A Self-Stabilizing Memory Efficient Algorithm for the Minimum Diameter Spanning Tree under an Omnipotent Daemon (original) (raw)
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A self stabilizing protocol for constructing a rooted spanning tree in an arbitrary asynchronous network of processors that communicate through sha~ed memory is presented. The processors have unique identifiers but are otherwise identical. The network topology is assumed to be dynamic, that is, edges can join or leave the computation before it eventually stabilizes.
Lecture Notes in Computer Science, 2009
A self-stabilizing protocol can eventually recover its intended behavior even when started from an arbitrary configuration. Most of self-stabilizing protocols require every pair of neighboring processes to communicate with each other repeatedly and forever even after converging to legitimate configurations. Such permanent communication impairs efficiency, but is necessary in nature of self-stabilization: if we allow a process to stop its communication with other processes, the process may initially start and remain forever at a state inconsistent with the states of other processes. So it is challenging to minimize the number of process pairs communicating after convergence. We investigate possibility of communication-efficient self-stabilization, which allows only O(n) pairs of neighboring processes to communicate repeatedly after convergence. For spanningtree construction, we show the following results: (a) communication-efficiency is attainable when a unique root is designated a priori, (b) communication-efficiency is impossible to attain when each process has a unique identifier but without a designated unique root, and (c) communication-efficiency becomes attainable with process identifiers if each process initially knows an upper bound of the network size.
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We present a uniform self-stabilizing algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of an arbitrary positively real-weighted graph. Our algorithm consists in two stages of stabilizing protocols. The first stage is a uniform randomized stabilizing unique naming protocol, and the second stage is a stabilizing MDST protocol, designed as a fair composition of Merlin-Segall's stabilizing protocol and a distributed deterministic stabilizing protocol solving the (MDST) problem. The resulting randomized distributed algorithm presented herein is a composition of the two stages; it stabilizes in O(nā + D 2 + n log log n) expected time, and uses O(n 2 log n + n log W) memory bits (where n is the order of the graph, ā is the maximum degree of the network, D is the diameter in terms of hops, and W is the largest edge weight). To our knowledge, our protocol is the very first distributed algorithm for the (MDST) problem. Moreover, it is fault-tolerant and works for any anonymous arbitrary network.
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Self-stabilization is a versatile approach to fault-tolerance since it permits a distributed system to recover from any transient fault that arbitrarily corrupts the contents of all memories in the system. Byzantine tolerance is an attractive feature of distributed systems that permits to cope with arbitrary malicious behaviors. Combining these two properties proved difficult: it is impossible to contain the spatial impact of Byzantine nodes in a self-stabilizing context for global tasks such as tree orientation and tree construction. We present and illustrate a new concept of Byzantine containment in stabilization. Our property, called Strong Stabilization enables to contain the impact of Byzantine nodes if they actually perform too many Byzantine actions. We derive impossibility results for strong stabilization and present strongly stabilizing protocols for tree orientation and tree construction that are optimal with respect to the number of Byzantine nodes that can be tolerated in a self-stabilizing context.
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We present results on the last topic we collaborate with our late friend, Professor Ajoy Kumar Datta (1958-2019). In this work, we shed new light on a self-stabilizing wave algorithm proposed by Colette Johnen in 1997. This algorithm constructs a BFS spanning tree in any connected rooted network. Nowadays, it is still the best existing self-stabilizing BFS spanning tree construction in terms of memory requirement, {\em i.e.}, it only requires Theta(1)\Theta(1)Theta(1) bits per edge. However, it has been proven assuming a weakly fair daemon. Moreover, its stabilization time was unknown. Here, we study the slightly modified version of this algorithm, still keeping the same memory requirement. We prove the self-stabilization of this variant under the distributed unfair daemon and show a stabilization time in O(D.n2)O(D.n^2)O(D.n2) rounds, where DDD is the network diameter and nnn the number of processes.
Silent Self-Stabilizing Scheme for Spanning-Tree-like Constructions
HAL (Le Centre pour la Communication Scientifique Directe), 2018
We propose a general scheme, called Algorithm STlC, to compute spanning-tree-like data structures on arbitrary networks. STlC is self-stabilizing and silent and, despite its generality, is also efficient. It is written in the locally shared memory model with composite atomicity assuming the distributed unfair daemon, the weakest scheduling assumption of the model. Its stabilization time is in O(nmaxCC) rounds, where nmaxCC is the maximum number of processes in a connected component. We also exhibit polynomial upper bounds on its stabilization time in steps and process moves holding for large classes of instantiations of Algorithm STlC. We illustrate the versatility of our approach by proposing several such instantiations that efficiently solve classical problems such as leader election, as well as, unconstrained and shortest-path spanning tree constructions.