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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.  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.  Upon initialization each row of $\sketch$ is an estimation of the $\kDom$ frequency for a given tuple represented by $\sketch$ across all possible worlds. 

To facilitate binning the $\kDom$ values for a given world $\wVec$, each of the $\sketchRows$ rows has two pairwise independent hash functions $\hash[i]:\pw \to [B]$ and $\pol[i]:\pw \to \{-1,1\}$, where all functions are independent of one another.  Finally, $\genV \in \pwK$ is simply a vector whose values are from the set $\kDom$, each of which denote the annotation of the tuple $t$ in its corresponding world.%defined as $\kMap{t} : \{0, 1\}^\numTup \rightarrow \kDom$ is used to determine the tuple's $\kDom$ annotation for a given world.

When a world $\wVec$'s $\kDom$ value is updated, it's $\kDom$ value is first retrieved via $\kMap{t}$ and then multiplied by the output of the $i^{th}$ row's polarity function $\pol$.  The resulting computation is then added to the current value contained in the bin mapping.  Formally:
$$\sketchJParam{\hashP{\wVec}} ~+=~ \polP{\wVec} \times \kMapParam{\wVec}$$

After initialization is complete we have that 
$$\sketchIj = \sum_{\{\wVec \st \hashP{\wVec} = j\}}\genVParam{\wVec} \polP{\wVec}.$$

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 $\wVec \in \{0, 1\}^\numTup$. 

