The complexity of data aggregation in static and dynamic wireless sensor networks

Abstract

The key feature of wireless sensor networks is to aggregate data collected by individual sensors in an energy efficient manner. We consider two techniques to save energy. The first one is to avoid collisions due to simultaneous transmissions among neighboring nodes. Second, when a node receives data from multiple neighbors, it aggregates these with its own data. Then, one transmission is sufficient to transmit all consolidated data to another neighbor. If the overall delay has to be kept as low as possible, scheduling sensors to avoid collisions while aggregating data becomes challenging. The contribution of this paper is threefold. First, we give tight bounds for the complexity of data aggregation in static networks. In more details, we show that the problem remains NP-complete when the graph is of degree at most three. As it is trivial to solve the problem in static graphs of degree at most two, our result implies that the problem is intrinsically difficult for any practical setting. Second, we investigate the complexity of the same problem in a dynamic network, that is, a network whose topology can evolve through time. In the case of dynamic networks, we show that the problem is NP-complete even in the case where the graph is of degree at most two (and it is trivial to solve the problem when the graph is of degree at most one). Third, we give the first lower and upper bounds for the minimum data aggregation time in a dynamic graph. We observe that even in a well-connected evolving graphs, the optimal solution cannot be found by a distributed algorithm or by a centralized algorithm that does not know the future. Thus we finally give the first approximation algorithm (centralized that knows the future) whose approximation factor is T (n − 1) if there exists a bound T such that there is a journey (a path in a dynamic graph) for all pairs of nodes in every time interval [t, t + T ].

Computing Parameters of Sequence-Based Dynamic Graphs

“…all nodes can recurrently reach each other through temporal paths called journeys), corresponding to class C 5 in the hierarchy of [12]. This property is very general, and it is used (implicitly or explicitly) in a number of recent studies addressing distributed problems in highly-dynamic environments [5][6][7]15]. Interestingly, this property was considered more than three decades ago by Awerbuch and Even [2].…”

“…A naive algorithm computes all of the rows of the composition hierarchy H using the fact that each element G (i,j) can be obtained from the composition of two elements below it in H, i.e. (4,6) in Figure 1. Since there are O(δ 2 ) elements in H, the total number of composition operations is O(δ 2 ).…”

“…Note that when the walk goes up in the minimization case, it is sometimes possible to compute the next composition without incrementing the current right ladder. For example, in Figure 3b, G 5 [6] could be constructed by composing G 4 [6] and G 1 [10]. However, we continue to increment the right ladder because it could be useful later if the walk moves right.…”

“…However, we continue to increment the right ladder because it could be useful later if the walk moves right. For example, in Figure 3b, G 3 [8] is not needed to compute G 5 [6], but G 4 [8] is needed to compute G 5 [7].…”

Distributed Online Data Aggregation in Dynamic Graphs

“…The best known algorithm is due to Nguyen et al [10]. More recently, Bramas et al [4] considered the generalization of the problem to dynamic wireless sensor networks (modeled by evolving graphs). Bramas et al [4] show that the problem remains NP-complete even when restricted to dynamic WSNs of degree at most 2 (compared to 3 in the static case).…”