2006-08-11 20:32:31 +03:00
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%% Nivio: 13/jan/06, 21/jan/06 29/jan/06
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% Time-stamp: <Sunday 29 Jan 2006 11:56:25pm EST yoshi@flare>
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\vspace{-3mm}
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\section{The algorithm}
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\label{sec:new-algorithm}
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\vspace{-2mm}
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\enlargethispage{2\baselineskip}
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The main idea supporting our algorithm is the classical divide and conquer technique.
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The algorithm is a two-step external memory based algorithm
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that generates a MPHF $h$ for a set $S$ of $n$ keys.
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Figure~\ref{fig:new-algo-main-steps} illustrates the two steps of the
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algorithm: the partitioning step and the searching step.
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\vspace{-2mm}
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\begin{figure}[ht]
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\centering
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\begin{picture}(0,0)%
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2006-09-20 07:05:40 +03:00
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\includegraphics{figs/brz}%
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2006-08-11 20:32:31 +03:00
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\end{picture}%
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\setlength{\unitlength}{4144sp}%
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%
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\begingroup\makeatletter\ifx\SetFigFont\undefined%
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\gdef\SetFigFont#1#2#3#4#5{%
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\reset@font\fontsize{#1}{#2pt}%
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\fontfamily{#3}\fontseries{#4}\fontshape{#5}%
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\selectfont}%
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\fi\endgroup%
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\begin{picture}(3704,2091)(1426,-5161)
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\put(2570,-4301){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}0}}}}
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\put(2782,-4301){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}1}}}}
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\put(2996,-4301){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}2}}}}
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\put(4060,-4006){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}Buckets}}}}
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\put(3776,-4301){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}${\lceil n/b\rceil - 1}$}}}}
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\put(4563,-3329){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}Key Set $S$}}}}
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\put(2009,-3160){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}0}}}}
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\put(2221,-3160){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}1}}}}
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\put(4315,-3160){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}n-1}}}}
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\put(1992,-5146){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}0}}}}
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\put(2204,-5146){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}1}}}}
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\put(4298,-5146){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}n-1}}}}
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\put(4546,-4977){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}Hash Table}}}}
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\put(1441,-3616){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}Partitioning}}}}
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\put(1441,-4426){\makebox(0,0)[lb]{\smash{{\SetFigFont{7}{8.4}{\familydefault}{\mddefault}{\updefault}Searching}}}}
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\put(1981,-4786){\makebox(0,0)[lb]{\smash{{\SetFigFont{5}{6.0}{\familydefault}{\mddefault}{\updefault}MPHF$_0$}}}}
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\put(2521,-4786){\makebox(0,0)[lb]{\smash{{\SetFigFont{5}{6.0}{\familydefault}{\mddefault}{\updefault}MPHF$_1$}}}}
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\put(3016,-4786){\makebox(0,0)[lb]{\smash{{\SetFigFont{5}{6.0}{\familydefault}{\mddefault}{\updefault}MPHF$_2$}}}}
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\put(3826,-4786){\makebox(0,0)[lb]{\smash{{\SetFigFont{5}{6.0}{\familydefault}{\mddefault}{\updefault}MPHF$_{\lceil n/b \rceil - 1}$}}}}
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\end{picture}%
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\vspace{-1mm}
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\caption{Main steps of our algorithm}
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\label{fig:new-algo-main-steps}
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\vspace{-3mm}
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\end{figure}
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The partitioning step takes a key set $S$ and uses a universal hash function
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$h_0$ proposed by Jenkins~\cite{j97}
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%for each key $k \in S$ of length $|k|$
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to transform each key~$k\in S$ into an integer~$h_0(k)$.
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Reducing~$h_0(k)$ modulo~$\lceil n/b\rceil$, we partition~$S$ into $\lceil n/b
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\rceil$ buckets containing at most 256 keys in each bucket (with high
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probability).
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The searching step generates a MPHF$_i$ for each bucket $i$,
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$0 \leq i < \lceil n/b \rceil$.
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The resulting MPHF $h(k)$, $k \in S$, is given by
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\begin{eqnarray}\label{eq:mphf}
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h(k) = \mathrm{MPHF}_i (k) + \mathit{offset}[i],
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\end{eqnarray}
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where~$i=h_0(k)\bmod\lceil n/b\rceil$.
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The $i$th entry~$\mathit{offset}[i]$ of the displacement vector
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$\mathit{offset}$, $0 \leq i < \lceil n/b \rceil$, contains the total number
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of keys in the buckets from 0 to $i-1$, that is, it gives the interval of the
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keys in the hash table addressed by the MPHF$_i$. In the following we explain
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each step in detail.
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