## Faster Deterministic Dictionaries (1999)

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Venue: | In 11 th Annual ACM Symposium on Discrete Algorithms (SODA |

Citations: | 9 - 5 self |

### BibTeX

@INPROCEEDINGS{Pagh99fasterdeterministic,

author = {Rasmus Pagh and Rasmus Pagh and Rasmus Pagh},

title = {Faster Deterministic Dictionaries},

booktitle = {In 11 th Annual ACM Symposium on Discrete Algorithms (SODA},

year = {1999},

pages = {487--493},

publisher = {ACM Press}

}

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### Abstract

We consider static dictionaries over the universe U = on a unit-cost RAM with word size w. Construction of a static dictionary with linear space consumption and constant lookup time can be done in linear expected time by a randomized algorithm. In contrast, the best previous deterministic algorithm for constructing such a dictionary with n elements runs in time O(n ) for # > 0. This paper narrows the gap between deterministic and randomized algorithms exponentially, from the factor of to an O(log n) factor. The algorithm is weakly non-uniform, i.e. requires certain precomputed constants dependent on w. A by-product of the result is a lookup time vs insertion time trade-o# for dynamic dictionaries, which is optimal for a certain class of deterministic hashing schemes.

### Citations

677 |
Universal classes of hash functions
- Carter, Wegman
- 1979
(Show Context)
Citation Context ... w = O(log n). In this case, it was shown how to construct the dictionary in time O(n 2 ). A breakthrough was made by Fredman, Komlos and Szemeredi [7], who showed how to use universal hash functions =-=[4]-=- to build an e#cient dictionary for any word size. Two construction algorithms were given: A randomized one running in expected time O(n), and a deterministic one with a running time of O(n 3 w). Rama... |

61 | Derandomization, witnesses for Boolean matrix multiplication and construction of perfect hash functions, to appear, Algorithmica. Available as Weizmann Institute TR
- Alon, Naor
(Show Context)
Citation Context ...nistic one with a running time of O(n 3 w). Raman [13] sped up the choice of universal hash functions in the deterministic algorithm, obtaining O(n 2 w) deterministic construction time. Alon and Naor =-=[1]-=- used small bias probability spaces to derandomize a variant of the FKS scheme, achieving construction time O(nw log 4 n). However, in this variant a lookup requires evaluation of a linear function in... |

61 | Optimal bounds for the predecessor problem
- Beame, Fich
- 1999
(Show Context)
Citation Context ...n time O(log O(1) n). The requirement l(n) = O( # log n) is not essential, but gives a simpler expression for the insertion time. For l(n) = # # log n/ log log n) the data structure of Beame and Fich =-=[3]-=- is superior to any approach using static dictionaries as a black box. 10 It might be argued that insertion time 2# # log n) makes Theorem 10 uninteresting. However, if the frequency of insertions com... |

46 |
Priority queues: Small, monotone and trans-dichotomous
- Raman
(Show Context)
Citation Context ...o build an e#cient dictionary for any word size. Two construction algorithms were given: A randomized one running in expected time O(n), and a deterministic one with a running time of O(n 3 w). Raman =-=[13]-=- sped up the choice of universal hash functions in the deterministic algorithm, obtaining O(n 2 w) deterministic construction time. Alon and Naor [1] used small bias probability spaces to derandomize ... |

43 |
Sorting and searching on the word RAM
- Hagerup
(Show Context)
Citation Context ...tructures. It is therefore of interest to have static dictionaries which are space economical, support fast lookups, and can be constructed e#ciently. Our model of computation is a unit-cost word RAM =-=[9]-=- with a standard instruction set, including multiplication and bit operations. Throughout this paper S will refer to an arbitrary set of n elements from the universe U = {0, 1} w , where w is the word... |

38 |
János Komlós, and Endre Szemerédi. Storing a sparse table with O(1) worst case access time
- Fredman
- 1984
(Show Context)
Citation Context ...uble displacement” dictionary which is efficient for w = O(log n). In this case, it was shown how to construct the dictionary in time O(n 2 ). A breakthrough was made by Fredman, Komlós and Szemerédi =-=[7]-=-, who showed how to use universal hash functions [4] to build an efficient dictionary for any word size. Two construction algorithms were given: A randomized one running in expected time O(n), and a d... |

29 |
Polynomial hash functions are reliable (extended abstract
- Dietzfelbinger, Gil, et al.
(Show Context)
Citation Context ... the dictionary is not e#cient unless w = O(log n). Another variant of the FKS scheme reduces the number of random bits to O(log n+log w), while achieving O(n) time construction with high probability =-=[5]-=-. For w = n#15 , fusion trees [8] improve the above deterministic bounds to n 1+# for any fixed # > 0, as observed by Andersson [2]. The fusion tree construction algorithm is weakly non-uniform in tha... |

22 |
Friedhelm Meyer auf der
- Dietzfelbinger, Karlin, et al.
- 1990
(Show Context)
Citation Context ...ord sizes. As a consequence of the result we are able to state an improved trade-o# between lookup time and insertion time for deterministic dynamic dictionaries. By a result of Dietzfelbinger et al. =-=[6]-=- the trade-o# is optimal for a certain class of data structures based on hashing. What is actually shown in the following is how to construct an e#cient perfect hash function for S. Definition 1 A fun... |

21 |
Endre Szemeredi. Storing a sparse table with O(1) worst case access time
- Fredman, Komlos
- 1984
(Show Context)
Citation Context ...double displacement" dictionary which is e#cient for w = O(log n). In this case, it was shown how to construct the dictionary in time O(n 2 ). A breakthrough was made by Fredman, Komlos and Szeme=-=redi [7]-=-, who showed how to use universal hash functions [4] to build an e#cient dictionary for any word size. Two construction algorithms were given: A randomized one running in expected time O(n), and a det... |

21 |
Worst-case optimal insertion and deletion methods for decomposable searching problems
- Overmars, Leeuwen
- 1981
(Show Context)
Citation Context ...ast updates and very fast queries seems considerably harder. Miltersen's static dictionary can be dynamized, giving update time O(n # ) and lookup time O(1). Hagerup uses the transformation result of =-=[12]-=- (Theorem A) to dynamize his static dictionary [10]. This gives a lookup time vs insertion time trade-o#. We follow the same path and obtain a dynamic dictionary with an improved trade-o#: Theorem 10 ... |

18 |
Endre Tarjan and Andrew Chi-Chih Yao. Storing a sparse table
- Robert
- 1979
(Show Context)
Citation Context ...ns. The aim of this paper is to show that randomized static dictionaries can be replaced with e#cient deterministic ones which have nearly the same construction time. 1.1 Related work. Tarjan and Yao =-=[15] gave a ri-=-gorous basis for understanding some heuristics which had been used for table compression, resulting in the "double displacement" dictionary which is e#cient for w = O(log n). In this case, i... |

17 | Error correcting codes, perfect hashing circuits, and deterministic dynamic dictionaries
- Miltersen
- 1998
(Show Context)
Citation Context ...weakly non-uniform in that it requires access to a constant number of precomputed word-size constants depending (only) on w. These constants may be thought of as computed at "compile time". =-=Miltersen [11]-=- introduced the use of error-correcting codes in a novel approach to the construction of perfect hash functions. For any fixed # > 0, this yields an e#cient dictionary requiring O(n 1+# ) construction... |

14 |
deterministic sorting and searching in linear space
- Anderson, ”Faster
- 1996
(Show Context)
Citation Context ...log w), while achieving O(n) time construction with high probability [5]. For w = n#15 , fusion trees [8] improve the above deterministic bounds to n 1+# for any fixed # > 0, as observed by Andersson =-=[2]-=-. The fusion tree construction algorithm is weakly non-uniform in that it requires access to a constant number of precomputed word-size constants depending (only) on w. These constants may be thought ... |

13 |
Surpassing the information-theoretic bound with fusion trees
- Fredman, Willard
- 1993
(Show Context)
Citation Context ...less w = O(log n). Another variant of the FKS scheme reduces the number of random bits to O(log n+log w), while achieving O(n) time construction with high probability [5]. For w = n#15 , fusion trees =-=[8]-=- improve the above deterministic bounds to n 1+# for any fixed # > 0, as observed by Andersson [2]. The fusion tree construction algorithm is weakly non-uniform in that it requires access to a constan... |

8 |
A lower bound on the cell probe complexity of the dictionary problem. Unpublished manuscript
- Sundar
- 1993
(Show Context)
Citation Context ...er natural question is whether weak non-uniformity is necessary in order to deal with large word lengths. A discussion of weak non-uniformity can be found in [11]. An unpublished manuscript by Sundar =-=[14]-=- claims an amorized lower bound of#9588 log n - log log w)/ log log log n) time per operation in a deterministic dynamic dictionary, so there is no hope of matching the expected performance of the bes... |

5 |
Fast deterministic construction of static dictionaries
- Hagerup
- 1999
(Show Context)
Citation Context ... time, also by a weakly nonuniform algorithm. The lookup time is inversely proportional to #. Extending the result of Miltersen, Hagerup exhibited a trade-o# between construction time and lookup time =-=[10]-=-. The algorithm achieves construction time O(n log n) and lookup time O(log log n) simultaneously. For lookup time o(log log n) it needs construction time n 2 (log n) 1-o(1) . 1.2 This work. We show h... |