Given an array @math of @math elements and a value @math , a frequent item or @math -majority element is an element occurring in @math more than @math times. The @math -majority problem requires finding all of the @math -majority elements. In this paper we deal with parallel shared-memory algorithms for frequent items; we present a shared-memory version of the Space Saving algorithm and we study its behavior with regard to accuracy and performance on many and multi-core processors, including the Intel Phi accelerator. We also investigate a hybrid MPI OpenMP version against a pure MPI based version. Through extensive experimental results we prove that the MPI OpenMP parallel version of the algorithm significantly enhances the performance of the earlier pure MPI version of the same algorithm. Results also prove that for this algorithm the Intel Phi accelerator does not introduce any improvement with respect to the Xeon octa-core processor.
The problem of merging two data summaries naturally arises in a distributed or parallel setting, in which a data set is partitioned between two or among several data sets. The goal in this context is to merge two data summaries into a single summary which provides candidate frequent items for the union of the input data sets. In particular, in order for the merged summary to be useful, it is required that its size and error bounds are those of the input data summaries. A few years ago we designed an algorithm @cite_10 @cite_7 for merging in parallel counter-based data summaries which are the output of the @cite_18 algorithm.
@cite_18 @cite_10 @cite_7
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We consider a router on the Internet analyzing the statistical properties of a TCP IP packet stream. A fundamental difficulty with measuring traffic behavior on the Internet is that there is simply too much data to be recorded for later analysis, on the order of gigabytes a second. As a result, network routers can collect only relatively few statistics about the data. The central problem addressed here is to use the limited memory of routers to determine essential features of the network traffic stream. A particularly difficult and representative subproblem is to determine the top k categories to which the most packets belong, for a desired value of k and for a given notion of categorization such as the destination IP address. We present an algorithm that deterministically finds (in particular) all categories having a frequency above 1 (m+1) using m counters, which we prove is best possible in the worst case. We also present a sampling-based algorithm for the case that packet categories follow an arbitrary distribution, but their order over time is permuted uniformly at random. Under this model, our algorithm identifies flows above a frequency threshold of roughly 1 √nm with high probability, where m is the number of counters and n is the number of packets observed. This guarantee is not far off from the ideal of identifying all flows (probability 1 n), and we prove that it is best possible up to a logarithmic factor. We show that the algorithm ranks the identified flows according to frequency within any desired constant factor of accuracy. We present a deterministic parallel algorithm for the k-majority problem, that can be used to find in parallel frequent items, i.e. those whose multiplicity is greater than a given threshold, and is therefore useful to process iceberg queries and in many other different contexts of applied mathematics and information theory. The algorithm can be used both in the online (stream) context and in the offline setting, the difference being that in the former case we are restricted to a single scan of the input elements, so that verifying the frequent items that have been determined is not allowed (e.g. network traffic streams passing through internet routers), while in the latter a parallel scan of the input can be used to determine the actual k-majority elements. To the best of our knowledge, this is the first parallel algorithm solving the proposed problem. Copyright © 2011 John Wiley & Sons, Ltd.