## 2 Problem Statement  
The problem we address in this work is the sensor-stream reduction algorithms as a data-centric mechanism to meet deadlines in real-time applications. We consider the data-stream sampling technique to perform data reduction [36]. Since we use a sensor-stream algorithm for data reduction, the scope, and the problem itself can be defined as follows. Let us consider a WSN monitoring physical or environmental conditions, such as temperature, sound, vibration, pressure, motion or pollutants, at different locations. Such a system is represented by the diagram [37]:  
\[ \begin{CD}\mathcal{N}@>{P}>{}>\mathbf{V}^{*}@>{S}>{}>\mathbf{V}@>{\Psi}>{}> \mathbf{V}^{\prime}\\ @V{R^{*}}V{}V@V{R}V{}V\\ D^{*}\end{CD} \]  
This diagram illustrates the following behavior:  
* The ideal behavior denoted by \(\mathcal{N}\rightarrow\mathbf{V}^{*}\to D^{*}\), where \(\mathcal{N}\) denotes the environment and the process to be measured, \(P\) is the phenomenon of interest, with \(\mathbf{V}^{*}\) their space-temporal domain. If complete and uncorrupted observation was possible, we could devise a set of ideal rules \(R^{*}\) leading to ideal decisions \(D^{*}\). * The sensed behavior is denoted by \(\mathcal{N}\rightarrow\mathbf{V}^{*}\rightarrow\mathbf{V}\to D\). In this case, we have a set of \(s\) sensors \(S=(S_{1},\ldots,S_{s})\), each one providing measurements of the phenomenon and producing a report in the domain \(V_{i}\), with \(1\leq i\leq s\); all possible domain sets are denoted \(V=(V_{1},\ldots,V_{s})\). Using such information, we can conceive the set of rules \(R\) leading to the set of decisions \(D\). We consider \(\mathbf{V}\) to be a sensor-stream, due to its \"time series\" characteristics. * The reduced behavior is denoted by \(\mathcal{N}\rightarrow\mathbf{V}^{*}\rightarrow\mathbf{V}\rightarrow\mathbf{V}^ {\prime}\to D^{\prime}\). Dealing with \(\mathbf{V}\) may be too expensive in terms of, for instance, power, bandwidth, computer resources usage, and, specially, time delivery to meet the deadline requirements. Since the level of redundancy is not negligible in most situations, we can reduce this information volume. Sensor-stream reduction techniques are denoted by \(\mathbf{\Psi}\), and they transform the complete domain \(\mathbf{V}\) into the smaller one \(\mathbf{V}^{\prime}\). New rules that use \(\mathbf{V}^{\prime}\) are denoted by \(R^{\prime}\), and they lead to the set of decisions \(D^{\prime}\). Based on these behaviors, the problem addressed in this work can be stated as follows:  
**Problem definition**: _Given a sensor-stream behavior, how can we use a data-centric data reduction algorithm (\(\mathbf{\Psi}\)) to meet application deadlines? Moreover, what is the impact over the decisions \(D\), when we use the \(\mathbf{\Psi}\) reduction over \(\mathbf{V}\) generated by \(S\)?_  
To address the data-centric reduction problem in real-time applications, we consider the following assumptions:  
* **WSN topology**: The set of sensors \(S=(S_{1},\ldots,S_{n})\) is distributed in a squared area \(A=L\times L\). There is only one sink node located at \((0,0)\) on the left bottom corner. The density is kept constant and all nodes have the same hardware configuration. * **Routing protocol**: The network communication is based on a multihop shortest-path tree [38] as the routing protocol. To evaluate only the data-centric stream reduction performance, the tree is built just once before the traffic starts and the network is kept static. The build tree process is depict in Figure 1. First in (a), the sink node sends a flooding message requesting to build tree. After this, in (b), the nodes sets your father node considering the first message received in flooding process (it is considered that the first packet received represents the shortest path to sink). Finally, in (c), we have the complete tree mounted.