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By Doina Bein, Ajoy K. Datta, Vincent Villain (auth.), Sébastien Tixeuil, Ted Herman (eds.)

This booklet constitutes the refereed lawsuits of the seventh foreign Symposium on Self-Stabilizing structures, SSS 2005, held in Barcelona, Spain, in October 2005.

The 15 revised complete papers offered have been conscientiously reviewed and chosen from 33 submissions. The papers handle classical themes of self-stabilization, triumphing extensions to the sphere, reminiscent of snap-stabilization, code stabilization, self-stabilization with both dynamic, defective or Byzantine parts, or care for purposes of self-stabilization, both with regards to working structures, protection, or cellular and advert hoc networks.

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Extra resources for Self-Stabilizing Systems: 7th International Symposium, SSS 2005, Barcelona, Spain, October 26-27, 2005. Proceedings

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Let kt = | ≡Vγ |. We denote Σ(γt ) by (V0t , V1t , . . , Vktt −1 ). Let p be an element t of V0t0 with the lowest δp = w. Of course, we have kt0 − 1 ≤ w. We will prove by induction that: ∀i ∈ N, B(p, i) ⊆ V0t0 +i . This will prove that for i = w, V0t0 +i = V . Hence, the time convergence is less than or equal to w, which is less than or equal to D. If i = 0, then B(p, 0) = {p} ⊆ V0t0 . Assume that i ≥ 0 and B(p, i) ⊆ V0t0 +i . Let q ∈ B(p, i + 1). Then, q is a neighbor of an element q of B(p, i).

This value forces all its neighbors satisfying S = idle to execute Que := R (Raction). When all the neighbors of p have reset, p also executes Quep := R. Then, A Snap-Stabilizing DFS with a Lower Space Requirement 41 the R values are propagated up as far as possible following the P ar variable (Raction). By this mechanism, the A values are deleted in the P ar paths of p and its neighbors (in particular, the A values present since the initial configuration). Thus, from now on, when a A value reaches a requesting processor, this value cannot come from anyone but r and the processor obviously belongs to the normal visiting phase.

I . . (i ≥ 0) be a maximum execution. There are two cases: 1. For every i > 0, there is no convergent state. Then, we show that for any maximal execution e, initial convergent states are possible only if i ∈ {0, 1}. Moreover, if for every i > 0 there exists no convergent state, then the system is in SU . So, the system is in SU and the No Lockout property is garanteed. Synchronous vs. Asynchronous Unison 31 2. There exists i > 0 with a convergent state. Let p be a process among processes in convergent state such that rp contains a minimal value in rp .

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