Approximate Range Emptiness in Constant Time for IoT Data Streams over Sliding Windows

Facilitating real-time query over massive IoT data streams becomes increasingly important nowadays, for that it can boost the performances of real-time network services significantly. Let delta = e(1), e(2), center dot center dot center dot, e(t), center dot center dot center dot represent an IoT data stream, where each element et arrives at time point t. In this paper, we consider the problem of how to support fast range emptiness querying over an IoT data stream delta in sliding window model with a space-efficient data structure, and we denote this problem as the (epsilon, L)-ARE-problem. To be more formally, subjected to the constraint of one-pass scan of stream delta, the main task of the (epsilon, L)-ARE-problem is to design a space-efficient data structure that is capable of always representing W(t, n), which are the n latest elements of stream delta until time point t (i.e., W(t, n) = e(max{1,t-n+1}), center dot center dot center dot, e(t-1), e(t)), and quickly answering an emptiness query of the form "W(t, n) boolean AND I = empty set?", with a false positive rate no larger than e, for any query interval I of length up to L. We design a space-efficient data structure D to solve the (epsilon, L)-ARE-problem and prove that D has constant time cost for querying an interval, inserting a stream element and evicting outdated elements. The efficiency is demonstrated with extensive simulation results as well.