67 lines
3.5 KiB
TeX
67 lines
3.5 KiB
TeX
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\section{Related Work}
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Czech, Havas and Majewski~\cite{chm97} provide a
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comprehensive survey of the most important theoretical results
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on perfect hashing.
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In the following, we review some of those results.
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Fredman, Koml\'os and Szemer\'edi~\cite{FKS84} showed that it is possible to
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construct space efficient perfect hash functions that can be evaluated in
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constant time with table sizes that are linear in the number of keys:
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$m=O(n)$. In their model of computation, an element of the universe~$U$ fits
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into one machine word, and arithmetic operations and memory accesses have unit
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cost. Randomized algorithms in the FKS model can construct a perfect hash
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function in expected time~$O(n)$:
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this is the case of our algorithm and the works in~\cite{chm92,p99}.
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Many methods for generating minimal perfect hash functions use a
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{\em mapping}, {\em ordering} and {\em searching}
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(MOS) approach,
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a description coined by Fox, Chen and Heath~\cite{fch92}.
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In the MOS approach, the construction of a minimal perfect hash function
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is accomplished in three steps.
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First, the mapping step transforms the key set from the original universe
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to a new universe.
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Second, the ordering step places the keys in a sequential order that
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determines the order in which hash values are assigned to keys.
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Third, the searching step attempts to assign hash values to the keys.
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Our algorithm and the algorithm presented in~\cite{chm92} use the
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MOS approach.
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Pagh~\cite{p99} proposed a family of randomized algorithms for
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constructing minimal perfect hash functions.
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The form of the resulting function is $h(x) = (f(x) + d_{g(x)}) \bmod n$,
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where $f$ and $g$ are universal hash functions and $d$ is a set of
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displacement values to resolve collisions that are caused by the function $f$.
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Pagh identified a set of conditions concerning $f$ and $g$ and showed
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that if these conditions are satisfied, then a minimal perfect hash
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function can be computed in expected time $O(n)$ and stored in
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$(2+\epsilon)n$ computer words.
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Dietzfelbinger and Hagerup~\cite{dh01} improved~\cite{p99},
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reducing from $(2+\epsilon)n$ to $(1+\epsilon)n$ the number of computer
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words required to store the function, but in their approach~$f$ and~$g$ must
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be chosen from a class
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of hash functions that meet additional requirements.
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Differently from the works in~\cite{p99,dh01}, our algorithm uses two
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universal hash functions $h_1$ and $h_2$ randomly selected from a class
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of universal hash functions that do not need to meet any additional
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requirements.
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The work in~\cite{chm92} presents an efficient and practical algorithm
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for generating order preserving minimal perfect hash functions.
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Their method involves the generation of acyclic random graphs
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$G = (V, E)$ with~$|V|=cn$ and $|E|=n$, with $c \ge 2.09$.
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They showed that an order preserving minimal perfect hash function
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can be found in optimal time if~$G$ is acyclic.
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To generate an acyclic graph, two vertices $h_1(x)$ and $h_2(x)$ are
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computed for each key $x \in S$.
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Thus, each set~$S$ has a corresponding graph~$G=(V,E)$, where $V=\{0,1,
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\ldots,t\}$ and $E=\big\{\{h_1(x),h_2(x)\}:x \in S\big\}$.
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In order to guarantee the acyclicity of~$G$, the algorithm repeatedly selects
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$h_1$ and $h_2$ from a family of universal hash functions
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until the corresponding graph is acyclic.
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Havas et al.~\cite{hmwc93} proved that if $|V(G)|=cn$ and $c>2$,
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then the probability that~$G$ is acyclic is $p=e^{1/c}\sqrt{(c-2)/c}$.
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For $c=2.09$, this probability is
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$p \simeq 0.342$, and
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the expected number of iterations to obtain an acyclic graph
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is~$1/p \simeq 2.92$.
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