The Online Event-Detection Problem
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Given a stream S = (s_1, s_2, ..., s_N), a ϕ-heavy hitter is an item s_i that occurs at least ϕ N times in S. The problem of finding heavy-hitters has been extensively studied in the database literature. In this paper, we study a related problem. We say that there is a ϕ-event at time t if s_t occurs exactly ϕ N times in (s_1, s_2, ..., s_t). Thus, for each ϕ-heavy hitter there is a single ϕ-event which occurs when its count reaches the reporting threshold ϕ N. We define the online event-detection problem (OEDP) as: given ϕ and a stream S, report all ϕ-events as soon as they occur. Many real-world monitoring systems demand event detection where all events must be reported (no false negatives), in a timely manner, with no non-events reported (no false positives), and a low reporting threshold. As a result, the OEDP requires a large amount of space (Omega(N) words) and is not solvable in the streaming model or via standard sampling-based approaches. Since OEDP requires large space, we focus on cache-efficient algorithms in the external-memory model. We provide algorithms for the OEDP that are within a log factor of optimal. Our algorithms are tunable: its parameters can be set to allow for a bounded false-positives and a bounded delay in reporting. None of our relaxations allow false negatives since reporting all events is a strict requirement of our applications. Finally, we show improved results when the count of items in the input stream follows a power-law distribution.
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