## Section: New Results

### Self-stabilizing distributed control

Participant : Sébastien Tixeuil.

We presented in [52] a 50 pages book chapter devoted to self-stabilizing algorithms in the Handbook of Theory of Computing and Algorithms, published by Taylor and Francis.

We generalized [16] the classic dining philosophers problem to separate the conflict and communication neighbors of each process. Communication neighbors may directly exchange information while conflict neighbors compete for the access to the exclusive critical section of code. This generalization is motivated by a number of practical problems in distributed systems including problems in wireless sensor networks. We presented a self-stabilizing deterministic algorithm — KDP that solves a restricted version of the generalized problem where the conflict set for each process is limited to its k-hop neighborhood. Our algorithm is terminating. We formally proved KDP correct and evaluated its performance. We then extended KDP to handle fully generalized problem. We further extended it to handle a similarly generalized drinking philosophers problem. We described how KDP can be implemented in wireless sensor networks and demonstrated that this implementation does not jeopardize its correctness or termination properties.

We quantified [20] the amount of “practical information
(*i.e.* views obtained from the neighbors, colors attributed to
the nodes and links) to obtain “theoretical information (*i.e.*
the local topology of the network up to distance k ) in anonymous
networks. In more details, we show that a coloring at distance 2k + 1
is necessary and sufficient to obtain the local topology at distance
k that includes outgoing links. This bound drops to 2k when
outgoing links are not needed. A second contribution deals with color
bootstrapping (from which local topology can be obtained using the
aforementioned mechanisms). On the negative side, we showed that
*(i)* with a distributed daemon, it is impossible to achieve
deterministic color bootstrap, even if the whole network topology can
be instantaneously obtained, and *(ii)* with a central daemon, it
is impossible to achieve distance m when instantaneous topology
knowledge is limited to m-1 . On the positive side, we showed that
*(i)* under the k -central daemon, deterministic
self-stabilizing bootstrap of colors up to distance k is possible
provided that k -local topology can be instantaneously obtained, and
*(ii)* under the distributed daemon, probabilistic
self-stabilizing bootstrap is possible for any range.

In [36] , our focus was to lower the communication complexity
of self-stabilizing protocols *below* the need of checking every
neighbor forever. In more details, our contribution was threefold: (i)
We provide new complexity measures for communication efficiency of
self-stabilizing protocols, especially in the stabilized phase or when
there are no faults, (ii) On the negative side, we show that for
non-trivial problems such as coloring, maximal matching, and maximal
independent set, it is impossible to get (deterministic or
probabilistic) self-stabilizing solutions where every participant
communicates with less than every neighbor in the stabilized phase,
and (iii) On the positive side, we present protocols for coloring,
maximal matching, and maximal independent set such that a fraction of
the participants communicates with exactly one neighbor in the
stabilized phase.

The maximal matching problem has received considerable attention in
the self-stabilizing community. Previous work has given different
self-stabilizing algorithms that solves the problem for both the
adversarial and fair distributed daemon, the sequential adversarial
daemon, as well as the synchronous daemon. In [19] we
presented a single self-stabilizing algorithm for this problem that
unites all of these algorithms in that it stabilizes in the same
number of moves as the previous best algorithms for the sequential
adversarial, the distributed fair, and the synchronous daemon. In
addition, the algorithm improves the previous best moves complexities
for the distributed adversarial daemon from O(n^{2}) and O(m)
to O(m) where n is the number of processes, m is the number of
edges, and is the maximum degree in the graph.

In large scale multihop wireless networks, flat architectures are typically not scalable. Clustering was introduced to support self-organization and enable hierarchical routing. When dealing with multihop wireless networks, robustness is a crucial issue due to the dynamism of such networks. Several algorithms have been designed for clustering but to date, none of them has investigated the self-stabilization features of the resulting structure.In [21] , we proved that a clustering algorithm that have previously exhibited good robustness properties, is actually self-stabilizing. We proposed several enhancements to the scheme to reduce the stabilization time and thus improve stability in a dynamic environment. The key technique to these enhancements is a localized self-stabilizing algorithm for Directed Acyclic Graph (DAG) construction. We provided extensive studies (both theoretical and experimental) that show that our approach enables efficient yet adaptive clustering in wireless multihop networks.

Unidirectional networks preclude many common techniques in self-stabilization from being used, such as preserving local predicates. In [28] , we investigated the intrinsic complexity of achievingself-stabilization in unidirectional anonymous general networks, and focused on the classical vertex coloring problem. In more details, we proved a lower bound of n states per process (where n is the network size) and a recovery time of at least n(n-1)/2 actions in total. We also provided a deterministic algorithm with matching upper bounds that performs in arbitrary unidirectional anonymous graphs.