## Membership in Constant Time and Minimum Space (1994)

Venue: | Lecture Notes in Computer Science |

Citations: | 18 - 5 self |

### BibTeX

@INPROCEEDINGS{Brodnik94membershipin,

author = {Andrej Brodnik and J. Ian Munro},

title = {Membership in Constant Time and Minimum Space},

booktitle = {Lecture Notes in Computer Science},

year = {1994},

pages = {72--81},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

. We investigate the problem of storing a subset of the elements of a boundeduniverse so that searches canbe performed in constant time and the space used is within a constant factor of the minimum required. Initially we focus on the static version of this problem and conclude with an enhancement that permits insertions and deletions. 1 Introduction Given a universal set M = f0; : : : ; M \Gamma 1g and any subset N = fe 1 ; : : : ; e N g the membership problem is to determine whether given query element in M is an element of N . There are two standard approaches to solve this problem: to list all elements of N (e.g. in a hash table) or to list all the answers (e.g. a bit map of size M ). When N is small the former approach comes close to the information theoretic lower bound on the number of bits needed to represent an arbitrary subset of the given size (i.e. a function of both N and M , l lg \Gamma M N \Delta m ). Similarly, when N is large (say ffM ) the later approach is near...

### Citations

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Storing a sparse table with O(1) worst case access time
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(Show Context)
Citation Context ...ime solution for N �� M or Ns1 4 p lg M , while Tarjan and Yao ([14]) presented a constant time solution using O(N lg M ) bits of space for N = O(M ffl ) and 0 ! ffls1. Fredman, Komlos and Szemere=-=di ([9]-=-) closed the gap developing a constant time algorithm which uses a data structure of size N lg M bits and O(N p log N + log (2) M ) additional bits of storage, for any N and M . Fiat, Naor, Schmidt an... |

148 |
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Citation Context ...ible configurations of these small ranges (table of small ranges) permit the encoding in the minimal space bound. Our technique is, then, an example of what we call word-size truncated recursions(cf. =-=[11, 12]). Namely,-=- the recursion needs only to continue down to a level of "small enough" subproblems, at which point indexing into a table of all solutions suffices. This happens to a large extent because at... |

136 | Should tables be sorted
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(Show Context)
Citation Context ...to answer a query whether a given element x 2 M is in N . The problem has the obvious dynamic extension which also includes the operations insert and delete. We use the standard cell probe model (cf. =-=[6, 10, 15]-=-) with word size dlg Me bits. 2 In other words, one register can be used to represent a single element from the entire domain M, specify all elements which are present in a set of a size lg M elements... |

128 |
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(Show Context)
Citation Context ...to answer a query whether a given element x 2 M is in N . The problem has the obvious dynamic extension which also includes the operations insert and delete. We use the standard cell probe model (cf. =-=[6, 10, 15]-=-) with word size dlg Me bits. 2 In other words, one register can be used to represent a single element from the entire domain M, specify all elements which are present in a set of a size lg M elements... |

126 | Dynamic perfect hashing: Upper and lower bounds
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Citation Context ...of an implicit search scheme for N =\Omega ((log M ) p ) we get a scheme which uses less than (1 + p) l log (2) M m +O(1) additional bits of memory. Moving to the dynamic case, Dietzfelbinger et al. (=-=[2]-=-) proved an\Omega (log N ) worst case lower bound for a class of realistic hashing schemes. In the same paper they also presented a scheme which, using results of [9] and a standard doublingtechnique,... |

77 |
Emde Boas. Machine models and simulations
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Citation Context ...to answer a query whether a given element x 2 M is in N . The problem has the obvious dynamic extension which also includes the operations insert and delete. We use the standard cell probe model (cf. =-=[6, 10, 15]-=-) with word size dlg Me bits. 2 In other words, one register can be used to represent a single element from the entire domain M, specify all elements which are present in a set of a size lg M elements... |

55 |
Storing a sparse table
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(Show Context)
Citation Context ...Omega ((log M ) ffl ). However, adding some storage changes the situation. For example, Yao ([15]) showed that there exists a constant time solution for N �� M or Ns1 4 p lg M , while Tarjan and Y=-=ao ([14]-=-) presented a constant time solution using O(N lg M ) bits of space for N = O(M ffl ) and 0 ! ffls1. Fredman, Komlos and Szemeredi ([9]) closed the gap developing a constant time algorithm which uses ... |

38 |
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Citation Context ...ved a constant amortized expected time per operation with a high probability. However, the worst case time per operation (non-amortized)was\Omega (N ). Later Dietzfelbinger and Meyer auf der Heide in =-=[3]-=- upgraded the scheme and achieved a constant worst case time per operation with a high probability. A similar result was also obtained by Dietzfelbinger, Gil, Matias and Pippenger in [1]. Finallywe tu... |

24 |
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Citation Context ... der Heide in [3] upgraded the scheme and achieved a constant worst case time per operation with a high probability. A similar result was also obtained by Dietzfelbinger, Gil, Matias and Pippenger in =-=[1]-=-. Finallywe turn to some information theoretic references. Elias ([4]) addressed a more general version of a static membership problem which involved several different types of queries. For these quer... |

23 |
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Citation Context ...ase time per operation with a high probability. A similar result was also obtained by Dietzfelbinger, Gil, Matias and Pippenger in [1]. Finallywe turn to some information theoretic references. Elias (=-=[4]-=-) addressed a more general version of a static membership problem which involved several different types of queries. For these queries he discussed a tradeoff between the size of the data structure an... |

18 | and M.Naor. Implicit O(1) probe search
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Citation Context ...such data structures Yao ([15]) showed that in a bounded universe there always exists some subset for which any implicit data structure requires at least logarithmic search time. Later Fiat and Naor (=-=[7]-=-) put a better bound on the size of the subset N . They proved that for this awkward case, N has to be larger than a quantity which lies somewhere between lg (O(1)) M and (lg M ) ffl (0sffl ! 1). 3 In... |

17 |
Non-oblivious hashing
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- 1992
(Show Context)
Citation Context ... gap developing a constant time algorithm which uses a data structure of size N lg M bits and O(N p log N + log (2) M ) additional bits of storage, for any N and M . Fiat, Naor, Schmidt and Siegel in =-=[8]-=- decreased the number of additional bits to dlg Ne + l log (2) M m + O(1). Moreover, combining their result with Fiat and Naor's ([7]) construction of an implicit search scheme for N =\Omega ((log M )... |

17 | The bit probe complexity measure revisited
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Citation Context ...tudied the relation between the size of the data structure and the number of bits probed, given the set of all possible queries. The same arrangement was later more rigorously studied by Miltersen in =-=[13]-=-. 3 log (p) x denotes the iterated logarithm, namely the logarithm function applied p times to x. 3 A Solution for the Static Case As mentioned earlier, if more than half the elements are present we c... |

11 |
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(Show Context)
Citation Context ... that is, using (2), B+o(B) -- bits which required an average of (1+ ffl) lg N +2 bit probes to answer a query. However, in the worst case, it was still necessary to probe N bits. Elias and Flower in =-=[5]-=- further generalized the notion of a query into a database. They defined the set of data and a set of queries, and in such a general setting they studied the relation between the size of the data stru... |

4 | Optimal Algorithms for Generating Discrete Random Variables with Changing Distributions," Max Planck Institute
- Hagerup, Mehlhorn, et al.
- 1992
(Show Context)
Citation Context ...ible configurations of these small ranges (table of small ranges) permit the encoding in the minimal space bound. Our technique is, then, an example of what we call word-size truncated recursions(cf. =-=[11, 12]). Namely,-=- the recursion needs only to continue down to a level of "small enough" subproblems, at which point indexing into a table of all solutions suffices. This happens to a large extent because at... |