## Optimal Bounds for the Predecessor Problem (0)

Venue: | In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing |

Citations: | 62 - 0 self |

### BibTeX

@INPROCEEDINGS{Beame_optimalbounds,

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

title = {Optimal Bounds for the Predecessor Problem},

booktitle = {In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing},

year = {},

pages = {295--304},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### 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 efficiently stored set. Our algorithms are for the unit-cost word-level RAM with multiplication and extend to give optimal dynamic algorithms. The lower bounds are proved in a much stronger communication game model, but they apply to the cell probe and RAM models and to both static and dynamic predecessor problems.

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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 =-=[8]-=-. Fredman, Komlos, and Szemeredi [16] improved the space to O(n) using two level perfect hashing. Their original algorithm used multiplication and division of log 2 N bit words to compute the hash fun... |

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Citation Context ...ent any set of n elements, then worst-case constant time for predecessor queries is impossible. Miltersen [21] observed that a cell-probe algorithm can be viewed as a two-party communication protocol =-=[28]-=- between a Querier who holds the input to a query and a Responder who holds the data structure. In each round of communication, the Querier sends the name of a memory cell to access and the Responder ... |

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Citation Context ...me, but they require W(N) space. However, there have been important algorithmic breakthroughs showing that such techniques have more general applicability. For example, with two level perfect hashing =-=[16]-=-, any n element set can be stored in O(n) space and constant time membership queries can be performed. Fusion trees fully exploit unit-cost word-level operations and the fact that data elements need t... |

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Citation Context ...f bits to represent individual elements is very small in comparison with the number of elements in the set. Otherwise, those algorithms consume huge amounts of space. For example, van Emde Boas trees =-=[25, 24]-=- can be used to perform predecessor queries on any set of integers from a universe of size N in O(loglogN) time, but they require W(N) space. However, there have been important algorithmic breakthroug... |

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Citation Context ...l condition that the sequence of locations and their values determine the answer is equivalent to the condition that the bits communicated alone determine the answer. The lower bound, in the style of =-=[19]-=-, works `top down', maintaining, for each player, a relatively large set of inputs on which the communication is fixed. Unlike [19], we actually have non-uniform distributions on the Responder's input... |

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Citation Context ... exploit unit-cost word-level operations and the fact that data elements need to fit in words of memory to store static sets of size n in O(n) space and perform predecessor queries in O( # logn) time =-=[18]-=-. For the static predecessor problem, it has been widely conjectured that the time complexities achieved by van Emde Boas trees and fusion trees are optimal for any data structure using a reasonable a... |

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Citation Context ...ost natural and general models for proving lower bounds for data structures problems, and one that is ideally suited for representing word-level operations, is the cell-probe model, introduced by Yao =-=[29]-=-. In this model, there is a memory consisting of cells, each of which is capable of storing some fixed number of bits. A cell-probe algorithm is a decision tree with one memory cell accessed at each n... |

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Citation Context ... that have small cost per query and do not run for very long can access only a small number of memory cells from a moderate size set of potential memory cells. Using static dictionary techniques from =-=[17]-=-, one can obtain an efficient solution to the static problem by beginning with the empty set and inserting elements one by one, recording the changes made to the memory in the dictionary. One then obt... |

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Citation Context ...f bits to represent individual elements is very small in comparison with the number of elements in the set. Otherwise, those algorithms consume huge amounts of space. For example, van Emde Boas trees =-=[25, 24]-=- can be used to perform predecessor queries on any set of integers from a universe of size N in O(loglogN) time, but they require W(N) space. However, there have been important algorithmic breakthroug... |

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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... |

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Citation Context ...can be made dynamic and the space improved to O(n). We also obtain matching lower bounds, improving Miltersen 's W( # loglogN) and W((logn) 1/3 ) lower bounds in the powerful communication game model =-=[21, 22]-=-. The key to our improved lower bounds is to use a more complicated input distribution. Unfortunately, this leads to a more complicated analysis. However, the understanding we obtained from identifyin... |

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Citation Context ...e membership queries, 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 =-=[18, 3, 7, 23]-=- can be time and space efficient. Specifically, using branching factor B # b/(1 + log 2 N), insertions, deletions, and membership and predecessor queries can be performed in O(logn/logB) steps using O... |

38 |
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Citation Context ...h functions. Recently it was shown that hash functions of the form h(x) = ax mod 2 #log 2 N# div 2 #log 2 N#-r for r # #log 2 n#, suffice for implementing the two level perfect hashing data structure =-=[11, 13]-=-. Notice that the evaluation of such functions does not depend on constant time division instructions; a right shift suffices. These algorithms can be made dynamic with the same constant time for memb... |

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Citation Context ...n constant time division instructions; a right shift suffices. These algorithms can be made dynamic with the same constant time for membership queries and with constant expected amortized update time =-=[13, 14, 12]-=-, by rehashing when necessary (using randomly chosen hash functions). For the static problem, Raman [23] proves that it is possible to choose such hash functions deterministically in O(n 2 logN) time.... |

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Citation Context ...n constant time division instructions; a right shift suffices. These algorithms can be made dynamic with the same constant time for membership queries and with constant expected amortized update time =-=[13, 14, 12]-=-, by rehashing when necessary (using randomly chosen hash functions). For the static problem, Raman [23] proves that it is possible to choose such hash functions deterministically in O(n 2 logN) time.... |

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Citation Context ...bed, so different probes to the same memory cell must always receive the same response. In fact, for many problems, the cell probe complexity is significantly larger than the communication complexity =-=[20]-=-. Miltersen [21] generalized Ajtai's proof to obtain an W( # loglogN) lower bound on time in this model for the problem of finding predecessors in a static set from a universe of size N. (Independent ... |

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Citation Context ...ry search is performed on the log 2 N bits used to represent individual elements. The major drawback is the use of an extremely large amount of space: W(N) words. Willard's x-fast trie data structure =-=[26]-=- uses the same approach, but reduces the space to O(nlogN) by using perfect hashing to represent the nodes at each level of the trie. In yfast tries, the space is further reduced to O(n) by only stori... |

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Citation Context ...eneralized Ajtai's proof to obtain an W( # loglogN) lower bound on time in this model for the problem of finding predecessors in a static set from a universe of size N. (Independent of our work, Xiao =-=[27]-=- has also improved this lower bound.) In [22], it was shown that for certain universe sizes, Ajtai's proof and its generalization in [21] also gives an W((logn) 1/3 ) lower bound on time. Furthermore,... |

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Citation Context ...t each memory cell can hold. Thus, lower bounds obtained in this model apply to all reasonable models of computation and give us insight into why certain problems are hard. 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... |

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Citation Context ... jumps and indirect addressing, as well as any finite set of AC 0 instructions (such as addition and shifts, but not multiplication or division). In this model, Andersson, Miltersen, Riis, and Thorup =-=[5]-=- proved that the time complexity of membership queries is Q( # logn/loglogn). Their algorithm uses O(n) words (of log 2 N bits each) and their lower bound holds even if 2 (logn) O(1) words are allowed... |

14 | Tables should be sorted (on random access machines
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Citation Context ...model is useful for proving the most generally applicable data structure lower bounds, it does not permit one to analyze the particular instructions necessary for these algorithms. Fich and Miltersen =-=[15]-=- have shown that, for the standard RAM model (which includes addition, multiplication, conditional jumps, and indirect addressing instructions, but not shifts, bitwise Boolean operations, or division)... |

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(Show Context)
Citation Context ...e membership queries, 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 =-=[18, 3, 7, 23]-=- can be time and space efficient. Specifically, using branching factor B # b/(1 + log 2 N), insertions, deletions, and membership and predecessor queries can be performed in O(logn/logB) steps using O... |

8 |
Probabilistic methods in graph theory
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Citation Context ...this reduction. We now follow through on this sketch, after stating a couple of preliminaries. Preliminaries The following form of the Chernoff-Hoeffding bound follows easily from the presentation in =-=[10]-=-. Proposition 10: Fix H #U with |H| # r|U | and let S #U with |S| = s be chosen uniformly at random. Then Pr[|H # S| # rs/4] # ( # 2/e 3/4 ) rss2 -rs/2 . The next result is a small modification and re... |

6 |
A good neighbor is hard to find
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(Show Context)
Citation Context ...rectly led to the development of our algorithm. This approach has also been used to obtain an W(loglogd/logloglogd) lower bound for the approximate nearest neighbour problem over the universe {0,1} d =-=[9]-=-. 2 Related Work The simplest model in which the dictionary and predecessor problems have been considered is the comparison model in which the only operations allowed that involve x are comparisons be... |

4 |
Sublogarithmic seaching without multiplications
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(Show Context)
Citation Context ...e membership queries, 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 =-=[18, 3, 7, 23]-=- can be time and space efficient. Specifically, using branching factor B # b/(1 + log 2 N), insertions, deletions, and membership and predecessor queries can be performed in O(logn/logB) steps using O... |

3 |
Faster deterministic sorting and seaching in linear space
- Andersson
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(Show Context)
Citation Context ...tegers from a universe of size N in n O(1) space and performs predecessor queries in O # min # loglogN/logloglogN, # logn/loglogn ## time. Using recent generic transformations of Andersson and Thorup =-=[4, 6]-=-, the algorithm can be made dynamic and the space improved to O(n). We also obtain matching lower bounds, improving Miltersen 's W( # loglogN) and W((logn) 1/3 ) lower bounds in the powerful communica... |

3 |
Exponential search trees for faster deterministic searching, sorting and priority queues in linear space
- Anderson, Thorup
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(Show Context)
Citation Context ...tegers from a universe of size N in n O(1) space and performs predecessor queries in O # min # loglogN/logloglogN, # logn/loglogn ## time. Using recent generic transformations of Andersson and Thorup =-=[4, 6]-=-, the algorithm can be made dynamic and the space improved to O(n). We also obtain matching lower bounds, improving Miltersen 's W( # loglogN) and W((logn) 1/3 ) lower bounds in the powerful communica... |

1 |
Dynamic pertect hashing: Upper and lower bounds
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(Show Context)
Citation Context ...h functions. Recently it was shown that hash functions of the form h(x) = ax mod 2 #log 2 N# div 2 #log 2 N#-r for r # #log 2 n#, suffice for implementing the two level perfect hashing data structure =-=[11, 13]-=-. Notice that the evaluation of such functions does not depend on constant time division instructions; a right shift suffices. These algorithms can be made dynamic with the same constant time for memb... |