32 lines
2.3 KiB
TeX
32 lines
2.3 KiB
TeX
|
|
\section{Generalizing Beyond Set Inputs}
|
|
\label{sec:gener-results-beyond}
|
|
|
|
\subsection{\abbrTIDB{}s}
|
|
\label{sec:abbrtidbs}
|
|
|
|
For results for \abbrTIDBs, we assumed a model of \abbrTIDBs where each input tuple is assigned a probability $p$ of having multiplicity $1$. That is, we assumed inputs to be sets, but interpret queries under bag semantics. Other sensible interpretations of what the generalization of \abbrTIDBs from sets to bags should be exist.
|
|
|
|
One important such generalization is to assign each input tuple $\tup$ a multiplicity $m_\tup$ and probability $p$: the tuple has probability $p$ to exists with multiplicity $m_\tup$, and otherwise has multiplicity $0$. If the maximal multiplicity of all tuples in the \abbrTIDB is bound by some constant, then a generalization of our hardness results and approximation algorithm can be achieved by changing the construction of lineage polynomials as follows:
|
|
|
|
\begin{align*}
|
|
\polyqdt{\rel}{\dbbase}{\tup} =&\begin{cases}
|
|
m_\tup X_\tup & \text{if }\dbbase.\rel\inparen{\tup} = m_\tup \\
|
|
0 &\text{otherwise.}\end{cases}
|
|
\end{align*}
|
|
That is the variable representing a tuple is multiplied by $m_\tup$ to encode the tuple's multiplicity $m_\tup$.
|
|
|
|
Yet another option would be to assign each tuple a probability distribution over multiplicities. It seems clear that our results would not extend to a model that allows arbitrary probability distributions for this purpose. However, we would like to note that the special case of a normal distribution over multiplicities can be handled as follows: we add an additional identifier attribute to each relation in the database. For a tuple $\tup$ with maximal multiplicity $m_\tup$, we create $m_\tup$ copies of $\tup$ with different identifiers. To answer a query over this encoding, we first project away the identifier attribute.
|
|
|
|
\subsection{\abbrBIDB{}s}
|
|
\label{sec:abbrbidbs}
|
|
|
|
The approach described above works for \abbrBIDB\xplural as well if we define the bag version of \abbrBIDB{}s to associate each tuple $\tup$ a multiplicity $m_\tup$. Recall that we associate each tuple in a block with a unique variable. Thus, the modified lineage polynomial construction shown above can be applied for \abbrBIDB{}s too.
|
|
|
|
|
|
|
|
%%% Local Variables:
|
|
%%% mode: latex
|
|
%%% TeX-master: "main"
|
|
%%% End:
|