# Membership in Constant Time and Almost-Minimum Space

@article{Brodnik1999MembershipIC,
title={Membership in Constant Time and Almost-Minimum Space},
author={Andrej Brodnik and J. Ian Munro},
journal={SIAM J. Comput.},
year={1999},
volume={28},
pages={1627-1640}
}
• Published 1 May 1999
• Computer Science
• SIAM J. Comput.
This paper deals with the problem of storing a subset of elements from the bounded universe $\mathcal{M} = \{0, \ldots, M-1\}$ so that membership queries can be performed efficiently. In particular, we introduce a data structure to represent a subset of $N$ elements of $\mathcal{M}$ in a number of bits close to the information-theoretic minimum, $B = \left\lceil \lg {M\choose N} \right\rceil$, and use the structure to answer membership queries in constant time.
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## References

SHOWING 1-10 OF 30 REFERENCES
Searching in constant time and minimum space
A data structure is introduced to represent a subset of N elements of $\lbrack0,\...,M-1\rbrack$ in a number of bits close to the information-theoretic minimum and use the structure to answer membership queries in constant time.
The Spatial Complexity of Oblivious k-Probe Hash Functions
• Computer Science, Mathematics
SIAM J. Comput.
• 1990
Nearly tight bounds on the spatial complexity of oblivious $O(1)$-probe hash functions, which are defined to depend solely on their search key argument are provided, establishing a significant gap between oblivious and nonoblivious search.
Membership in Constant Time and Minimum Space
• Computer Science
ESA
• 1994
We investigate the problem of storing a subset of the elements of a bounded universe so that searches can be performed in constant time and the space used is within a constant factor of the minimum
Should tables be sorted?
• A. Yao
• Computer Science
19th Annual Symposium on Foundations of Computer Science (sfcs 1978)
• 1978
It is shown that, in a rather general model including al1 the commonly-used schemes, $\lceil$ lg(n+l) $\rceil$ probes to the table are needed in the worst case, provided the key space is sufficiently large.
The program complexity of searching a table
• Harry G. Mairson
• Computer Science
24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
• 1983
Under a model combining perfect hashing and binary search methods, it is shown that for k probes to the table, nk/2k+1(1 + o(1)) bits are necessary and sufficient to describe a table searching algorithm.
Dynamic perfect hashing: upper and lower bounds
• Computer Science
[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
• 1988
An Omega (log n) lower bound is proved for the amortized worst-case time complexity of any deterministic algorithm in a class of algorithms encompassing realistic hashing-based schemes.
The Complexity of Some Simple Retrieval Problems
• Computer Science
JACM
• 1975
The cost of a complete updating algorithm is taken to be the number of bits it reads and/or writes in updating the representation of a data base, and lower bounds to measures of this cost are cited.
Storing a sparse table with O(1) worst case access time
• Computer Science
23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
• 1982
A data structure for representing a set of n items from a universe of m items, which uses space n+o(n) and accommodates membership queries in constant time and is easy to implement.
Efficient Storage and Retrieval by Content and Address of Static Files
Firm lower bounds are given to minimax measures of bits stored and bits accessed for each of four retrieval questions, and representations and algorithms for a bit-addressable machine which come within factors of two or three of attaining all four bounds at once for files of any size.
Storing a sparse table
• Computer Science
CACM
• 1979
This work proposes a good worst-case method for storing a static table of n entries, each an integer between 0 and N - 1, and analysis shows why a simpler algorithm used for compressing LR parsing tables works so well.