# Self-Stabilizing Disconnected Components Detection and Rooted Shortest-Path Tree Maintenance in Polynomial Steps

**Affiliations:**

^{1}LaBRI,

^{2}LaBRI,

^{3}LaBRI

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, Vr, 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, where edges have strictly positive weights. This algorithm does not require any a priori knowledge about global parameters of the network. We prove its correctness assuming the distributed unfair daemon, the most general daemon. Its stabilization time in rounds is at most 3nmaxCC + D, where nmaxCC is the maximum number of non-root processes in a connected component and D is the hop-diameter of Vr. Furthermore, if we additionally assume that edge weights are positive integers, then it stabilizes in a polynomial number of steps: namely, we exhibit a bound in O(WmaxnmaxCC 3 n), where Wmax is the maximum weight of an edge and n is the number of processes.

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