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 $\delta=e_1,e_2,\cdots,e_t,\cdots$ represent an IoT data stream, where each element $e_t$ 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 \textbf{$(\varepsilon,L)$-$\text{ARE}$-problem}. To be more formally, subjected to the constraint of one-pass scan of stream $\delta$, the main task of the \textbf{$(\varepsilon,L)$-$\text{ARE}$-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., $W(t,n)=e_{{\rm{max}}\{1,t-n+1\}},\cdots,e_{t-1},e_t$), and quickly answering an emptiness query of the form "$W(t,n)\cap I=\emptyset?$", with a false positive rate no larger than $\varepsilon$, for any query interval $I$ of length up to ℒ. We design a space-efficient data structure \textbf{D} to solve the \textbf{$(\varepsilon,L)$-$\text{ARE}$-problem} and prove that \textbf{D} has constant time cost for querying an interval, inserting a stream element and evicting outdated elements.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.