From b991beb75727f9e97ac45cd3f7fac18e64ab463d Mon Sep 17 00:00:00 2001 From: Atri Rudra Date: Sat, 11 Sep 2021 16:45:18 -0400 Subject: [PATCH] finally finished the lb table --- intro-rewrite-070921.tex | 18 ++++++++++++------ macros.tex | 1 + 2 files changed, 13 insertions(+), 6 deletions(-) diff --git a/intro-rewrite-070921.tex b/intro-rewrite-070921.tex index 2502e0a..32cf072 100644 --- a/intro-rewrite-070921.tex +++ b/intro-rewrite-070921.tex @@ -149,18 +149,24 @@ Given an $\raPlus$ query $\query$ and \abbrTIDB\AR{Changed this to \abbrTIDB: we We note that the above is a special case of \Cref{prob:bag-pdb-query-eval} since we are asking whether the query evaluation over \abbrBPDB is {\em linear} in the runtime of deterministic query processing time. We stress that this question is very well motivated. In particular, we note that an answer in the affirmative for~\Cref{prob:informal} indicates that bag-probabilistic databases can be competitive with classical deterministic databases, opening the door for deployment in practice. -Unfortunately, we prove the negative. In fact in \Cref{tab:lbs} we show that depending on what hardness result/conjecture we assume, we get various emphatic versions of {\em no} as an answer to \Cref{prob:informal}. +Unfortunately, we prove the negative. In fact in Table~\ref{tab:lbs}\AR{Cref was not formatting Table correct so added Table in explicitly.} we show that depending on what hardness result/conjecture we assume, we get various emphatic versions of {\em no} as an answer to \Cref{prob:informal}. \begin{table} -\begin{tabular}{|c|c|c|} +\begin{tabular}{|p{0.4\textwidth}|p{0.1\textwidth}|p{0.5\textwidth}|} \hline -Lower bound on $\timeOf{}^*(Q,\pdb)$ & How many $\pdb$s & Hardness Assumption\\ +Lower bound on $\timeOf{}^*(Q,\pdb)$ & Num. $\pd$s & Hardness Assumption\\ \hline -$(\qruntime{Q, \dbbase})^{1+\eps_0}$ for {\em some} $\eps_0$ & Single & Triangle Detection hypothesis\\ +$\Omega\inparen{\inparen{\qruntime{Q, \dbbase}}^{1+\eps_0}}$ for {\em some} $\eps_0>0$ & Single & Triangle Detection hypothesis\\ +\hline +$\omega\inparen{\inparen{\qruntime{Q, \dbbase}}^{C_0}}$ for {\em all} $C_0>0$ & Multiple &$\sharpwzero\ne\sharpwone$\\ +\hline +$\Omega\inparen{\inparen{\qruntime{Q, \dbbase}}^{c_0\cdot k}}$ for {\em some} $c_0>0$ & Multiple & Current algorithms for counting $k$-matchings\\ \hline -$(\qruntime{Q, \dbbase})^{c_0}$ for {\em all} $\eps_0$ & Multiple & Triangle Detection hypothesis\\ \end{tabular} +\caption{Our lower bounds for a specific hard query $Q$ parameterized by $k$. The $\pdb$ is over the same $\dbbase$ and those with `Multiple' in the second column need the algorithm to be able to handle multiple $\pd$. The last column states the hardness assumptions that imply the lower bounds in the first column (all of $\eps_o,C_0,c_0$ are all constants independent of $k$).} +\label{tab:lbs} \end{table} - +Note that the lower bound in the first row by itself is enough to refute \Cref{prob:informal}. +To make some sense of the other lower bounds in Table~\ref{tab:lbs}, we note that it is not too hard to show that $\timeOf{}^*(Q,\pdb) \le O\inparen{\inparen{\qruntime{Q, \dbbase}}^k})$, where $k$ is the largest degree of the polynomial $\apolyqdt$ over all result tuple $\tup$ (which is the parameter that defines our family of hard queries). What our lower bound in the third rows says that one cannot get more than a polynomial improvement over essentially the trivial algorithm for \Cref{prob:informal}. However, this result assumes a hardness conjecture that is not as well studied as those in the first two rows of the table (see \Cref{sec:hard} for more discussion on the hardness assumptions). To put these hardness results in context, we will next take a short detour to review the existing hardness results for \abbrPDB\xplural under set semantics. % Atri: Converting sub-section to para since it saves space diff --git a/macros.tex b/macros.tex index a0a6c5f..03b6c9b 100644 --- a/macros.tex +++ b/macros.tex @@ -293,6 +293,7 @@ \newcommand{\sharpp}{\#{\sf P}\xspace} \newcommand{\sharpphard}{\#{\sf P}-hard\xspace} \newcommand{\sharpwone}{\#{\sf W}[1]\xspace} +\newcommand{\sharpwzero}{\#{\sf W}[0]\xspace} \newcommand{\sharpwonehard}{\#{\sf W}[1]-hard\xspace} \newcommand{\ptime}{{\sf PTIME}\xspace} \newcommand{\timeOf}[1]{T_{#1}}