Uniform Hashing in Constant Time and Optimal Space

Many algorithms and data structures employing hashing have been analyzed under the uniform hashing assumption, i.e., the assumption that hash functions behave like truly random functions. Starting with the discovery of universal hash functions, many researchers have studied to what extent this theor...

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Veröffentlicht in:SIAM journal on computing Jg. 38; H. 1; S. 85 - 96
Hauptverfasser: Pagh, Anna, Pagh, Rasmus
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2008
Schlagworte:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Computer science; control theory; systems
Data processing. List processing. Character string processing
Exact sciences and technology
Memory organisation. Data processing
Miscellaneous
Software
Theoretical computing
Lower bound
Ideal
Probability
Entropy
Space time
Probability distribution
Algorithm
Input
hash function
Hashing
randomized algorithms; 68P5
Random function
Data structure
uniform hashing
68P10
68P20
ISSN:0097-5397, 1095-7111
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Zusammenfassung:Many algorithms and data structures employing hashing have been analyzed under the uniform hashing assumption, i.e., the assumption that hash functions behave like truly random functions. Starting with the discovery of universal hash functions, many researchers have studied to what extent this theoretical ideal can be realized by hash functions that do not take up too much space and can be evaluated quickly. In this paper we present an almost ideal solution to this problem: a hash function $h: U\rightarrow V$ that, on any set of $n$ inputs, behaves like a truly random function with high probability, can be evaluated in constant time on a RAM and can be stored in $(1+\epsilon)n\log |V| + O(n+\log\log |U|)$ bits. Here $\epsilon$ can be chosen to be any positive constant, so this essentially matches the entropy lower bound. For many hashing schemes this is the first hash function that makes their uniform hashing analysis come true, with high probability, without incurring overhead in time or space.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0097-5397
1095-7111
DOI:10.1137/060658400