We consider the problem of sparse matrix multiplication by the column row method in a distributed setting where the matrix product is not necessarily sparse. We present a surprisingly simple method for "consistent" parallel processing of sparse outer products (column-row vector products) over several processors, in a communication-avoiding setting where each processor has a copy of the input. The method is consistent in the sense that a given output entry is always assigned to the same processor independently of the specific structure of the outer product. We show guarantees on the work done by each processor, and achieve linear speedup down to the point where the cost is dominated by reading the input. Our method gives a way of distributing (or parallelizing) matrix product computations in settings where the main bottlenecks are storing the result matrix, and inter-processor communication. Motivated by observations on real data that often the absolute values of the entries in the product adhere to a power law, we combine our approach with frequent items mining algorithms and show how to obtain a tight approximation of the weight of the heaviest entries in the product matrix. As a case study we present the application of our approach to frequent pair mining in transactional data streams, a problem that can be phrased in terms of sparse @math -integer matrix multiplication by the column-row method. Experimental evaluation of the proposed method on real-life data supports the theoretical findings.
Mining frequent itemsets from transactional data streams is challenging due to the nature of the exponential explosion of itemsets and the limit memory space required for mining frequent itemsets. Given a domain of I unique items, the possible number of itemsets can be up to 2^I-1. When the length of data streams approaches to a very large number N, the possibility of an itemset to be frequent becomes larger and difficult to track with limited memory. The existing studies on finding frequent items from high speed data streams are false-positive oriented. That is, they control memory consumption in the counting processes by an error parameter @e, and allow items with support below the specified minimum support s but above [email protected] counted as frequent ones. However, such false-positive oriented approaches cannot be effectively applied to frequent itemsets mining for two reasons. First, false-positive items found increase the number of false-positive frequent itemsets exponentially. Second, minimization of the number of false-positive items found, by using a small @e, will make memory consumption large. Therefore, such approaches may make the problem computationally intractable with bounded memory consumption. In this paper, we developed algorithms that can effectively mine frequent item(set)s from high speed transactional data streams with a bound of memory consumption. Our algorithms are based on Chernoff bound in which we use a running error parameter to prune item(set)s and use a reliability parameter to control memory. While our algorithms are false-negative oriented, that is, certain frequent itemsets may not appear in the results, the number of false-negative itemsets can be controlled by a predefined parameter so that desired recall rate of frequent itemsets can be guaranteed. Our extensive experimental studies show that the proposed algorithms have high accuracy, require less memory, and consume less CPU time. They significantly outperform the existing false-positive algorithms.  Research in data stream algorithms has blossomed since late 90s. The talk will trace the history of the Approximate Frequency Counts paper, how it was conceptualized and how it influenced data stream research. The talk will also touch upon a recent development: analysis of personal data streams for improving our quality of lives.