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\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.  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.

