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In this paper we propose a sufficient condition for minimal routing in 3-dimensional (3-D) meshes with faulty nodes. It is based on an early work of the author on minimal routing in 2-dimensional (2-D) meshes. Unlike many traditional models that assume all the nodes know global fault distribution or just adjacent fault information, our approach is based on the concept of limited global fault information. First, we propose a fault model called faulty cube in which all faulty nodes in the system are contained in a set of faulty cubes. Fault information is then distributed to limited number of nodes while it is still sufficient to support minimal routing. The limited fault information collected at each node is represented by a vector called extended safety level. The extended safety level associated with a node can be used to determine the existence of a minimal path from this node to a given destination. Specifically, we study the existence of minimal paths at a given source node, limited distribution of fault information, minimal routing, and deadlock-free and livelock-free routing. Our results show that any minimal routing that is partially adaptive can be applied in our model as long as the destination node meets a certain condition. We also propose a dynamic planar-adaptive routing scheme that offers better fault tolerance and adaptivity than the planar-adaptive routing scheme in 3-D meshes. Our approach is the first attempt to address adaptive and minimal routing in 3-D meshes with faulty nodes using limited fault information.

A simple fault-tolerant adaptive and minimal routing approach in 3-D meshes