## 5 Discussion  
The traffic overhead footprint on the total network load and the need for special purpose cooperation from the network elements involved in network monitoring motivates approaches of network tomography as a fast and lightweight alternative to traditional techniques. In this review, we focused on two families of algebraic-based approaches, namely network coding and compressed sensing, which capitalize on concepts from linear algebra for building more advanced tomographic techniques. Network coding, although developed for different goals (i.e., improving the throughput of networks), can be exploited in network tomography due to the introduced topology-dependent correlations and information that are implicitly present in the obtained measurements. Classic tomography problems such as link loss inference and topology recovery can be revisited with approaches that leverage the network coding capabilities of the intermediate nodes in order to improve the estimation accuracy and to reduce the probing cost and the complexity of selecting measurement/monitoring paths. On the other hand, compressed sensing enables the recovering of sparse signals from lower-dimensional linear measurements (i.e., the inference of the sparse vector \(\mathbf{X}\) from measurements \(\mathbf{Y}=\mathbf{A}\mathbf{X}\), where the dimension of \(\mathbf{Y}\) is smaller than the dimension of \(\mathbf{X}\)). Furthermore, it can be applied in cases where measurements are noisy (noise is usually assumed to be Gaussian distributed) and hard/expensive to collect directly from the sources. Although it is a flexible and data-efficient method, its application has been restricted by the stringent assumption of sparsity. However, in the context of network tomography, this sparsity assumption is not so restrictive as in other fields and it is in fact appropriate, since often path-level performance characteristics (e.g., delays, packet losses, etc.) can be attributed to a few \"bottleneck\" links. Therefore, the unknown vector \(\mathbf{X}_{t}\) can be considered sparse with only a few nonzero elements (the components corresponding to the \"bottleneck\" links) and it can be recovered nearly perfectly with high probability from the ill-posed routing matrix (or measurement matrix in CS terminology) and measurement vector \(\mathbf{Y}_{t}\) based on CS theory [58; 59]. Figure 2 illustrates the classification of some of the examined tomographic methods based on the network tomography category in which they belong and the enabling technology they employ. It can be used for a quick overview of positioning of each included approach in the state-of-the-art and as a potential guidance for the novice reader who wants to start becoming acquainted with the field. Similarly, Table 1 summarizes the key features of the approaches. It can be used as a quick reference for each approach separately, providing succinctly the key features of each method while at the same time providing the overall bigger picture at a glance in terms of features. A more educated overview of the key ideas, strong points, and performance of the presented approaches is provided in Table 2, in a comparative manner when possible. This table can be used for choosing among the available approaches, weighing the pros and cons of the included techniques and identifying the most appropriate for more specific applications and operational objectives. Finally, Table 3 lists a selection of quantitative performance evaluation results, each in their relevant context, as compiled from the simulation-based evaluations presented in the respective original works that proposed them. This table can be used as a performance guide for selecting among the various approaches given performance operational requirements, i.e., the RMSE error, entropy, FPR and DR.