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Figure 9.9. Finding the minimum enclosing circle for six application nodes [12]. The base station marked as a triangle is placed at the center of the smallest disk that contains all application nodes (the small circles).
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algorithm is further extended for a two-tiered WSN where special application nodes are designated as cluster heads [12]. The application node interfaces the cluster with the base station. The minimum enclosing circle is thus found for the application nodes as shown in Figure 9.9, redrawn from reference 12. Although the time complexity for these approaches are quite promising, they imply a centralized network management strategy, including routing and/or MAC, which may not be desirable particularly in large-scale settings. Positioning Multiple Base-Stations. Multi-base-station positioning is even more challenging given the higher scale and the fact that individual sensors can select among multiple destinations to which they send the data. The positioning problem is typically de ned as the optimal layout for a known number of base stations in order to maximize some performance metric such as total communication energy and throughput [11] or area coverage [44]. In some cases, the number of base stations may not be known in advance, and thus the optimal number and location of base stations are to be found [40]. In the context of wireless sensor and actor networks, the base stations (actors) need to be positioned for maximized area coverage and reduced data delivery latency [44]. In general, the complexity of the multi-base-station positioning problem varies based on the planned network architecture. When a at network topology is pursued, the problem stays NP-complete [7]. The same applies to clustered networks in which some sensors are designated as cluster heads forming a two-tier topology. However, when the nodes are grouped into clusters that are led by the individual base stations, the complexity depends on the order of the network clustering and base-station positioning procedures. If the sensors are assigned to base stations prior to placing or nalizing the positions of base stations, the scope of the problem becomes local to the individual clusters and concerns only each base station independently from the others [40]. In other words, the problem becomes similar to a single base-station positioning. However, if the base-station positioning precedes the network clustering, the complexity remains NP-complete [11].
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NODE POSITIONING FOR INCREASED DEPENDABILITY OF WIRELESS SENSOR NETWORKS
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To counter the high complexity, most published approaches try to limit the search space (i.e., the cardinality of the set of candidate locations) in the hope of converging to some locations that achieve near-optimal performance. For example, in reference 41, candidate locations are determined a priori by the application designers. Meanwhile, in reference 11 the feasible set is restricted to where sensors are. The latter case also applies when the base-station positioning is combined with the network clustering scheme. For example, it may be desired for the base stations to be placed so that each sensor would reach a base station in at most K hops. By deciding to place base stations next to some sensors, the positioning/clustering problem in that case becomes nding the K-dominating set that has some polynomial time solution [45]. It is worth noting that if the number of base stations is xed, the problem stays NP-complete. Possible approaches for the multi-base-station positioning include approximation algorithms [11] and integer programming [41]. To maximize the achievable rate of collecting sensor s data, the base-station positioning problem is solved by using two approximation strategies, namely, greedy and local search [11]. In the greedy algorithm the base-station position is restricted to the location of sensors and then the base stations are individually placed in an arbitrary order for increased data rate. The local search starts with a random con guration of base stations and is then followed by a search for a better layout that boosts the data rate. An example for local search for four base stations on a 10 10 grid with a sensor transmission radius of 2.2 units is shown in Figure 9.10, which is redrawn from [11]. On the other hand, the Integer Linear Programming (ILP) formulation proposed in reference 41 is geared for splitting the data routing load among the sensors as evenly as possible. The objective function is to minimize the maximum energy consumption at the individual sensors while minimizing the total communication energy. The constraints of the ILP formulation include a bound on the total energy consumed by a node in a data collection round and a restriction on the candidate base-station location to be picked from a set of predetermined positions. Other constraints are also speci ed to ensure a balanced
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