Optimal Space Lower Bound for Deterministic Self-Stabilizing Leader Election Algorithms

Abstract

Given a boolean predicate $\Pi$ on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for $\Pi$ is a distributed algorithm that can start from any initial configuration of the network (i.e., every node has an arbitrary value assigned to each of its variables), and eventually converge to a configuration satisfying $\Pi$. It is known that leader election does not have a deterministic self-stabilizing algorithm using a constant-size register at each node, i.e., for some networks, some of their nodes must have registers whose sizes grow with the size $n$ of the networks. On the other hand, it is also known that leader election can be solved by a deterministic self-stabilizing algorithm using registers of $O(\log \log n)$ bits per node in any $n$-node bounded-degree network. We show that this latter space complexity is optimal. Specifically, we prove that every deterministic self-stabilizing algorithm solving leader election must use $\Omega(\log \log n)$-bit per node registers in some $n$-node networks. In addition, we show that our lower bounds go beyond leader election, and apply to all problems that cannot be solved by anonymous algorithms.

Optimal Memory Requirement for Self-stabilizing Token Circulation

“…We prove that even under a central strongly fair scheduler, or a synchronous scheduler, on a tree topology, and without the requirement of being selfstabilizing, no algorithm can solve the fair token circulation problem in anonymous network. The result established in [3] concludes to the space optimality of our algorithm.…”

“…Finally, we extend the results of [3] to prove that the space complexity of our algorithm is optimal (see Appendix C.4). More precisely, we prove that no algorithm can solve the fair token circulation problem using o(log log n) bits of memory per node, even under least challenging hypothesis than ours.…”

“…This indicates that some tasks can be achieved in S with only sublogarithmic memory. As suggested by the results established in [3], it is much harder to reach o(log log n) memory. No previous work addresses the self-stabilizing DFTC in S without pointers.…”

“…The part of the specification of the token circulation problem which corresponds to the existence and unicity of the token is not complicated to state, and is basically what is made in [3] when the authors give a specification for the leader election problem. It boils down to defining a local predicate on nodes, which represents the fact that the token is held by that node, and guaranteeing that the predicate is evaluated to true on exactly one node in each configuration.…”

“…Finally, in Section C.4 we use the results of [3] to prove that the space complexity of our algorithm is optimal.…”