Abstract
In this paper we compare two methods for generating finite families of random subsets according to some sequence of independent random variables ζ 1, ..., ζ n distributed uniformly over the interval [0,1]. The first method called uniform split uses ζ i values straightforwardly to determine points of division of [0,1] into subintervals. The second method called binary split uses ζ i only to perform subsequent divisions of already existing subintervals into exact halves. We show that the variance of lengthes of obtained intervals in the first method is approximately \(\frac{1}{n^2}\) and that the variance of lengthes of obtained intervals in the second method is approximately \(\frac{1}{n^2}(\frac{1}{\ln 2}-1)\).
The uniform split is used in the Chord peer-to-peer protocol while the binary split is used in the CAN protocol. Therefore our analysis applies to this protocols and shows that CAN has a better probabilistic properties than Chord. We propose also a simple modification of the Chord protocol which improves its statistical properties.
Partially supported by the EU within the 6th Framework Programme under contract 001907 (DELIS).
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Cichoń, J., Klonowski, M., Krzywiecki, Ł., Różański, B., Zieliński, P. (2007). Random Subsets of the Interval and P2P Protocols. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_30
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DOI: https://doi.org/10.1007/978-3-540-74208-1_30
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