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