Self-Stabilizing Disconnected Components Detection and Rooted Shortest-Path Tree Maintenance in Polynomial Steps
Résumé
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 {\em 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 $O(\texttt{W}_{\max} {n_{\texttt{maxCC}}}^3 n)$, where
$\texttt{W}_{\max}$ is the maximum weight of an edge, ${n_{\texttt{maxCC}}}$ 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~$3{n_{\texttt{maxCC}}}+D$, where $D$ is the hop-diameter of $V_r$.
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