{d_\ell}$.  Then $\gamma \leq 1 - \inparen{c + 1}^{d_\tup - 1}$, since for each element in the set of cross products$\inset{\bigtimes_{i\in\pbox{d_\tup}, j_i\in\pbox{0, \bound}}X_{\tup, j_i}}$ there are \emph{exactly} $\bound+1$ surviving elements with $j_1=\cdots=j_{d_\tup}$, i.e. $X_j^{d_\tup}$ for each $j\in\pbox{0, \bound}$.  The rest of the $\inparen{\bound + 1}^{d_\tup-1}$ cross terms cancel.  Regarding the whole monomial \monom it is the case that the proportion of non-cancellations across each $X_\tup^{d_\tup}\in\encMon$ multiply as non-cancelling terms for $X_\tup$ can only be joined with non-cancelling terms of $X_{\tup'}^{d_{\tup'}}$.  This then yields the fraction of cancelled monomials $1 - \prod_{i = 1}^{\ell}\inparen{c +1}^{d_i - 1}\leq \gamma \leq 1 - \inparen{c + 1}^{-\inparen{k - 1}}$ where the inequalities take into account the fact that $\sum_{i = 1}^\ell d_i \leq k$.
 
 Since this is true for arbitrary \monom, the bound follows for $\polyf\inparen{\circuit'}$.
 \end{proof}

macros.tex

@@ -152,7 +152,7 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %Binary-BIDB Notation																		%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\newcommand{\onebidbworlds}[1]{\bigtimes_{\tup\in #1}\inset{0,\bound_\tup}}
+\newcommand{\onebidbworlds}[1]{\bigtimes_{\tup\in #1}\inset{0, 1}}
 
 %PDB Abbreviations
 \newcommand{\abbrOneBIDB}{\text{Binary-BIDB}\xspace}