## Dynamic Ordered Sets with Exponential Search Trees (2001)

Venue: | Combination of results presented in FOCS 1996, STOC 2000 and SODA |

Citations: | 26 - 1 self |

### BibTeX

@INPROCEEDINGS{Andersson01dynamicordered,

author = {Arne Andersson and Mikkel Thorup},

title = {Dynamic Ordered Sets with Exponential Search Trees},

booktitle = {Combination of results presented in FOCS 1996, STOC 2000 and SODA},

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully-dynamic linear space data structures. This leads to an optimal bound of O ( √ log n/log log n) for searching and updating a dynamic set of n integer keys in linear space. Here searching an integer y means finding the maximum key in the set which is smaller than or equal to y. This problem is equivalent to the standard text book problem of maintaining an ordered set (see, e.g., Cormen, Leiserson, Rivest, and Stein: Introduction to Algorithms, 2nd ed., MIT Press, 2001). The best previous deterministic linear space bound was O(log n/log log n) due Fredman and Willard from STOC 1990. No better deterministic search bound was known using polynomial space.

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Citation Context ... bound is best possible. We achieve this by introducing a new kind of search trees, called exponential search trees, illustrated in Figure 1.1. The lower bound follows from a result of Beame and Fich =-=[7]-=-. It shows that even if we just want to support insert and predecessor operations in polynomial space, one of these two operations have a worst-case bound of Ω( � log n/ log log n), matching our commo... |

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Citation Context ...alized to finger searching and string searching, providing optimal results for both in terms of n. ∗ This paper combines results presented by the authors at the 37th FOCS 1996 [2], the 32nd STOC 2000 =-=[5]-=-, and the 12th SODA 2001 [6] 1s1 Introduction 1.1 The Textbook Problem Maintaining a dynamic ordered set is a fundamental textbook problem (see, e.g., [13, Part III]). Starting from an empty set X, th... |

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Citation Context ...ch, or look-up, identifies the key to be deleted, if any. 1.3 Model of computation Our algorithms run on a RAM that reflects what we can program in standard imperative programming languages such as C =-=[23]-=-. The memory is divided into addressable words of length W . Addresses are themselves contained in words, so W ≥ log n. Moreover, we have a constant number of registers, each with capacity for one wor... |

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Citation Context ...nistic dynamic search time measured in terms of the universe size U, that is, when all keys are in [U]? The current O( � log n/ log log n) bound is only optimal in terms of the number n of keys. From =-=[29]-=- we have a lower bound of Ω(log log U). Using randomized hashing, this lower bound is matched by van Emde Boas’ data structure [26, 34, 35]. Can this be derandomized? In log log U Corollary 4, we prov... |

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Citation Context ...l. Our results are generalized to finger searching and string searching, providing optimal results for both in terms of n. ∗ This paper combines results presented by the authors at the 37th FOCS 1996 =-=[2]-=-, the 32nd STOC 2000 [5], and the 12th SODA 2001 [6] 1s1 Introduction 1.1 The Textbook Problem Maintaining a dynamic ordered set is a fundamental textbook problem (see, e.g., [13, Part III]). Starting... |

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Citation Context ... of O( √ log n/ log log n) for dynamic integer searching. We note that this also provides the best bounds for the simpler problem of membership and look-up queries if one wants updates in n o(1) time =-=[21]-=-. Our results also extend to optimal finger search with constant finger update, and to optimal string searching. 1.4 Model of computation Our algorithms runs on a RAM, which models what we program in ... |

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Citation Context ...log log n)(log log log n)) and sorting in O(n(log log n)(log log log n)) time. However, exponential search trees are not used in Han’s recent deterministic O(n log log n) time sorting in linear space =-=[22]-=- or in Thorup’s [32] corresponding priority queue with O(log log n) update time. Since (1) cannot give bounds below O(log log n) per key, it seems that the role of exponential search trees is played o... |

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Citation Context ...joining and splitting nodes. By locally we mean that the joining and splitting is done just by joining and splitting the children sequences. This type of data structure is for example used by Willard =-=[37]-=- to obtain a worst-case version of fusion trees. One problem with the previous definition of exponential search trees is that the criteria for when subtrees are too large or too small depend on their ... |

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Citation Context ...s, which is obtained below using a combination of techniques. We will use a tabulation technique for the lower levels of the exponential search tree, and a scheduling idea of Levcopoulos and Overmars =-=[27]-=- for the upper levels. 5.1 Constant update cost for small trees on the lower levels In this subsection, we will consider small trees induced by lower levels of the multiway tree from Theorem 24. One p... |

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Citation Context ...ince (1) cannot give bounds below O(log log n) per key, it seems that the role of exponential search trees is played out in the context of integer sorting and priority queues. Bender, Cole, and Raman =-=[8]-=- have used the techniques to derive worst-case efficient cache-oblivious algorithms for several data structure problem. This nicely highlights that the exponential search trees themselves are not rest... |

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Citation Context ...og n)) and sorting in O(n(log log n)(log log log n)) time. However, exponential search trees are not used in Han’s recent deterministic O(n log log n) time sorting in linear space [22] or in Thorup’s =-=[32]-=- corresponding priority queue with O(log log n) update time. Since (1) cannot give bounds below O(log log n) per key, it seems that the role of exponential search trees is played out in the context of... |

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Citation Context ...the lower bound. (log log U)2 log log log U ). AC 0 RAM Another interesting open problem is to find the complexity for searching with standard, or even non-standard, AC 0 operations? Andersson et.al. =-=[3]-=-, have shown that even if we allow non-standard AC 0 operations, the exact complexity of membership queries is Θ( � log n/ log log n). This contrast the situation at the RAM, where we can get down to ... |

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Citation Context ...been applied in other contexts. Variants of exponential search trees have been instrumental in many of the previous strongest results on deterministic linear integer space sorting and priority queues =-=[2, 31, 5, 21]-=-. Here a priority queue is a dynamic set for which we maintain the minimum element. When first introduced by Andersson [2], they provided the then strongest time bounds of O( √ log n) for priority que... |

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Citation Context ... in Corollary 21, with the same restructuring cost of O(log log n) per update. The only difference is that we change the search algorithm from the proof of Lemma 20. Applying an idea of Chen and Reif =-=[11]-=-, we replace the binary search for the longest matching (distinguishing) prefix by an exponential-and-binary search. Then, at each node in the exponential search tree, the search cost will decrease fr... |

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Citation Context ... pointer as the finger. The finger delete is just a regular delete as defined above, i.e. we are given a pointer to the key to be deleted. In the comparison-based model of computation Dietz and Raman =-=[14]-=- have provided optimal bounds, supporting finger searches in O(log q) time while supporting finger updates in constant time. Brodal et al. [9] managed to match these results on a pointer machine. In t... |

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Citation Context ...ft, and bit-wise Boolean operations are all AC 0 operations. On the other hand, multiplication is not. Fredman and Willard’s own techniques [18] were heavily based on multiplication, but, as shown in =-=[4]-=- they can be implemented with AC 0 operations if we allow some non-standard operations that are not part of the usual instruction set. However, as mentioned previously, here we only allow standard ope... |

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Citation Context ...log n/ log W, log log n · }). The last bound should be compared log log log U with van Emde Boas’ bound of O(log log U) [34, 35] that requires randomization (hashing) in order to achieve linear space =-=[26]-=-. 1.6 AC 0 operations As an additional challenge, Fredman and Willard [18] asked how quickly we can search on a RAM if all the computational instructions are AC 0 operations. A computational instructi... |

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Citation Context ...he second author [32] has recently used our Proposition 15 in a general reduction from priority queue to sorting, providing a priority queue whose update cost is the per key cost of sorting. Also, he =-=[33]-=- has recently used Theorem 8 in a space efficient solution to dynamic stabbing, i.e., the problem of maintaining a dynamic set of intervals where the query is to find an interval containing a given po... |

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Citation Context ...tic AC 0 dictionary with membership queries in time t, then in polynomial time and space, we can build a static search structure with operation time O(mini{it + log n/i}). In addition, Brodnik et.al. =-=[10]-=- have shown that such a static dictionary, using only standard AC 0 operations, can be built with membership queries in time t = O((log n) 1/2+o(1) ). We get the desired static search time by setting ... |

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Citation Context ...he comparison-based model of computation Dietz and Raman [14] have provided optimal bounds, supporting finger searches in O(log q) time while supporting finger updates in constant time. Brodal et al. =-=[9]-=- managed to match these results on a pointer machine. In this paper we present optimal bounds on the RAM; namely O( � log q/ log log q) for finger search with constant time finger updates. Also, we pr... |

8 |
Fast integer sorting in linear space
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(Show Context)
Citation Context ...been applied in other contexts. Variants of exponential search trees have been instrumental in many of the previous strongest results on deterministic linear integer space sorting and priority queues =-=[2, 31, 5, 21]-=-. Here a priority queue is a dynamic set for which we maintain the minimum element. When first introduced by Andersson [2], they provided the then strongest time bounds of O( √ log n) for priority que... |

6 |
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Citation Context ...cally ordered, we can apply classical non-comparison based techniques such as radix sort and hashing. Historically, radix sort dates back at least to 1929 [12] and hashing dates back at least to 1956 =-=[15]-=-, whereas the focus on general comparison based methods only date back to 1959 [16]. In this paper, we consider the above basic data types of integers and floating point numbers. Our main result is th... |

6 |
Planar point location in sublogarithmic time
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Citation Context ...ture problem. This nicely highlights that the exponential search trees themselves are not restricted to integer domains. It just happens that our applications in this paper are for integers. Patrascu =-=[28]-=- has very recently used exponential search trees to get the first sublogarithmic query time for planar point location. Theorem 8 provides a general tool for maintaining balance in multiway trees. Thes... |

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2 |
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Citation Context ... such data types, represented as lexicographically ordered, we can apply classical non-comparison based techniques such as radix sort and hashing. Historically, radix sort dates back at least to 1929 =-=[12]-=- and hashing dates back at least to 1956 [15], whereas the focus on general comparison based methods only date back to 1959 [16]. In this paper, we consider the above basic data types of integers and ... |