## Binary search trees: How low can you go? (1996)

Venue: | SWAT'96, LNCS |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Fagerberg96binarysearch,

author = {Rolf Fagerberg},

title = {Binary search trees: How low can you go?},

booktitle = {SWAT'96, LNCS},

year = {1996},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

We prove that no algorithm for balanced binary search trees performing insertions and deletions in amortized time O(f(n)) can guarantee a height smaller than dlog(n + 1) + 1=f(n)e for all n. We improve the existing upper bound to dlog(n + 1) + log 2 (f(n))=f(n)e, thus almost matching our lower bound. We also improve the existing upper bound for worst case algorithms, and give a lower bound for the semi-dynamic case.

### Citations

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Citation Context ...es for keeping the height within some constant factor of this optimum while inserting and deleting in logarithmic time. For instance, in AVL-trees [1], the constant factor is 1:44, in red-black trees =-=[12]-=-, it is 2, etc. A natural question to ask is: Can we do better than a constant factor? Or, more fundamentally: What is the smallest height maintainable with a given update time? In a paper from 1976 [... |

153 |
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Citation Context ... with regular intervals, and spread the resulting work on the top tree incrementally over the updates between the splits. As in [7], the following lemma, versions of which are proved independently in =-=[9]-=- and [15], will allow us to control the size of the buckets. It concerns a combinatorial game on n real variables x 1 ; x 2 ; : : : ; xn , all initially zero. The game consists of a possibly infinite ... |

129 |
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(Show Context)
Citation Context ...orem 6], and is omitted. ut We note that the updates in the top tree after a bucket split, as well as the global rebuilding, must be done incrementally, a by now standard technique first described in =-=[17]-=-. The basic idea is to have two copies of the data structure. Updates are done on one copy, while the other is being restructured. An amount of restructuring work is done for each update, such that th... |

23 |
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Citation Context ...in a worst case setting is that several consecutive insertions each can trigger !(log n) work on the top tree (e.g. when buckets are split). Here, we indicate how to adapt an idea employed in [7] and =-=[15]-=- to the present setting, enabling us to sustain ffl(n) 2 o(1) in logarithmic worst case time. The idea is simple: Split buckets with regular intervals, and spread the resulting work on the top tree in... |

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(Show Context)
Citation Context ...pacing can vary. ! 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 T /(T ) To determine the gap into which we will insert, we use a variation (actually, a dual) of a rather fundamental lemma on density, given in =-=[8]-=-. For completeness, we give a proof here, although it follows the same lines as in [8]. We let [a; b] denote the integers a; a + 1; : : : ; b, and define its length l by l([a; b]) = b \Gamma a + 1. Le... |

19 |
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Citation Context ...nodes is dlog(n + 1)e. There is an abundance of schemes for keeping the height within some constant factor of this optimum while inserting and deleting in logarithmic time. For instance, in AVL-trees =-=[1]-=-, the constant factor is 1:44, in red-black trees [12], it is 2, etc. A natural question to ask is: Can we do better than a constant factor? Or, more fundamentally: What is the smallest height maintai... |

18 | A simple balanced search tree with O(1) worst-case update time
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(Show Context)
Citation Context ...ic case. Again, we believe the lower bound to be tight (this will follow if the lower bound in the fully dynamic case is tight, by a technique in [5]). Finally, it is an open question, also raised in =-=[11]-=-, whether a height of dlog(n + 1)e + O(1) can be maintained in \Theta(1) worst case rebalancing time. As mentioned in the introduction, an amortized solution is given in [6]. Worst case time O(log n) ... |

17 |
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Citation Context ...results in a worst case setting is that several consecutive insertions each can trigger !(log n) work on the top tree (e.g. when buckets are split). Here, we indicate how to adapt an idea employed in =-=[7]-=- and [15] to the present setting, enabling us to sustain ffl(n) 2 o(1) in logarithmic worst case time. The idea is simple: Split buckets with regular intervals, and spread the resulting work on the to... |

17 |
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Citation Context ...tion occurs, the partial rebuilding can be viewed as the even redistribution of the occupied slots on some segment of the lowest level (hence, it can be seen as a smooth list labeling algorithm---see =-=[8, 10, 18]-=- for more on this). In our tree, we group the buckets into groups of size \Theta(1=ffl), and use the algorithm on each group with a small, constant ~ffl (unrelated to the global ffl), in a way to be d... |

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9 |
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Citation Context ...], it is 2, etc. A natural question to ask is: Can we do better than a constant factor? Or, more fundamentally: What is the smallest height maintainable with a given update time? In a paper from 1976 =-=[16]-=-, Maurer et al. present a search tree, the k-neighbor tree, with a maximal height of c \Delta log(n), where c can be chosen arbitrarily close to 1. The update time is O(log n), with the constant depen... |

6 |
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(Show Context)
Citation Context ...e constant depending on c. Amazingly, no further progress on the problem seems to have been made until around 1990, when Lai and Andersson addressed it in their theses [3, 13] and in resulting papers =-=[2, 4, 5, 6, 14]-=-, some of which are with additional collaborators. They give a series of schemes for maintaining height dlog(n + 1)e + 1, using amortized O(log 2 n) rebalancing work per update for the simplest, impro... |

5 |
E cient Search Trees
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(Show Context)
Citation Context ...update time is O(log n), with the constant depending on c. Amazingly, no further progress on the problem seems to have been made until around 1990, when Lai and Andersson addressed it in their theses =-=[3, 13]-=- and in resulting papers [2, 4, 5, 6, 14], some of which are with additional collaborators. They give a series of schemes for maintaining height dlog(n + 1)e + 1, using amortized O(log 2 n) rebalancin... |

4 | Optimal bounds on the dictionary problem
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(Show Context)
Citation Context ...e constant depending on c. Amazingly, no further progress on the problem seems to have been made until around 1990, when Lai and Andersson addressed it in their theses [3, 13] and in resulting papers =-=[2, 4, 5, 6, 14]-=-, some of which are with additional collaborators. They give a series of schemes for maintaining height dlog(n + 1)e + 1, using amortized O(log 2 n) rebalancing work per update for the simplest, impro... |

4 | Comparison-efficient and write-optimal searching and sorting
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(Show Context)
Citation Context ...he update time in search trees is logarithmic, as this is the time spent on searching. In this case, the result above only allows constant ffl, in contrast to the amortized results in section 3.1 and =-=[6]-=-, where ffl(n) 2 o(1) is possible. The problem with using the techniques of these results in a worst case setting is that several consecutive insertions each can trigger !(log n) work on the top tree ... |

4 |
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(Show Context)
Citation Context ...update time is O(log n), with the constant depending on c. Amazingly, no further progress on the problem seems to have been made until around 1990, when Lai and Andersson addressed it in their theses =-=[3, 13]-=- and in resulting papers [2, 4, 5, 6, 14], some of which are with additional collaborators. They give a series of schemes for maintaining height dlog(n + 1)e + 1, using amortized O(log 2 n) rebalancin... |

4 |
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- 1993
(Show Context)
Citation Context ...tion occurs, the partial rebuilding can be viewed as the even redistribution of the occupied slots on some segment of the lowest level (hence, it can be seen as a smooth list labeling algorithm---see =-=[8, 10, 18]-=- for more on this). In our tree, we group the buckets into groups of size \Theta(1=ffl), and use the algorithm on each group with a small, constant ~ffl (unrelated to the global ffl), in a way to be d... |

3 |
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(Show Context)
Citation Context ...e constant depending on c. Amazingly, no further progress on the problem seems to have been made until around 1990, when Lai and Andersson addressed it in their theses [3, 13] and in resulting papers =-=[2, 4, 5, 6, 14]-=-, some of which are with additional collaborators. They give a series of schemes for maintaining height dlog(n + 1)e + 1, using amortized O(log 2 n) rebalancing work per update for the simplest, impro... |