## 1 Introduction
### Related Work  
CS, which unifies sampling and compression, has been enthusiastically promoted and studied [12; 13]. It states that when the original signal is sparse itself or sparse on some basis, we can reconstruct it from compressed measurements. Making use of its advantages on simple encoding and efficient compressing, many researchers apply CS in WSN to achieve advanced compression compared with conventional compression algorithms [14; 15; 16]. CS could also be utilized for sensor selection; that is, only part of nodes will be chosen to transmit the product of its data with a random coefficient to the sink [17; 18; 19]. In the context that all nodes transmitted their data directly to the sink node in one hop, [PERSON] et al. [20] propose both centralized and decentralized algorithms for sensor selection under multi-variated noise condition. In [21], a tree-based algorithm named CDG is designed to reduce the payload falling on nodes close to the BS. In [22; 23], a WSN is partitioned into clusters. Sensor readings are sent to CHs and the CHs send the received data to the BS. However, the measurement matrices they adopt are all full Gaussian ones which consume more power than the sparse measurement matrix used in our work. In the other word, they do not combine the routing and nodes selection with the help of CS. PSO is a computational method that can be used to improve the optimization of candidate solutions through iteration. The way to adopt PSO for data routing [24] has been studied in several literatures [25; 26; 27]. [PERSON]. [25] proposes a novel bi-velocity discrete PSO approach which extends PSO from the continuous version to the binary or discrete domain to find routing paths for WSN. [PERSON] et al. [26] presents a PSO based method to find routings for the network with energy harvesting. Targeting on the problem that some of the nodes would be left out during cluster formation, ref. [27]introduces to apply the concepts of PSO and gravitational search algorithm to prevent residual nodes. All of these studies consider the application of PSO either in the clustering or in the routing, i.e., only one tier. And they do not consider the application of CS. To the best of our knowledge, there are few works besides the following literatures that have considered the application of CS into combination of data routing and nodes selection in WSNs, especially in UWSNs. In [28], the network is partitioned into clusters. Then each CH collects the sensor readings within its cluster and generates CS measurements to be forwarded directly to the base station. As a result, the overall CS measurement matrix at the base station is a block diagonal matrix. Later on, in [29], an improved method is introduced which allows the multi-hop routing of generated CS measurements through intermediate CHs. The authors generalize the two methods in [30]. Compared to our work, they obtain CS measurements in each CH and then forward to sink node while we postpone this operation to be done in the sink node to reduce the burden in CHs. On the other hand, while choosing multi-hop routing paths from the CH to the sink node, they use traditional tree-based algorithm while we design a PSO-optimization based scheme.Specifically targeting on three dimensional UWSNs as well, distributed multi-hop CS has been proposed by [PERSON] et al. in [31]. This algorithm randomly chooses a subset with size \(M\) of \(N\) nodes in the first place, then each chosen node finds a tour to the sink node. During the traveling of the message through the tour, each node computes the product of its sensing data and a random weighted coefficient and adds the product to the intermediate result it received. Finally the sink node obtains \(M\) measurements and reconstructs the whole data.Compared to our work, they ignore the noise and select the nodes which participate in sensing by Bernoulli generator.