33 lines
1.6 KiB
TeX
33 lines
1.6 KiB
TeX
% -*- root: main.tex -*-
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\section{Hash Function Construction}
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As with world identification, bucket identification can be viewed as a binary vector. This vector is of length $\lenB = \log\sketchCols$, where $\buck \in \{0, 1\}^\lenB$. Similarly, we can define a set of hash vectors $\matrixH$ as a matrix of $\lenB$ precomputed vectors $\hVec$ where each $\hVec \in \{0, 1\}^\numTup$, formally
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\begin{equation*}
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\begin{pmatrix*}[l]
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h_{i, 0, 0}&\cdots &h_{0, \numTup} \\
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\vdots \\
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h_{i, \lenB, 0} &\cdots &h_{\lenB, \numTup}\\
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\end{pmatrix*}.
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\end{equation*}
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The row hash function $\sketchHash$ that maps input to buckets is defined as the multiplication of the matrix $\matrixH \cdot \wVec = \jVec$ , as
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\begin{equation*}
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\hVecMatrix \cdot \vecCol{w} = \vecCol{j},
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\end{equation*}
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or equivalently
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\begin{equation*}
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\sketchHash = \buck = \forall i \in [\lenB], \buck_i = \langle\textbf{h}_i, \wVec\rangle
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\end{equation*}
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Polarity function $\sketchPolar$ is analogously defined as the inner product of a precomputed vector (abusing notation) $\mathbf{\sketchPolar}$ and $\wVec$,
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\begin{equation*}
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\sketchPolar = \langle\mathbf{\sketchPolar}, \wVec\rangle
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\end{equation*}
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Finally, we augment $\matrixH$ to $\matrixH$' by adding $\mathbf{\sketchPolar}$ as an additional row in $\matrixH$
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\begin{equation*}
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\matrixH' = \begin{pmatrix*}[l]
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h_{i, 0, 0}&\cdots &h_{0, \numTup} \\
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\vdots \\
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h_{i, \lenB, 0} &\cdots &h_{\lenB, \numTup}\\
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s_{i, 0} &\cdots &s_{i, \numTup}
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\end{pmatrix*}.
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\end{equation*}
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Note that this also turns $\buck$ into a $b + 1$ size column vector, with the last element being the polarity of the hashed world vector. |