Ressource Efficient Stabilization for Local Tasks despite Unknown Capacity Links

Abstract: Self-stabilizing protocols enable distributed systems to recover correct behavior starting from any arbitrary configuration. In particular, when processors communicate by message passing, fake messages may be placed in communication links by an adversary. When the number of such fake messages is unknown, self-stabilization may require huge resources: (a) generic solutions (a.k.a. data link protocols) require unbounded resources, which makes them unrealistic to deploy and (b) specific solutions (e.g., census or tree construction) require $O(n\log n)$ or $O(\Delta\log n)$ bits of memory per node, where $n$ denotes the network size and $\Delta$ its maximum degree, which may prevent scalability. We investigate the possibility of resource efficient self-stabilizing protocols in this context. Specifically, we present a self-stabilizing protocol for $(\Delta+1)$-coloring in any n-node graph, under the asynchronous message-passing model. It is deterministic, it converges in $O(k\Delta n^2\log n)$ message exchanges, where $k$ is the bound of the link capacity in terms of number of messages, and it uses messages on $O(\log\log n+\log\Delta)$ bits with a memory of $O(\Delta\log\Delta+\log\log n)$ bits at each node. The resource consumption of our protocol is thus almost oblivious to the number of nodes, enabling scalability. Moreover, a striking property of our protocol is that the nodes do not need to know the number, or any bound on the number of messages initially present in each communication link of the initial (potentially corrupted) network configuration. This permits our protocol to handle any future network with unknown message capacity communication links. A key building block of our coloring scheme is a spanning directed acyclic graph construction, that is of independent interest, and can serve as a useful tool for solving other tasks in this challenging setting.