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Arne Andersson

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13 papers
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13

FOCS Conference 1996 Conference Paper

Faster Deterministic Sorting and Searching in Linear Space

  • Arne Andersson

We present a significant improvement on linear space deterministic sorting and searching. On a unit-cost RAM with word size w, an ordered set of n w-bit keys (viewed as binary strings or integers) can be maintained in O(min{[/spl radic/(logn)][logn/logw+loglogn][logwloglogn]}) time per operation, including insert, delete, member search, and neighbour search. The cost for searching is worst-case while the cost for updates is amortized. As an application, n keys can be sorted in linear at O(n/spl radic/(logn)) worst-case cost. The best previous method for deterministic sorting and searching in linear space has been the fusion trees which supports updates and queries in O(logn/loglogn) amortized time and sorting in O(nlogn/loglogn) worst-case time. We also make two minor observations on adapting our data structure to the input distribution and on the complexity of perfect hashing.

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FOCS Conference 1996 Conference Paper

Static Dictionaries on AC 0 RAMs: Query Time Theta(sqrt(log n/log log n)) is Necessary and Sufficient

  • Arne Andersson
  • Peter Bro Miltersen
  • Søren Riis
  • Mikkel Thorup

In this paper we consider solutions to the static dictionary problem on AC/sup 0/ RAMs, i. e. random access machines where the only restriction on the finite instruction set is that all computational instructions are in AC/sup 0/. Our main result is a tight upper and lower bound of /spl theta/(/spl radic/log n/log log n) on the time for answering membership queries in a set of size n when reasonable space is used for the data structure storing the set; the upper bound can be obtained using O(n) space, and the lower bound holds even if we allow space 2/sup polylog n/. Several variations of this result are also obtained. Among others, we show a tradeoff between time and circuit depth under the unit-cost assumption: any RAM instruction set which permits a linear space, constant query time solution to the static dictionary problem must have an instruction of depth /spl Omega/(log w/log log to), where w is the word size of the machine (and log the size of the universe). This matches the depth of multiplication and integer division, used in the perfect hashing scheme by M. L. Fredman, J. Komlos and E. Szemeredi (1984).

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FOCS Conference 1995 Conference Paper

Sublogarithmic Searching without Multiplications

  • Arne Andersson

We show that a unit-cost RAM with word length w can maintain an ordered set of w-bit integers (or binary strings) under the operations search, insert, delete, nearest neighbour in O(/spl radic/(logn)) worst-case time and range queries in O(/spl radic/(logn)+size of output) worst-case time. The operations rely on AC/sup 0/ instructions only, thereby solving an open problem posed by Fredman and Willard. The data structure is simple. We also present a static data structure that can process a set of /spl Theta/O(logn) searches in O(lognloglogn) time.

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FOCS Conference 1994 Conference Paper

A New Efficient Radix Sort

  • Arne Andersson
  • Stefan Nilsson

We present new improved algorithms for the sorting problem. The algorithms are not only efficient but also clear and simple. First, we introduce Forward Radix Sort which combines the advantages of traditional left-to-right and right-to-left radix sort in a simple manner. We argue that this algorithm will work very well in practice. Adding a preprocessing step, we obtain an algorithm with attractive theoretical properties. For example, n binary strings can be sorted in /spl Theta/ (n log(B/(n log n)+2)) time, where B is the minimum number of bits that have to be inspected to distinguish the strings. This is an improvement over the previously best known result by Paige and Tarjan (1987). The complexity may also be expressed in terms of H, the entropy of the input: n strings from a stationary ergodic process can be sorted in /spl Theta/ (n log(1/H+1)) time an improvement over the result recently presented by Chen and Reif (1993). >

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FOCS Conference 1991 Conference Paper

Faster Uniquely Represented Dictionaries

  • Arne Andersson
  • Thomas Ottmann

The authors present a solution to the dictionary problem where each subset of size n of an ordered universe is represented by a unique structure, containing a (unique) binary search tree. The structure permits the execution of search, insert, and delete operations in O(n/sup 1/3/) time in the worst case. They also give a general lower bound, stating that for any unique representation of a set in a graph of, bounded outdegree, one of the operations search or update must require a cost of Omega (n/sup 1/3/) Therefore, the result sheds new light on previously claimed lower bounds for unique binary search tree representations. >

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