Regular Article
A Distributed Formation of Smallest Faulty Orthogonal Convex Polygons in 2-D Meshes

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Abstract

The rectangular faulty block model is the most commonly used fault model for designing fault-tolerant and deadlock-free routing algorithms in mesh-connected multicomputers. The convexity of a rectangle facilitates simple and efficient ways to route messages around fault regions using relatively few virtual channels to avoid deadlock. However, such a faulty block may include many nonfaulty nodes which are disabled, i.e., they are not involved in the routing process. Therefore, it is important to define a fault region that is convex, and at the same time, to include a minimum number of nonfaulty nodes. In this paper, we propose a simple and efficient distributed algorithm that can quickly construct a set of special convex polygons, called orthogonal convex polygons, from a given set of rectangular faulty blocks in a 2-D mesh (or 2-D torus). The formation of orthogonal convex polygons is done through a labeling scheme based on iterative message exchanges among neighboring nodes. For a given faulty block, after some nonfaulty nodes have been removed, the block is split into a set of orthogonal convex polygons, each of which is the smallest orthogonal convex polygon that contains all the faults it covers. Moreover, we show that the number of nonfaulty nodes covered in these polygons is no more than the one in the smallest orthogonal convex polygon that includes all the faulty nodes in the original faulty block. Finally, we discuss one open problem and present extensions to multidimensional meshes.

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This work was supported in part by NSF Grant CCR 9900646 and Grant ANI 0073736.

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