Tight bounds for distributed selection

Complexity of Data Collection, Aggregation, and Selection for Wireless Sensor Networks

“…On the other hand, the time complexity of finding the median for a wireless network is at least as expensive as that of finding the median at a corresponding wired network (by assuming that no interferences exist among all transmission links). It has been proved in [11] that any deterministic algorithm for finding median in a wired network G of n nodes with diameter D, and total N data items has time complexity at least Ω (D log D N ). Finding k th smallest element…”

“…We then present our method (Algorithm 3) for distributed data selection in WSNs. In our method, we first collect data from dominatee nodes to corresponding dominator nodes, then we will run the distributed selection method for wired networks (from [11] and is summarized in Algorithm 4 for completeness of presentation) over the CDS. Algorithm 4 will be run by the sink node and the basic idea is as follows:…”

“…2: Then the median is found using only the connected dominating set, i.e., only nodes in CDS will participate. We run the randomized Algorithm 4 with t = 8λD with a constant λ > 1 (see [11] for details). This method has time complexity O(D log D N ) in wired communication model.…”

“…The time-costs of our algorithm are as follows (1) the first step has time-complexity Θ(∆); (2) the second step will cost O(D log D N ) rounds of communications with high probability [11], i.e., O(D log D N ) time-slots w.h.p., since each round is composed of β time-slots.…”

“…Similarly, if we run the best deterministic algorithm for data selection for wired networks [11], we have the following theorem.…”