IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 24, DECEMBER 15, 2013 6355 Joint Sensor Selection and Multihop Routing for Distributed Estimation in Ad-hoc Wireless Sensor Networks Santosh Shah, Student Member, IEEE, and Baltasar Beferull-Lozano, Senior Member, IEEE Abstract—This paper considers the problem of power-efficient distributed estimation of vector parameters related to localized phenomena so that both sensor selection and routing structure in a Wireless Sensor Network (WSN) are jointly optimized to obtain the best possible estimation performance at a given querying node, for a given total power budget. First, we formulate our problem as an optimization problem and show that it is an NP-Hard problem. Then, we design two algorithms: a Fixed-Tree Relaxation-Based Algorithm (FTRA) and a very efficient Iterative Distributed Algo- rithm (IDA) to optimize the sensor selection and routing structure. We also provide a lower bound for our optimization problem and show that our IDA provides a performance that is close to this bound, and it is substantially superior to the previous approaches presented in the literature. An important result from our work is the fact that because of the interplay between communication cost and estimation gain when fusing measurements from different sensors, the traditional Shortest Path Tree (SPT) routing struc- ture, widely used in practice, is no longer optimal. To be specific, our routing structure provides a better trade-off between the overall power efficiency and estimation accuracy. Comparing to more conventional sensor selection and fixed routing algorithms, our proposed algorithms yield a significant amount of energy saving for the same estimation accuracy. Index Terms—Distributed estimation, joint sensor selection and routing, lower bound, NP-hard, vector parameter estimation. I. INTRODUCTION W IRELESS SENSOR NETWORKs (WSNs) can provide several services in very important applications for the society. Among the various existing constraints in the design of Manuscript received December 24, 2012; revised May 04, 2013 and Au- gust 08, 2013; accepted September 22, 2013. Date of publication October 04, 2013; date of current version November 14, 2013. The associate editor coordi- nating the review of this manuscript and approving it for publication was Prof. Huaiyu Dai. This work was supported by the Spanish MEC grants TEC2010- 19545-C04-04 “COSIMA,” CONSOLIDER-INGENIO 2010 CSD2008-00010, “COMONSENS”, the European STREP project “HYDROBIONETS” grant no. 287613 within the FP7 Framework Programme, and by a Telefonica Chair. Part of the material in this paper was presented at the IEEE International Conference on Distributed Computing in Sensor Systems, Hangzhou, China, May 2012, where [4] was the Best Student Paper Award. S. Shah is with the Group of Information and Communication Systems, Insti- tuto de Robótica y Tecnologías de la Información y las Comunicaciones, Univer- sidad de Valencia, 46980, Paterna (Valencia), Spain (e-mail: Santosh.Shah@uv. es). B. Beferull-Lozano is with the Group of Information and Communication Systems, Instituto de Robótica y Tecnologías de la Información y las Comuni- caciones, Universidad de Valencia, 46980, Paterna (Valencia), Spain, and also with the Department of Information and Communication Technology and the Center for Integrated Emergency Management, University of Agder, NO-4898, Grimstad, Norway (e-mail: Baltasar.Beferull@uv.es). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2013.2284486 WSNs a very important one is usually the power efficiency since sensors are usually dependent on a limited battery power. Since radio is the main power consumption in a sensor, transmission and reception of the information should be limited as much as possible. Given a certain task to be performed, selecting (ac- tivating) a proper subset of sensors from which information is taken and optimizing the routing structure can lead to very im- portant power savings. A fundamental problem that arises in WSN applications is the distributed vector parameter estimation associated to the localized phenomena, such as the energy captured by acoustic amplitude sensors where the sound source is localized in a certain spatial point [1], the direction-of-arrival sensors for localization [2], or any other locally generated diffusive source. A commonly used network scenario for distributed estimation involves a set of spatially distributed sensors linked to a fusion center (sink node) using direct wireless transmissions. How- ever, intuitively, direct wireless transmission may not be power efficient taking into account a stringent power budget, thus may not be convenient in WSNs. Because of the enforcement of power efficiency, only a subset of sensors should be selected and each sensor should fuse all other measurements that are re- ceived from its child sensors together with its own measurement in order to perform the vector parameter estimation, and then send only one flow of data to its parent sensor in the chosen routing structure (see Fig. 1(b) and Fig. 1(c)). We call this scheme an Estimate-and-Forward (EF) [3]–[5]. On the other hand, a scheme, in which we simply forward the measurements to the sink node, is denoted as a Measure-and-Forward (MF) [3] (see Fig. 1(a)). Given a WSN with a certain underlying connectivity graph, a certain querying (sink) node, and a localized source target (see Fig. 1), we consider the problem of jointly optimizing the sensor selection and routing structure in order to minimize the esti- mation distortion (estimation error) under a given certain total power budget. On the other hand, in this paper, we show that the Shortest Path Tree (SPT) is not the optimal routing structure when both the total communication cost and estimation accu- racy are taken into account. Fig. 1 illustrates a simple example where the SPT based only on Communication Cost (SPT-CC) is not the optimal routing structure when both the EF and MF schemes are used. In this particular case of Fig. 1, three sen- sors are chosen, the same estimation accuracy is obtained, how- ever taking into account that the communication cost is increasing with a certain power of the distance, the total com- munication cost corresponding to the SPT-CC routing structures 1053-587X © 2013 IEEE