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Energy-Efficient Shortest Path Algorithms for Convergecast in Sensor Networks

Abstract

We introduce a variant of the capacitated vehicle routing problem that is encountered in sensor networks for scientific data collection. Consider an undirected graph G=(V{sink},E)G=(V \cup \{\mathbf{sink}\},E). Each vertex vVv \in V holds a constant-sized reading normalized to 1 byte that needs to be communicated to the sink\mathbf{sink}. The communication protocol is defined such that readings travel in packets. The packets have a capacity of kk bytes. We define a {\em packet hop} to be the communication of a packet from a vertex to its neighbor. Each packet hop drains one unit of energy and therefore, we need to communicate the readings to the sink\mathbf{sink} with the fewest number of hops. We show this problem to be NP-hard and counter it with a simple distributed (232k)(2-\frac{3}{2k})-approximation algorithm called {\tt SPT} that uses the shortest path tree rooted at the sink\mathbf{sink}. We also show that {\tt SPT} is absolutely optimal when GG is a tree and asymptotically optimal when GG is a grid. Furthermore, {\tt SPT} has two nice properties. Firstly, the readings always travel along a shortest path toward the sink\mathbf{sink}, which makes it an appealing solution to the convergecast problem as it fits the natural intuition. Secondly, each node employs a very elementary packing strategy. Given all the readings that enter into the node, it sends out as many fully packed packets as possible followed by at most 1 partial packet. We show that any solution that has either one of the two properties cannot be a (2ϵ)(2-\epsilon)-approximation, for any fixed ϵ>0\epsilon > 0. This makes \spt optimal for the class of algorithms that obey either one of those properties.

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