## 6 Future Research Directions  
In what follows we briefly outline a number of immediate research directions we plan to pursue.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Distance** & **Mean Comps** & **Messages** \\ \hline \(l_{2}\) & 1491 & 119,280 \\ \(l_{\infty}\) & 1388 & 111,040 \\ \(l_{1}\) & 1304 & 104,320 \\ \hline \hline \end{tabular}
\end{table}
Table 4: number of mean computations, messages, and crossings per norm for feature “febru” with threshold \(r=0.0025\), and stream dependent local constraint \(\delta_{j}\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Distance** & **Mean Comps** & **Messages** \\ \hline \(l_{2}\) & 7656 & 612,480 \\ \(l_{\infty}\) & 7377 & 590,160 \\ \(l_{1}\) & 6309 & 504,720 \\ \hline \hline \end{tabular}
\end{table}
Table 5: number of mean computations, messages, and crossings per norm for feature “ipo” with threshold \(r=0.0025\), and stream dependent local constraint \(\delta_{j}\).  
The local constraints introduced in this paper depend on history of a data stream at each node, and variations \(\|\mathbf{v}_{j}(t_{i+1})-\mathbf{v}_{j}(t_{i})\|\) over time contribute uniformly to local constraints. Attaching more weight to recent changes than to older ones may contribute to further improvement of monitoring process.  
Table 6 (borrowed from [10]) shows that in about 75% of instances (\(3034\) out of \(4006\)) the mean \(\mathbf{v}(t)\) is updated because of a single node violation. This observation naturally leads to the idea of clustering nodes, and independent monitoring of the node clusters equipped with a coordinator. The monitoring will become a two step procedure. At the first step node violations are checked in each node separately. If a node violates its local constraint, the corresponding cluster computes updated cluster coordinator. At the second step, violations of local constraints by coordinators are checked, and if at least one violation is detected the root is updated. Table 6 indicates that in most of the instances only one coordinator will be effected, and, since communication within cluster requires less messages, the two step procedure briefly described above has a potential to bring additional savings.  
We note that a standard clustering problem is often described as \"...finding and describing cohesive or homogeneous chunks in data, the clusters\" (see e.g., [17]). The monitoring data streams problem requires to assign to the same cluster \(i\) nodes \(\mathbf{N}_{i}\) so that the total change within cluster is minimized, _i.e._, nodes with **different** variations \(\mathbf{v}-\mathbf{v}(t_{j})\) that cancel out each other as much as possible should be assigned to the same cluster. Hence, unlike classical clustering procedures, one needs to combine \"dissimilar\" nodes together. This is a challenging new type of a difficult clustering problem.  
Realistically, verification of inequality \(f(\mathbf{x})-r>0\) should be conducted with an error margin (_i.e._, the inequality \(f(\mathbf{x})-r-\epsilon>0\) should be investigated, see [9]). A possible effect of an error margin on the required communication load is another direction of future research.  
\begin{table}
\begin{tabular}{c r} \hline \hline
**nodes** & **violations** \\ \hline \(1\) & 3034 \\ \(2\) & 620 \\ \(3\) & 162 \\ \(4\) & 70 \\ \(5\) & 38 \\ \(6\) & 26 \\ \(7\) & 34 \\ \(8\) & 17 \\ \(9\) & 5 \\ \(10\) & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 6: number of nodes simultaneously violating local constraints. for feature “bosnia” with threshold \(r=0.0025\), and \(l_{2}\) norm