A geometric approach to approximate continuous k-median query

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We revisit the classic k-median problem in continuous distributed model. The rapid advance in electronic miniaturization, wireless communication and position technologies makes a significant contribution to pervasive applications of continuous distributed model. Data sets acquired in continuous distributed model are automatically and continuously updated, or even distributed over a wide area in typical cases. The sequence of k-median at each time stamp in the continuous distributed model forms a k-median series, which is called continuous k-median. Our main idea is to transform continuous k-median problem to continuous k-median query, which applies a selection operation on continuous k-median. Because the result of this selection is a subset of k-median series, time and communication efficiency in the continuous distributed model can be achieved. The continuous k-median query provides an insightful structure of data sets along time dimension and widely applied in various cases such as location-based services, sensor network monitor, and etc. In this paper, the time-efficiency of continuous k-median query in a central paradigm is first studied where an efficient indicator function is designed to suppress unnecessary re-evaluations. Then, communication-efficiency of continuous k-median query is addressed in a distributed paradigm where a geometric approach is applied to suppress unnecessary communications between nodes. Our approach to continuous k-median query distinguishes itself in two aspects. First, the indicator function is built on the aggregation distribution of data sets instead of prevailing safe region of individuals and time-efficiency can therefore be achieved. Second, a geometric approach is explored so that a single local node can trigger a re-valuation and therefore communication-efficiency can be obtained. Experiments are done to empirically demonstrate the time and communication efficiency of our approach on various data sets.