# Data Gathering Techniques in WSN: A Cross-Layer View
## 3 Compressed Sensing
This clustering approach strives to minimize the average compression ratio of all clusters by greedily selecting the set of nodes based on the compression ratio. It then tries to maximize the number of compressible clusters so as to determine the suitable transmission mode for each cluster using a mode threshold that is a function of the number of nodes and the number of hops from a cluster-head to a sink. To reduce the number of sensors involved in each CS measurement, [PERSON] et al. [71] propose a sparsest random scheduling scheme for compressive data gathering in large-scale wireless sensor networks (WSNs), leveraging the spatial-temporal properties in the sensory data. The central theme of this study is that the measurement matrix is designed based on the representation basis and sensory data and according to the sensor network requirements rather than the network environment. By combining compressed sensing and network coding in the data-gathering scheme, [PERSON] et al. [72] introduce a multi-hop topology in which the sink node adaptively adjusts the measurement formation according to the reconstruction of received measurements at each data-gathering epoch. The sink node dictates the data aggregation performed to balance the energy consumption among sensor nodes. [PERSON] et al. [73] exploit the CS paradigm for network tomography. Specifically, [PERSON]. [73] leverages the fact that, typically, only a small fraction of network entities such as links or nodes are responsible for anomalies or degradation in network performance, as a limited number of congested links can be responsible for significant delays or high packet drop rates, and suggests utilizing CS theory in order to identify these few entities based on end-to-end measurements. [PERSON] et al. [74] provide an analysis of the capacity and delay of data gathering with compressive sensing in wireless sensor networks. The paper considers a random topology where sensor nodes are randomly deployed in a region, for both single-sink and multi-sink scenarios, and characterizes the capacity and delay performance improvement that the CS paradigm can achieve for data gathering. In particular, for the single sink, a simple routing scheme for data gathering with CS is suggested, and a tight capacity in the order sense is presented. In particular, the suggested routing scheme with pipelining scheduling algorithm for data gathering shows that the proposed single-sink scheme can achieve a capacity gain of \(\Theta(\frac{n}{M})\) over the baseline transmission scheme, and the delay can be reduced by a factor of \(\Theta(\frac{\sqrt{mlogn}}{M})\), where \(M\) is the number of random projections required for reconstructing a snapshot, and \(n\) is the number of randomly deployed nodes. For the multi-sink case, their architecture shows that the regression capacity of data gathering with CS is \(\Theta(\frac{n\sqrt{n}W}{Mn_{d}\sqrt{n_{logn}}})\), and the per-session delay is \(\Theta(M\sqrt{\frac{n}{logn}})\), where \(W\) is the data rate. The number of sinks present in the network is \(n_{d}\), and the number of randomly selected source nodes is \(n_{s}\). They validate the theoretical results with simulations for the scaling laws of the capacity in both single-sink and multi-sink networks.