From a05758aa389651e6a899fe49958c3168529916d5 Mon Sep 17 00:00:00 2001 From: Atri Rudra Date: Wed, 15 May 2019 19:59:49 -0400 Subject: [PATCH] Started adding comments. Also added .gitignore --- .gitignore | 5 +++++ macros.tex | 1 + notation.tex | 2 +- 3 files changed, 7 insertions(+), 1 deletion(-) create mode 100644 .gitignore diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..6d22ad1 --- /dev/null +++ b/.gitignore @@ -0,0 +1,5 @@ +*.aux +*.log +*.bbl +*.pdf +*.out diff --git a/macros.tex b/macros.tex index 64465ab..db3ee32 100644 --- a/macros.tex +++ b/macros.tex @@ -27,6 +27,7 @@ \newcommand{\SF}[1]{\todo[inline]{\textbf{Su says:$\,$} #1}} \newcommand{\OK}[1]{\todo[inline]{\textbf{Oliver says:$\,$} #1}} \newcommand{\AH}[1]{\todo[inline]{\textbf{Aaron says:$\,$} #1}} +\newcommand{\AR}[1]{\todo[inline]{\textbf{Atri says:$\,$} #1}} %\newcommand{\comment}[1]{} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% diff --git a/notation.tex b/notation.tex index 6bedefe..f2a9eb3 100644 --- a/notation.tex +++ b/notation.tex @@ -2,7 +2,7 @@ \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 $\wVec$ are binned by a pairwise independent hash function $\sketchHash{i}$, where for each $i, j \in \sketchRows ~s.t.~ i \neq j, \sketchHash{i}$ is independent of $\sketchHash{j}$. Thus each row can be viewed as an independent estimation. When a $\kDom$ value is binned, it is first multiplied by the ouput of the $i^{th}$ row's polarity function $\sketchPolar$. 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. 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. When a $\kDom$ value is binned, it is first multiplied by the ouput of the $i^{th}$ row's polarity function $\sketchPolar$. 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 $\wIndicator$ defined as $\wIndicator : \{0, 1\}^\numTup \rightarrow \kDom$ is used to determine the tuple's $\kDom$ annotation for a given world.