Space-Optimal Time-Efficient Silent Self-Stabilizing Constructions of Constrained Spanning Trees

Abstract

International audienceSelf-stabilizing algorithms are distributed algorithms supporting transient failures. Starting from any configuration, they allow the system to detect whether the actual configuration is legal, and, if not, they allow the system to eventually reach a legal configuration. In the context of network computing, it is known that, for every task, there is a self-stabilizing algorithm solving that task, with optimal space-complexity, but converging in an exponential number of rounds. On the other hand, it is also known that, for every task, there is a self-stabilizing algorithm solving that task in a linear number of rounds, but with large space-complexity. It is however not known whether for every task there exists a self-stabilizing algorithm that is simultaneously space-efficient and time-efficient. In this paper, we make a first attempt for answering the question of whether such an efficient algorithm exists for every task, by focussing on constrained spanning tree construction tasks. We present a general roadmap for the design of silent space-optimal self-stabilizing algorithms solving such tasks, converging in polynomially many rounds under the unfair scheduler. By applying our roadmap to the task of constructing minimum-weight spanning tree (MST), and to the task of constructing minimum-degree spanning tree (MDST), we provide algorithms that outperform previously known algorithms designed and optimized specifically for solving each of these two tasks

Silent MST approximation for tiny memory

“…The main ingredient to achieve this result is an exact algorithm for MST (that is selfstabilizing, silent and polynomial-time), that uses O(log n • s) space, where s is the number of bits used to encode an edge weight. This matches the performance of the O(log 2 n)-space algorithm of [2], for polynomial weights, but applies to any weight range. We conjecture that this result is tight, because for weights polynomially large, an Ω(log n • s) bound is known [15], and it seems likely that it is also the right answer for smaller weights.…”

“…But an algorithm in constant space cannot exist for this problem because one cannot break symmetry in constant space [1]. On the positive side, [2] shows that for various tree construction problems, one can match the space bound and have polynomial-time stabilization. In particular, one can get down to Θ(log 2 n) for minimum spanning tree, which is optimal when the edge weight are in a polynomial range.…”

Redundancy in distributed proofs

“…Such objects can be the outcome of an algorithm that might be subject to failures, or be a-priori correctly given objects but subject to later corruption. There are several mechanisms for checking the correctness of distributed objects (see, e.g., [2,3,7,[10][11][12]), and here we focus on one classical mechanism which is both simple and versatile, known as proof-labeling schemes [37], or as locally checkable proofs [30]. 1 Roughly, a proof-labeling scheme assigns certificates to each node of the network.…”

Redundancy in Distributed Proofs

“…Indeed, such data structures can be the outcome of an algorithm that might be subject to failures, or be a-priori correctly given data-structures but subject to later corruption. Several mechanisms exist enabling checking the correctness of distributed data structures (see, e.g., [2,5,9,10,11]). For its simplicity and versatility, we shall focus on one classical mechanism known as proof-labeling schemes [31], a.k.a.…”