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Improved Bounds for Finger Search on a RAM

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Abstract

We present a new finger search tree with O(loglogd) expected search time in the Random Access Machine (RAM) model of computation for a large class of input distributions. The parameter d represents the number of elements (distance) between the search element and an element pointed to by a finger, in a finger search tree that stores n elements. Our data structure improves upon a previous result by Andersson and Mattsson that exhibits expected O(loglogn) search time by incorporating the distance d into the search time complexity, and thus removing the dependence on n. We are also able to show that the search time is O(loglogd+ϕ(n)) with high probability, where ϕ(n) is any slowly growing function of n. For the need of the analysis we model the updates by a “balls and bins” combinatorial game that is interesting in its own right as it involves insertions and deletions of balls according to an unknown distribution.

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Notes

  1. A density μ is regular if there are constants b 1,b 2,b 3,b 4 such that μ(x)=0 for x<b 1 or x>b 2, and μ(x)≥b 3>0 and |μ′(x)|≤b 4 for b 1≤x≤b 2.

  2. Throughout the paper, we say that an event E occurs with high probability if Pr[E]=1−o(1).

  3. Whenever H(n) refers to height, R(n) refers to out-degree, and I(n) refers to number of equal parts, we mean ⌈H(n)⌉, ⌈R(n)⌉, and ⌈I(n)⌉, respectively.

  4. Actually, \(\frac{P_{T}(n_{1})}{r_{1} n_{1}} + WI_{T}(n_{1}(1+r_{1}))\) work per insertion, and \(\frac{P_{T}(n_{1})}{r_{1} n_{1}} + WD_{T}(n_{1}(1+r_{1}))\) work per deletion [27]. As it is also shown in [27] and adopted in our case, any \(r_{1} <\frac {1}{3}\) suffices for the global rebuilding technique.

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Acknowledgements

We are indebted to the anonymous referees for their valuable comments that improved considerably the presentation of the paper, as well as for their constant insistence and patience which helped us to resolve several subtle points and thus improving the quality of the paper. We would also like to thank the editor for his persistence in clarifying various involved issues and for his guidance during the whole revision process.

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Authors and Affiliations

  1. Department of Information and Communication Systems Engineering, University of the Aegean, Karlovassi, Samos, 83200, Greece

    Alexis Kaporis

  2. Department of Computer Engineering and Informatics, University of Patras, 26500, Patras, Greece

    Christos Makris, Athanasios Tsakalidis & Christos Zaroliagis

  3. Computer Technology Institute, Patras University Campus, N. Kazantzaki Str, 26504, Patras, Greece

    Christos Makris, Athanasios Tsakalidis, Kostas Tsichlas & Christos Zaroliagis

  4. Department of Informatics, Ionian University, Corfu, Greece

    Spyros Sioutas

  5. Department of Informatics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Kostas Tsichlas

Authors
  1. Alexis Kaporis
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  2. Christos Makris
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  3. Spyros Sioutas
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  4. Athanasios Tsakalidis
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  5. Kostas Tsichlas
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  6. Christos Zaroliagis
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Corresponding author

Correspondence to Kostas Tsichlas.

Additional information

This work was partially supported by the Future and Emerging Technologies Unit of EC (IST priority—6th FP), under contracts no. IST-2002-001907 (integrated project DELIS) and no. FP6-021235-2 (project ARRIVAL), and by the Carathéodory project of the University of Patras.

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Kaporis, A., Makris, C., Sioutas, S. et al. Improved Bounds for Finger Search on a RAM. Algorithmica 66, 249–286 (2013). https://doi.org/10.1007/s00453-012-9636-4

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