Energy-Efficient Shortest Path Algorithms for Convergecast in Sensor Networks

We introduce a variant of the capacitated vehicle routing problem that is encountered in sensor networks for scientific data collection. Consider an undirected graph $G=(V \cup \{\mathbf{sink}\},E)$. Each vertex $v \in V$ holds a constant-sized reading normalized to 1 byte that needs to be communicated to the $\mathbf{sink}$. The communication protocol is defined such that readings travel in packets. The packets have a capacity of $k$ 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 $\mathbf{sink}$ with the fewest number of hops. We show this problem to be NP-hard and counter it with a simple distributed $(2-\frac{3}{2k})$-approximation algorithm called {\tt SPT} that uses the shortest path tree rooted at the $\mathbf{sink}$. We also show that {\tt SPT} is absolutely optimal when $G$ is a tree and asymptotically optimal when $G$ is a grid. Furthermore, {\tt SPT} has two nice properties. Firstly, the readings always travel along a shortest path toward the $\mathbf{sink}$, 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 $(2-\epsilon)$-approximation, for any fixed $\epsilon > 0$. This makes \spt optimal for the class of algorithms that obey either one of those properties.Comment: 15 pages, 7 figure

Energy-Efficient Shortest Path Algorithms for Convergecast in Sensor Networks

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). Each vertex v ∈ V holds a constantsized reading normalized to 1 byte that needs to be communicated to the sink. The communication protocol is defined such that readings travel in packets. The packets have a capacity of k bytes. We define a 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 with the fewest number of hops. We show this problem to be NP-hard and counter it with a simple distributed (2 − 3 2k)-approximation algorithm called SPT that uses the shortest path tree rooted at the sink. We also show that SPT is absolutely optimal when G is a tree and asymptotically optimal when G is a grid. Furthermore, SPT has two nice properties. Firstly, the readings always travel along a shortest path toward the 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 (2 − ɛ)-approximation, for any fixed ɛ> 0. This makes SPT optimal for the class of algorithms that obey either one of those properties