Wireless Aggregation at Nearly Constant Rate

One of the most fundamental tasks in sensor networks is the computation of a (compressible) aggregation function of the input measurements. What rate of computation can be maintained, by properly choosing the aggregation tree, the TDMA schedule of the tree edges, and the transmission powers? This can be viewed as the convergecast capacity of a wireless network. We show here that the optimal rate is effectively a constant. This holds even in arbitrary networks, under the physical model of interference. This compares with previous bounds that are logarithmic (e.g., Ω(1/ log n)). Namely, we show that a rate of Ω(1/ log∗ ∆) is possible, where ∆ is the length diversity (ratio between the furthest to the shortest distance between nodes). It also implies that the scheduling complexity of wireless connectivity is O(log∗ ∆). This is achieved using the natural minimum spanning tree (MST). Our method crucially depends on choosing the appropriate power assignment for the instance at hand, since without power control, only a trivial linear rate can be guaranteed. We also show that there is a fixed power assignment that allows for a rate of Ω(1/ log log ∆). Surprisingly, these bounds are best possible. No aggregation network can guarantee a rate better than O(1/ log log ∆) using fixed power assignment. Also, when using arbitrary power control, there are instances whose MSTs cannot be scheduled in fewer than Ω(1/ log∗ ∆) slots. ar X iv :1 71 2. 03 05 3v 1 [ cs .N I] 8 D ec 2 01 7