## Optimal Bounds for the Predecessor Problem and Related Problems (2001)

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Venue: | Journal of Computer and System Sciences |

Citations: | 57 - 0 self |

### BibTeX

@ARTICLE{Beame01optimalbounds,

author = {Paul Beame and Faith E. Fich},

title = {Optimal Bounds for the Predecessor Problem and Related Problems},

journal = {Journal of Computer and System Sciences},

year = {2001},

volume = {65},

pages = {2002}

}

### Years of Citing Articles

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### Abstract

We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unit-cost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved for a large class of problems, including both static and dynamic predecessor problems, in a much stronger communication game model, but they apply to the cell probe and RAM models.

### Citations

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(Show Context)
Citation Context ...res with these properties: Constant time membership queries can be obtained for any set of size n using an O(n 2 ) word hash table and a hash function randomly chosen from a suitable universal family =-=[13]-=-. Fredman, Komlos, and Szemeredi [25] improved the space to O(n) using two level perfect hashing. Their data structure can be constructed in O(n) expected time. To evaluate the hash functions, multipl... |

102 |
The Discrepancy Method: Randomness and Complexity
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Citation Context ...e reduced to the point separation problem, as follows. Given an instance z 2 f0;1;?g N of the prefix problem, an instance of the point separation problem can be constructed as follows, using ideas in =-=[16, -=-17]. Suppose i 1s si s 1 denote the indices of the non-? characters of z. Let i s = N + 1 and P = fp i 1 ; x i 1 i 2 ; p i 2 ; : : : ; x i s 1 i s ; p i s g, where p i = (2i j ; 4i 2 j ) for j = 1; :... |

49 |
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Citation Context ...e (i.e. n W(1) bits), then any set of size n can be stored, using a trie, in O(n) words so that predecessor queries can be performed in constant time in the cell probe model. On the other hand, Ajtai =-=[2]-=- proved that, if the word length is sufficiently small (i.e. O(logn) bits), and only n O(1) words of memory are used to represent any set of n elements, then worst-case constant time for predecessor q... |

49 | Marked ancestor problems
- Alstrup, Husfeldt, et al.
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(Show Context)
Citation Context ...given set of integers are less than or equal to a given integer. It is interesting that, for static sets, predecessor queries and rank queries are equally difficult [28]. Alstrup, Husfeldt, and Rauhe =-=[-=-4] considered a generalization of the dynamic predecessor problem in a universe of size N: the marked ancestor problem in a tree of N nodes. They proved a tradeoff, t 2 W logN log(ublogN) ; between ... |

41 | Tight(er) worst-case bounds on dynamic searching and priority queues
- Andersson, Thorup
- 2000
(Show Context)
Citation Context ...egers from a universe of size N in n O(1) space and performs predecessor queries in time O min ( loglogN log loglogN ; s logn loglogn )! : Using recent generic transformations of Andersson and Thorup =-=[7, -=-10], the algorithm can be made dynamic and the space improved to O(n), although the time increases to O min n loglog N logloglog N loglogn; q logn loglogn o We also obtain matching lower bounds for ... |

27 |
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Citation Context ...hing, rehashing when necessary (with hash functions randomly chosen from an appropriate universal family), to perform membership queries in constant time and perform updates in expected constant time =-=[21, 22, 19]-=-. Miltersen [36] has constructed a deterministic dynamic dictionary using error correcting codes and clustering. It performs membership queries in constant time and performs updates in time O(n e ) fo... |

17 |
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Citation Context ...for query time. They also construct a RAM algorithm which matches this query time while using only O(log logN) time per update and O(N) words, each containing O(logN) bits. Ajtai, Fredman, and Komlos =-=[1]-=- showed that, if the word length is sufficiently large (i.e. n W(1) bits), then any set of size n can be stored, using a trie, in O(n) words so that predecessor queries can be performed in constant ti... |

17 | Optimal static range reporting in one dimension
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Citation Context ...roblem that can be solved using predecessor queries is one dimensional range search (i.e. find some element of S contained within a query interval, if one exists). However, Alstrup, Brodal, and Rauhe =-=[3]-=- have recently obtained a static data structure of size O(n) for storing a set S of n integers that supports range search in constant time. Using this data structure, they can approximately count (i.e... |

16 |
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Citation Context ... and updates in O(1 + logn=log b) time. Andersson, Miltersen, and Thorup [9] showed how to efficiently implement fusion trees on an AC 0 RAM. In the same model, Andersson, Miltersen, Riis, and Thorup =-=[8]-=- proved that the time complexity of the static dictionary problem is Q( p log n=log logn). Their algorithm uses O(n) words and their lower bound holds even if 2 (log n) O(1) words are allowed. It is i... |

13 | Trans-dichotomous algorithms without multiplication - some upper and lower bounds
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Citation Context ...In this special case, updates to the table can also be performed in constant time. If the universe size N is significantly less than 2 b , where b is the number of bits in a word, then packed B-trees =-=[27, 6, 12, 40-=-] are time and space efficient. Specifically, using branching factor B b=(1 + log N), insertions, deletions, and membership and predecessor queries can be performed in O(logn=logB) steps using O(n=B)... |

12 | Efficient regular data structures and algorithms for location and proximity problems
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(Show Context)
Citation Context ...Further connections have been made between the approximate nearest neighbour problem for inputs in [1; N] d , for constant d, and our bounds for the predecessor problem. Amir, Efrat, Indyk, and Samet =-=[5]-=- give a reduction from the predecessor problem to the approximate nearest neighbour problem in one dimension to derive an W(log logN=logloglogN) lower bound for the approximate nearest neighbour probl... |

10 |
M.: Tight(er) worst-case bounds on dynamic searching and priority queues
- Anderson, Thorup
- 2000
(Show Context)
Citation Context ...egers from a universe of size N in n O(1) space and performs predecessor queries in time O min ( loglogN log loglogN ; s logn loglogn )! : Using recent generic transformations of Andersson and Thorup =-=[7, -=-10], the algorithm can be made dynamic and the space improved to O(n), although the time increases to O min n loglog N logloglog N loglogn; q logn loglogn o We also obtain matching lower bounds for ... |

8 |
Probabilistic methods in graph theory
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(Show Context)
Citation Context ...state two combinatorial results which are important for the lower bound proofs given in the next subsection. The following form of the Chernoff-Hoeffding bound follows easily from the presentation in [18]. Proposition 3.1: Fix H U with jHj rjU j and let S U with jSj = s be chosen uniformly at random. Then Pr[jH \Sj rs=4] ( p 2=e 3=4 ) rss2 rs=2 : 8 The next result is a small modification and r... |

6 |
A good neighbor is hard to find
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Citation Context ...his requires a more complicated analysis. The same approach has also been used to obtain an W(log logd=log loglogd) lower bound for the approximate nearest neighbour problem over the universe f0;1g d =-=[15]-=-. A description of related work is given in Section 2, followed by our lower bounds in Section 3.4, and our new algorithms in Section 4. Throughout the paper log will denote a logarithm to the base 2 ... |

5 | Predecessor queries in dynamic integer sets
- Brodal
- 1997
(Show Context)
Citation Context ...nking dictionary in O(n) time, given the elements in sorted order. His data structure uses O(n) space and performs rank, selection, and, hence, predecessor queries, in O(1+ (logn)=log b) time. Brodal =-=[11]-=- constructs a data structure that is similar to Andersson's but uses buffers to delay insertions and deletions to the packed B-tree. In the worst case, it uses O( f (n)) time to perform updates and O(... |

4 |
Sublogarithmic seaching without multiplications
- Andersson
- 1995
(Show Context)
Citation Context ...In this special case, updates to the table can also be performed in constant time. If the universe size N is significantly less than 2 b , where b is the number of bits in a word, then packed B-trees =-=[27, 6, 12, 40-=-] are time and space efficient. Specifically, using branching factor B b=(1 + log N), insertions, deletions, and membership and predecessor queries can be performed in O(logn=logB) steps using O(n=B)... |

4 |
Fusion trees can be implemented with AC instructions only
- Andersson, Miltersen, et al.
- 1996
(Show Context)
Citation Context ...ivision). Hagerup [28] gave an AC 0 RAM algorithm for the dynamic predecessor problem that uses O(n) words and performs queries and updates in O(1 + logn=log b) time. Andersson, Miltersen, and Thorup =-=[9]-=- showed how to efficiently implement fusion trees on an AC 0 RAM. In the same model, Andersson, Miltersen, Riis, and Thorup [8] proved that the time complexity of the static dictionary problem is Q( p... |

4 | Geometric searching over the rationals
- Chazelle
(Show Context)
Citation Context ...e reduced to the point separation problem, as follows. Given an instance z 2 f0;1;?g N of the prefix problem, an instance of the point separation problem can be constructed as follows, using ideas in =-=[16, -=-17]. Suppose i 1s si s 1 denote the indices of the non-? characters of z. Let i s = N + 1 and P = fp i 1 ; x i 1 i 2 ; p i 2 ; : : : ; x i s 1 i s ; p i s g, where p i = (2i j ; 4i 2 j ) for j = 1; :... |

3 |
Faster deterministic sorting and seaching in linear space
- Andersson
- 1996
(Show Context)
Citation Context ...egers from a universe of size N in n O(1) space and performs predecessor queries in time O min ( loglogN log loglogN ; s logn loglogn )! : Using recent generic transformations of Andersson and Thorup =-=[7, -=-10], the algorithm can be made dynamic and the space improved to O(n), although the time increases to O min n loglog N logloglog N loglogn; q logn loglogn o We also obtain matching lower bounds for ... |

3 |
Exponential search trees for faster deterministic searching, sorting and priority queues in linear space
- Anderson, Thorup
- 1997
(Show Context)
Citation Context ...egers from a universe of size N in n O(1) space and performs predecessor queries in time O min ( loglogN log loglogN ; s logn loglogn )! : Using recent generic transformations of Andersson and Thorup =-=[5,-=- 8], the algorithm can be made dynamic and the space improved to O(n), although the time increases to O min n loglog N logloglog N loglogn; q logn loglogn o We also obtain matching lower bounds for ... |

2 |
Toward optimal e-approximate nearest neighbor algorithms
- Cary
(Show Context)
Citation Context ...he approximate nearest neighbour problem in one dimension to derive an W(log logN=logloglogN) lower bound for the approximate nearest neighbour problem. Using an extension of our data structure, Cary =-=[14]-=- gives a data structure that matches this lower bound for any constant number of dimensions. It would be nice to remove the log logn factor in the numerator in the first term of the minimum in Corolla... |