%% Nivio: 13/jan/06, 21/jan/06 29/jan/06
% Time-stamp: <Sunday 29 Jan 2006 11:56:25pm EST yoshi@flare>
\vspace{-3mm}
\section{The algorithm}
\label{sec:new-algorithm}
\vspace{-2mm}

\enlargethispage{2\baselineskip}
The main idea supporting our algorithm is the classical divide and conquer technique.
The algorithm is a two-step external memory based algorithm 
that generates a MPHF $h$ for a set $S$ of $n$ keys.
Figure~\ref{fig:new-algo-main-steps} illustrates the two steps of the
algorithm: the partitioning step and the searching step.

\vspace{-2mm}
\begin{figure}[ht]
\centering
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\vspace{-1mm}
\caption{Main steps of our algorithm}
\label{fig:new-algo-main-steps}
\vspace{-3mm}
\end{figure}

The partitioning step takes a key set $S$ and uses a universal hash function 
$h_0$ proposed by Jenkins~\cite{j97} 
%for each key $k \in S$ of length $|k|$ 
to transform each key~$k\in S$ into an integer~$h_0(k)$.
Reducing~$h_0(k)$ modulo~$\lceil n/b\rceil$, we partition~$S$ into $\lceil n/b
\rceil$ buckets containing at most 256 keys in each bucket (with high
probability).  

The searching step generates a MPHF$_i$ for each bucket $i$, 
$0 \leq i < \lceil n/b \rceil$.
The resulting MPHF $h(k)$, $k \in S$, is given by
\begin{eqnarray}\label{eq:mphf}
h(k) = \mathrm{MPHF}_i (k) + \mathit{offset}[i], 
\end{eqnarray}
where~$i=h_0(k)\bmod\lceil n/b\rceil$.
The $i$th entry~$\mathit{offset}[i]$ of the displacement vector
$\mathit{offset}$, $0 \leq i < \lceil n/b \rceil$, contains the total number
of keys in the buckets from 0 to $i-1$, that is, it gives the interval of the
keys in the hash table addressed by the MPHF$_i$.  In the following we explain
each step in detail.