56 lines
2.6 KiB
Plaintext
56 lines
2.6 KiB
Plaintext
Minimal Perfect Hash Functions - Introduction
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%!includeconf: CONFIG.t2t
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----------------------------------------
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==Basic Concepts==
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Suppose [figs/img14.png] is a universe of //keys//.
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Let [figs/img15.png] be a //hash function// that maps the keys from [figs/img14.png] to a given interval of integers [figs/img16.png].
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Let [figs/img17.png] be a set of [figs/img8.png] keys from [figs/img14.png].
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Given a key [figs/img18.png], the hash function [figs/img7.png] computes an
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integer in [figs/img19.png] for the storage or retrieval of [figs/img11.png] in
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a //hash table//.
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Hashing methods for //non-static sets// of keys can be used to construct
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data structures storing [figs/img20.png] and supporting membership queries
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"[figs/img18.png]?" in expected time [figs/img21.png].
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However, they involve a certain amount of wasted space owing to unused
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locations in the table and waisted time to resolve collisions when
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two keys are hashed to the same table location.
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For //static sets// of keys it is possible to compute a function
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to find any key in a table in one probe; such hash functions are called
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//perfect//.
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More precisely, given a set of keys [figs/img20.png], we shall say that a
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hash function [figs/img15.png] is a //perfect hash function//
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for [figs/img20.png] if [figs/img7.png] is an injection on [figs/img20.png],
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that is, there are no //collisions// among the keys in [figs/img20.png]:
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if [figs/img11.png] and [figs/img22.png] are in [figs/img20.png] and [figs/img23.png],
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then [figs/img24.png].
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Figure 1(a) illustrates a perfect hash function.
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Since no collisions occur, each key can be retrieved from the table
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with a single probe.
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If [figs/img25.png], that is, the table has the same size as [figs/img20.png],
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then we say that [figs/img7.png] is a //minimal perfect hash function//
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for [figs/img20.png].
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Figure 1(b) illustrates a minimal perfect hash function.
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Minimal perfect hash functions totally avoid the problem of wasted
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space and time. A perfect hash function [figs/img7.png] is //order preserving//
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if the keys in [figs/img20.png] are arranged in some given order
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and [figs/img7.png] preserves this order in the hash table.
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| [figs/img26.png]
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| **Figure 1:** (a) Perfect hash function. (b) Minimal perfect hash function.
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Minimal perfect hash functions are widely used for memory efficient
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storage and fast retrieval of items from static sets, such as words in natural
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languages, reserved words in programming languages or interactive systems,
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universal resource locations (URLs) in Web search engines, or item sets in
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data mining techniques.
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%!include: ALGORITHMS.t2t
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%!include: FOOTER.t2t
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%!include(html): ''GOOGLEANALYTICS.t2t'' |