Online Routing in Faulty Meshes with Sub-linear Comparative Time and Traffic Ratio

Abstract

We consider the problem of routing a message in a mesh network with faulty nodes. The number and positions of faulty nodes is unknown. It is known that a flooding strategy like expanding ring search can route a message in the minimum number of steps h while it causes a traffic (i.e. the total number of messages) of ${\mathcal O}(h^{2})$. For optimizing traffic a single-path strategy is optimal producing traffic ${\mathcal O}(p + h)$, where p is the perimeter length of the barriers formed by the faulty nodes. Therefore, we define the comparative traffic ratio as a quotient over p+h and the competitive time ratio as a quotient over h. Optimal algorithms with constant ratios are known for time and traffic, but not for both. We are interested in optimizing both parameters and define the combined comparative ratio as the maximum of competitive time ratio and comparative traffic ratio. Single-path strategies using the right-hand rule for traversing barriers as well as multi-path strategies like expanding ring search have a combined comparative ratio of Θ(h). It is an open question whether there exists an online routing strategy optimizing time and traffic for meshes with an unknown set of faulty nodes. We present an online strategy for routing with faulty nodes providing sub-linear combined comparative ratio of $h^{{\mathcal O}(\sqrt{\frac{{\rm log log}h}{{\rm log}h}})}$.