  We introduce 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. The straggler identification problem has
applications, for example, in determining the set of 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 and is
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 show that
there is 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.
