ODIn/paper-BagRelationalPDBsAreHard
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One minor fix in Approx Alg results.

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Aaron Huber 2021-02-02 11:42:24 -05:00
parent c9bb2c40cc
commit c302088621
2 changed files with 4 additions and 4 deletions
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6
approx_alg.tex
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@ -140,7 +140,7 @@ Let \revision{\circuit be a circuit} for a UCQ over \bi and define $\poly(\vct{X
%Let $\poly(\vct{X})$ be a query polynomial corresponding to the output of a UCQ in a \bi.
Then an estimate $\mathcal{E}$ %=\approxq(\circuit, P_1,\dots,p_\numvar), \conf, \error')$
of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ can be computed in time
\[O\left(\revision{\size(\circuit)^2} + \frac{\log{\frac{1}{\conf}}\cdot \abs{\circuit}^2(1,\ldots, 1)\cdot k\cdot \log{k} \cdot depth(\circuit))}{\inparen{\error'}^2\cdot\rpoly^2(\prob_1,\ldots, \prob_\numvar)}\right)\]
\[O\left(\revision{\size(\circuit)} + \frac{\log{\frac{1}{\conf}}\cdot \abs{\circuit}^2(1,\ldots, 1)\cdot k\cdot \log{k} \cdot depth(\circuit))}{\inparen{\error'}^2\cdot\rpoly^2(\prob_1,\ldots, \prob_\numvar)}\right)\]
such that
\begin{equation}
\label{eq:approx-algo-bound}
@ -164,8 +164,8 @@ Given an expression tree $\circuit$, define
\begin{Corollary}
\label{cor:approx-algo-const-p}
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
\[O\left(\revision{\size(\circuit)^2} + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot depth(\circuit))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\]
In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\revision{\size(\circuit)^2} + \frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$.
\[O\left(\revision{\size(\circuit)} + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot depth(\circuit))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\]
In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$.
\end{Corollary}
The proof for~\Cref{cor:approx-algo-const-p} can be seen in~\Cref{sec:proofs-approx-alg}.

2
hardness-app.tex
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@ -437,7 +437,7 @@ Further, since each $\prob_i\ge \prob_0$ and $\poly(\vct{X})$ (and hence $\rpoly
\[ \rpoly(1,\dots,1) \ge \prob_0^k\cdot \rpoly(1,\dots,1).\]
The above two inequalities implies $\rpoly(1,\dots,1) \ge \prob_0^k\cdot (1-\gamma)\cdot \abs{\revision{\circuit}}(1,\dots,1)$.
%\AH{This looks really nice!}
Applying this bound in the runtime bound in~\Cref{lem:approx-alg} gives the first claimed runtime. The final runtime of $O_k\left(\revision{\size(\circuit)^2 +}\frac 1{\eps^2}\cdot\size(\revision{\circuit})\cdot \log{\frac{1}{\conf}}\right)$ follows by noting that $depth(\revision{\circuit})\le \size(\revision{\circuit})$ and absorbing all factors that just depend on $k$.
Applying this bound in the runtime bound in~\Cref{lem:approx-alg} gives the first claimed runtime. The final runtime of $O_k\left(\frac 1{\eps^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$ follows by noting that $depth(\revision{\circuit})\le \size(\revision{\circuit})$ and absorbing all factors that just depend on $k$.