Resource 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 and the communication links are unbounded, 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:

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  generic solutions (a.k.a. data link protocols) require unbounded resources, which makes them unrealistic to deploy,

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  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 Δ its maximum degree, which may prevent scalability.

We investigate the possibility of resource-efficient self-stabilizing protocols in this context.

Specifically, we present self-stabilizing protocols for $(\Delta +1)$-coloring and maximal independent set construction in any n-node graph, under the asynchronous message-passing model. The problems of $(\Delta +1)$-coloring and maximal independent set construction are widely regarded as benchmark problems for evaluating local algorithms. Our protocols offer many desirable features. They are deterministic, converge in $O(k\Delta n^2\log n)$ message exchanges, where k is the (unknown) initial number of (possibly corrupted) messages in a communication link. They use messages of $O(\log \log n+\log \Delta )$ bits with a memory of $O(\Delta \log \Delta +\log \log n)$ bits at each node. The resource consumption of our protocols is thus almost oblivious to the number of nodes, enabling scalability. Moreover, a striking property of our protocols 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 protocols to handle any future network with unknown message capacity communication links.

A key building block of our coloring and maximal independent set schemes is an algorithm to obtain an acyclic orientation of graph edges, that is of independent interest, and can serve as a useful tool for solving other tasks in this challenging setting.