Abstract.
We give a complete characterization of the complexity of the element distinctness problem for n elements of \(m\ge\log n\) bits each on deterministic and nondeterministic one-tape Turing machines. We present an algorithm running in time \(O(n^2m(m+2-\log n))\) for deterministic machines and nondeterministic solutions that are of time complexity \(O(nm(n + \log m))\). For elements of logarithmic size \(m = O(\log n)\), on nondeterministic machines, these results close the gap between the known lower bound \(\Omega(n^2\log n)\) and the previous upper bound \(O(n^2(\log n)^{3/2}(\log\log n)^{1/2})\). Additional lower bounds are given to show that the upper bounds are optimal for all other possible relations between m and n. The upper bounds employ hashing techniques, while the lower bounds make use of the communication complexity of set disjointness.
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Received: 23 April 2001, Published online: 2 September 2003
Holger Petersen: Supported by “Deutsche Akademie der Naturforscher Leopoldina”, grant number BMBF-LPD 9901/8-1 of “Bundesministerium für Bildung und Forschung”.
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Ben-Amram, A.M., Berkman, O. & Petersen, H. Element distinctness on one-tape Turing machines: a complete solution. Acta Informatica 40, 81–94 (2003). https://doi.org/10.1007/s00236-003-0125-8
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DOI: https://doi.org/10.1007/s00236-003-0125-8