Random Subsets of the Interval and P2P Protocols

In this paper we compare two methods for generating finite families of random subsets according to some sequence of independent random variables ζ1, . . . , ζn distributed uniformly over the interval [0, 1]. The first method called uniform split uses ζi values straightforwardly to determine points of division of [0, 1] into subintervals. The second method called binary split uses ζi 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 1 n2 and that the variance of lengthes of obtained intervals in the second method is approximately 1 n2 ( 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.