We deal with the problem of maintaining a shortest-path tree rooted at some process r in a network that may be disconnected after topological changes. The goal is then to maintain a shortest-path tree rooted at r in its connected component, V_r, and make all processes of other components detecting that r is not part of their connected component. We propose, in the composite atomicity model, a silent self-stabilizing algorithm for this problem working in semi-anonymous networks under the distributed unfair daemon (the most general daemon) without requiring any a priori knowledge about global parameters of the network. This is the first algorithm for this problem that is proven to achieve a polynomial stabilization time in steps. Namely, we exhibit a bound in O(W_{max} * n_{maxCC}^3 * n), where W_{max} is the maximum weight of an edge, n_{maxCC} is the maximum number of non-root processes in a connected component, and n is the number of processes. The stabilization time in rounds is at most 3n_{maxCC} + D, where D is the hop-diameter of V_r.
@InProceedings{devismes_et_al:LIPIcs.OPODIS.2016.10, author = {Devismes, St\'{e}phane and Ilcinkas, David and Johnen, Colette}, title = {{Self-Stabilizing Disconnected Components Detection and Rooted Shortest-Path Tree Maintenance in Polynomial Steps}}, booktitle = {20th International Conference on Principles of Distributed Systems (OPODIS 2016)}, pages = {10:1--10:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-031-6}, ISSN = {1868-8969}, year = {2017}, volume = {70}, editor = {Fatourou, Panagiota and Jim\'{e}nez, Ernesto and Pedone, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2016.10}, URN = {urn:nbn:de:0030-drops-70792}, doi = {10.4230/LIPIcs.OPODIS.2016.10}, annote = {Keywords: distributed algorithm, self-stabilization, routing algorithm, shortest path, disconnected network, shortest-path tree} }
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