Document Zbl 0938.90068

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We say a network (graph) can tolerate $\ell$ faulty nodes for a routing problem if after removing at most $\ell$ arbitrary faulty nodes from the graph the routing paths exist or the routing problem. However, the bound $\ell$ is usually a worst-case measure and it is of great interest to find the routing paths when more than $\ell$ faulty nodes exist.<br class="zbmathjax-paragraph">Cluster Fault-Tolerant (CFT) routing was proposed as an approach for this purpose. In the CFT routing, we reduce the number of “faults” that a routing problem has to deal with using subgraphs to cover the faulty nodes. In particular, we consider the number and the size (diameter) of faulty subgraphs rather than the number of faulty nodes that a graph can tolerate.<br class="zbmathjax-paragraph">In this paper, we show that a subgraph of diameter 2 can be viewed as a single “fault” for the following routing problems in the star graph: Given a source node $s$ and $t$ target nodes $t_1,\dots,t_k$, find $k$ node-disjoint paths from $s$ to $t_i(1\leq i\leq k)$, and given $k$ node pairs $(s_1,t_1), \dots, (s_k,t_k)$, find $k$ node-disjoint paths $s_i\to t_i$ $(1\leq i\leq k)$. Since a subgraph of diameter 2 of the $n$-dimensional star graph $G_n$ may have $n$ nodes, the above result implies that the number of faulty nodes that $G_n$ can tolerate is $n$ times larger than the worst-case measure if the faulty nodes can be covered by certain subgraphs.<br class="zbmathjax-paragraph">We also give algorithms which find the paths for the two routing problems.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1420493">Cited in <strong>6</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90B18" title="MSC2020">90B18</a> </td> <td class="space"> Communication networks in operations research </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68M10" title="MSC2020">68M10</a> </td> <td class="space"> Network design and communication in computer systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C08" title="MSC2020">90C08</a> </td> <td class="space"> Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C40" title="MSC2020">05C40</a> </td> <td class="space"> Connectivity </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afault-tolerant+routing">fault-tolerant routing</a>; <a href="/?q=ut%3Ainterconnection+network">interconnection network</a>; <a href="/?q=ut%3Anode-disjoint+paths">node-disjoint paths</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0938.90068" data-ciurl="/ci/01420493" data-biburl="/bibtex/01420493.bib" data-amsurl="/amsrefs/01420493.bib" data-xmlurl="/xml/01420493.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01420493.pdf" title="Zbl 0938.90068 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1002/(SICI)1097-0037(200001)35:1<83::AID-NET7>3.0.CO;2-D" aria-label="DOI for “Cluster fault-tolerant routing in star graphs”" title="10.1002/(SICI)1097-0037(200001)35:1<83::AID-NET7>3.0.CO;2-D">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Akers, IEEE Trans Comput 38 pp 555– (1989)</td> </tr><tr> <td>[2]</td> <td class="space">Akl, Networks 23 pp 215– (1993)</td> </tr><tr> <td>[3]</td> <td class="space">Bermond, Discr Appl Math (1992)</td> </tr><tr> <td>[4]</td> <td class="space">Bermond, Graphs Combin 5 (1989)</td> </tr><tr> <td>[5]</td> <td class="space">Day, IEEE Trans Parallel Distrib Syst 5 pp 31– (1994)</td> </tr><tr> <td>[6]</td> <td class="space">and ?Three disjoint path paradigms in star networks,? 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