  Consider the continuous distributed monitoring model in which $n$ distributed
nodes, receiving individual data streams, are connected to a designated server.
The server is asked to continuously monitor a function defined over the values
observed across all streams while minimizing the communication. We study a
variant in which the server is equipped with a broadcast channel and is
supposed to keep track of an approximation of the set of nodes currently
observing the $k$ largest values. Such an approximate set is exact except for
some imprecision in an $\varepsilon$-neighborhood of the $k$-th largest value.
This approximation of the Top-$k$-Position Monitoring Problem is of interest in
cases where marginal changes (e.g. due to noise) in observed values can be
ignored so that monitoring an approximation is sufficient and can reduce
communication.
  This paper extends our results from [IPDPS'15], where we have developed a
filter-based online algorithm for the (exact) Top-k-Position Monitoring
Problem. There we have presented a competitive analysis of our algorithm
against an offline adversary that also is restricted to filter-based
algorithms. Our new algorithms as well as their analyses use new methods. We
analyze their competitiveness against adversaries that use both exact and
approximate filter-based algorithms, and observe severe differences between the
respective powers of these adversaries.
