S. Unconstrained and . Forest, Given an input set of processes rootSet, and assuming (strictly) positive integer weights for each edge, Algorithms Forest and SPF are the instantiations of Scheme with the parameters given in Algorithm 2 Algorithm 2: Parameters for any process u in Algorithms Forest and SPF Inputs: (1) canBeRoot u is true if and only if u ? rootSet, ) pname u is ?, and (3) ? u (v) = ? v (u) ? N * , for every v ? ?(u)

A. Forest, SPF) computes (in a self-stabilizing manner) an unconstrained (resp. shortestpath ) spanning forest in each connected component of G containing at least one process of rootSet. The forest consists of trees rooted at each process of rootSet. Moreover, in any component containing no process of rootSet, the processes eventually detect the absence of roots by taking the status I (Isolated) By Theorem 2 (resp. Theorem 3), Algorithms Forest and SPF self-stabilize to a terminal legitimate configuration in at most O(n maxCC n)

L. Algorithm and L. Bfs, the instantiations of Scheme with the parameters given in Algorithm 3 Algorithm 3: Parameters for any process u in Algorithm LEM and LEM BFS Inputs: (1) canBeRoot u is true for any process, ) pname u is the identifier of u (n.b., pname u ? N) (3) ? u (v) = (?, 1) for every v ? ?(u)

=. Distset and ×. Ids, DistSet, we let d.id = a and d.h = b; (2) (id 1 , i 1 ) ? (id 2 , i 2 ) = (id 1 , i 1 + i 2 ), ) ? (id 1 < id 2 ) ? [(id 1 = id 2 ) ? (i 1 < i 2 )]; (4) distRoot(u) = (pname u

. K. References, A. Altisen, S. Cournier, A. Devismes, F. Durand et al., Self-stabilizing leader election in polynomial steps Information and Computation [Cou09] A. Cournier. A new polynomial silent stabilizing spanning-tree construction algorithm Self-stabilizing disconnected components detection and rooted shortest-path tree maintenance in polynomial steps, SIROCCO'09 20th Int. Conf. on Principles of Distributed Systems, OPODIS'16, pp.330-366, 2009.

S. Devismes and C. C. Gärtner, Silent self-stabilizing BFS tree algorithms revisited, Swiss Federal Institute of Technolog (EPFL), pp.11-23, 2003.
DOI : 10.1016/j.jpdc.2016.06.003

URL : https://hal.archives-ouvertes.fr/hal-01411862