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"... This work would not have been possible without the understanding of my friends, the encouragement of my parents and the support of all the colleagues with whom I worked together. I have to thank them all and I am especially indebted to the following people. I would like to sincerely thank my advisor ..."

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This work would not have been possible without the understanding of my friends, the encouragement of my parents and the support of all the colleagues with whom I worked together. I have to thank them all and I am especially indebted to the following people. I would like to sincerely thank my advisor Friedhelm Meyer auf der Heide for giving me the opportunity to choose the direction of my research, for his optimism and his encouragement. I am also grateful to my co-advisor Christian Schindelhauer for the good cooperation in research and teaching, for having patience with me in numerous discussions and for his invaluable help. I also thank Inés Bebea González for burning the midnight oil when working on our research project and the good cooperation despite the thousand miles between Paderborn and Madrid. ¡Muchas gracias! Special thanks to Silke Geisen and Peter Birkner who didn’t shrink away from reading parts of this work in a preliminary state. Their comments helped a lot in improving its readability. Paderborn, July 2006 Stefan Rührup

### 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 to ..."

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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 O(h 2). For optimizing traffic a single-path strategy is optimal producing traffic 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 O “q ” log log h log h 1

### Online Multi-Path Routing in a Maze

, 2005

"... 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 linear in the minimum number of steps h while it causes a traffic (i.e. ..."

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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 linear in the minimum number of steps h while it causes a traffic (i.e.

### Broadcasting and routing in faulty mesh networks

"... Abstract — Broadcasting is a data communication task in which one processor sends the same message to all other processors. Routing is a task where a source processor sends a message to a destination processor. A faulty node is in an error state and cannot participate in the activities or the commun ..."

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Abstract — Broadcasting is a data communication task in which one processor sends the same message to all other processors. Routing is a task where a source processor sends a message to a destination processor. A faulty node is in an error state and cannot participate in the activities or the communication in a given network. In this paper, we consider the family of mesh networks, which include the mesh connected computer (MCC), k-dimensional mesh, torus, and k-ary n-cube. Our goal is to design routing and broadcasting algorithms which will use local knowledge of faults, no additional resources, will work for an arbitrary number and structure of faults, will guarantee delivery to all nodes connected to the source, and will remain optimal in a fault free mesh. We did not find any solution in literature to satisfy these desirable properties. Our routing and broadcasting schemes for MCCs and tori, and our broadcasting algorithm for the all-port model on any faulty mesh network satisfy all of these properties. For routing and broadcasting in a one-port model in higher dimensions, a condition on fault structure needs to be met. We propose a new broadcasting algorithm which guarantees delivery to all processors connected to the source in the all-port model of faulty meshes. We then describe a routing algorithm that guarantees delivery in faulty MCCs and tori, the connectivity of the source and destination being the only obvious requirement. The algorithm can be extended to faulty k-D meshes and k-ary n-cubes, where the delivery will be guaranteed if healthy nodes in every 2-D submesh (sub-tori) remain connected. We then describe broadcasting algorithms for the one-port model, which again guarantee delivery to all connected processors in two-dimensional cases, and guarantee delivery in k-dimensional cases if healthy processors in every 2-D submesh (sub-tori) remain connected.