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Issue No. 02 - February (2011 vol. 23)
ISSN: 1041-4347
pp: 297-306
Michael T. Goodrich , University of California, Irvine
David Eppstein , University of California, Irvine
In this paper, we study the straggler identification problem, in which an algorithm must determine the identities of the remaining members of a set after it has had a large number of insertion and deletion operations performed on it, and now has relatively few remaining members. The goal is to do this in o(n) space, where n is the total number of identities. Straggler identification has applications, for example, in determining the unacknowledged packets in a high-bandwidth multicast data stream. We provide a deterministic solution to the straggler identification problem that uses only O(d\log n) bits, based on a novel application of Newton's identities for symmetric polynomials. This solution can identify any subset of d stragglers from a set of n O(\log n)-bit identifiers, assuming that there are no false deletions of identities not already in the set. Indeed, we give a lower bound argument that shows that any small-space deterministic solution to the straggler identification problem cannot be guaranteed to handle false deletions. Nevertheless, we provide a simple randomized solution, using O(d\log n\log (1/\epsilon )) bits that can maintain a multiset and solve the straggler identification problem, tolerating false deletions, where \epsilon >0 is a user-defined parameter bounding the probability of an incorrect response. This randomized solution is based on a new type of Bloom filter, which we call the invertible Bloom filter.
Straggler identification, Newton's identities, Bloom filters, data streams.
Michael T. Goodrich, David Eppstein, "Straggler Identification in Round-Trip Data Streams via Newton's Identities and Invertible Bloom Filters", IEEE Transactions on Knowledge & Data Engineering, vol. 23, no. , pp. 297-306, February 2011, doi:10.1109/TKDE.2010.132
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