ODIn/paper-BagRelationalPDBsAreHard
1
0
Fork 0
Code Issues Pull requests Projects Releases Wiki Activity

Implemented Atri's comments

Browse source
This commit is contained in:
Aaron Huber 2019-05-20 10:36:38 -04:00
parent c4ed3a8663
commit cadc973d8f
3 changed files with 16 additions and 3 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

2
macros.tex
View file

@ -18,7 +18,7 @@
\newcommand{\numTup}{N}
%\newcommand{\wIndicator}{v_t}
\newcommand{\wIndicator}[1]{v_{#1}}
\newcommand{\wVec}{\vec{w}}
\newcommand{\wVec}{\textbf{w}}
\newcommand{\relation}{R}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% COMMENTS

BIN
main.synctex.gz Normal file
View file

Binary file not shown.

17
notation.tex
View file

@ -2,7 +2,20 @@
\section{Notation}
\label{sec:notation}
The following notation is used to reason about the sketching of world membership for a given tuple. A given sketch $\sketch$ can be viewed as a matrix with $\sketchRows$ rows and $\sketchCols$ columns. Each row of $\sketch$ is an estimation of the of $\kDom$ frequency for the given tuple represented by $\sketch$ across all possible worlds. The $\kDom$ values for a given world $\wVec$ \AR{I do not like this notation. I prefer vectors being typeset in bold, i.e. $\mathbf{w}$. $\wVec$ is good for writing on the board but it is more standard to bold vectors in linear algebra. Also the $\kDom$ values are not binned by $\sketchHash{i}$ but the actual $\wVec$s are.} are binned by a pairwise independent hash function $\sketchHash{i}$, where for each $i, j \in \sketchRows \text{ s.t. } i \neq j, \sketchHash{i}$ is independent of $\sketchHash{j}$. Thus each row can be viewed as an independent estimation. \AR{While in general I'm a fan of using English to define things, one of the exceptions if when you are defining a function. It would be better to explicit state that $\sketchHash{i}:W\to [B]$ and $\sketchPolar{i}:W\to \{-1,1\}$. Of course for these definitions you need to define $W$ upfront.} When a $\kDom$ value is binned, it is first multiplied by the output of the $i^{th}$ row's polarity function $\sketchPolar{i}$. The resulting computation is then added to the current value contained in the bin mapping.
The following notation is used to reason about the sketching of world membership for a given tuple. We denote the set of all possible worlds as $\pw$. A given sketch $\sketch$ can be viewed as an $\sketchRows \times \sketchCols$ matrix, i.e. a matrix with $\sketchRows$ rows and $\sketchCols$ columns. Each row of $\sketch$ is an estimation of the of $\kDom$ frequency for the given tuple represented by $\sketch$ across all possible worlds. The $\kDom$ values for a given world $\wVec$
When referring to Tuple Independent Databases (TIDB), a database $\relation$ contains $\numTup$ tuples, with $\numWorlds$ possible worlds $\pw$. $\pw$ is denoted as $\{0, 1\}^\numTup$, where a specific world $\wVec$ is defined as $\wVec \in \{0, 1\}^\numTup$. An indicator function $\wIndicator{t}$ defined as $\wIndicator{t} : \{0, 1\}^\numTup \rightarrow \kDom$ is used to determine the tuple's $\kDom$ annotation for a given world. \AR{I'm fine $\wIndicator{t}$ defined as a function instead of a vector in $\kDom^W$ but I'm not sure if one would be easier than the other to write arguments. I guess we can re-consider this later as it is defined as a macro.}
\AR{I do not like this notation. I prefer vectors being typeset in bold, i.e. $\mathbf{w}$. $\wVec$ is good for writing on the board but it is more standard to bold vectors in linear algebra. Also the $\kDom$ values are not binned by $\sketchHash{i}$ but the actual $\wVec$s are.}
\AH{Done. The only thing I was unsure about is the binning process.}
are binned by a pairwise independent hash function $\sketchHash{i}:\pw \to [B]$, where for each $i, j \in \sketchRows \text{ s.t. } i \neq j, \sketchHash{i}$ is independent of $\sketchHash{j}$. Thus each row can be viewed as an independent estimation.
\AR{While in general I'm a fan of using English to define things, one of the exceptions if when you are defining a function. It would be better to explicit state that $\sketchHash{i}:W\to [B]$ and $\sketchPolar{i}:W\to \{-1,1\}$. Of course for these definitions you need to define $W$ upfront.}
\AH{Done}
When a $\kDom$ value is binned, it is first multiplied by the output of the $i^{th}$ row's polarity function $\sketchPolar{i}:\pw \to \{-1,1\}$. The resulting computation is then added to the current value contained in the bin mapping.
When referring to Tuple Independent Databases (TIDB), a database $\relation$ contains $\numTup$ tuples, with $\numWorlds$ possible worlds $\pw$. $\pw$ is denoted as $\{0, 1\}^\numTup$, where a specific world $\wVec$ is defined as $\wVec \in \{0, 1\}^\numTup$. An indicator function $\wIndicator{t}$ defined as $\wIndicator{t} : \{0, 1\}^\numTup \rightarrow \kDom$ is used to determine the tuple's $\kDom$ annotation for a given world.
\AR{I'm fine $\wIndicator{t}$ defined as a function instead of a vector in $\kDom^W$ but I'm not sure if one would be easier than the other to write arguments. I guess we can re-consider this later as it is defined as a macro.}
\AH{I too am unsure of which way would be best to go on this. I think originally we had proposed to define $\wVec$ as a mapping to the tuple's $\kDom$ annotation.}