Fast and Compact Distributed Verification and Self-Stabilization of a DFS Tree

We present algorithms for distributed verification and silent-stabilization of a DFS(Depth First Search) spanning tree of a connected network. Computing and maintaining such a DFS tree is an important task, e.g., for constructing efficient routing schemes. Our algorithm improves upon previous work in various ways. Comparable previous work has space and time complexities of $O(n\log \Delta)$ bits per node and $O(nD)$ respectively, where $\Delta$ is the highest degree of a node, $n$ is the number of nodes and $D$ is the diameter of the network. In contrast, our algorithm has a space complexity of $O(\log n)$ bits per node, which is optimal for silent-stabilizing spanning trees and runs in $O(n)$ time. In addition, our solution is modular since it utilizes the distributed verification algorithm as an independent subtask of the overall solution. It is possible to use the verification algorithm as a stand alone task or as a subtask in another algorithm. To demonstrate the simplicity of constructing efficient DFS algorithms using the modular approach, We also present a (non-sielnt) self-stabilizing DFS token circulation algorithm for general networks based on our silent-stabilizing DFS tree. The complexities of this token circulation algorithm are comparable to the known ones

Self-stabilizing K-out-of-L exclusion on tree network

In this paper, we address the problem of K-out-of-L exclusion, a generalization of the mutual exclusion problem, in which there are $\ell$ units of a shared resource, and any process can request up to $\mathtt k$ units ($1\leq\mathtt k\leq\ell$). We propose the first deterministic self-stabilizing distributed K-out-of-L exclusion protocol in message-passing systems for asynchronous oriented tree networks which assumes bounded local memory for each process.Comment: 15 page

Self-stabilizing network orientation algorithms in arbitrary rooted networks

Network orientation is the problem of assigning different labels to the edges at each processor, in a globally consistent manner. A self-stabilizing protocol guarantees that the system will arrive at a legitimate state in finite time, irrespective of the initial state of the system. Two deterministic distributed network orientation protocols on arbitrary rooted, asynchronous networks are proposed in this work. Both protocols set up a chordal sense of direction in the network. The protocols are self-stabilizing, meaning that starting from an arbitrary state, the protocols are guaranteed to reach a state in which every processor has a valid node label and every link has a valid edge label. The first protocol assumes an underlying depth-first token circulation protocol; it orients the network as the token is passed among the nodes and stabilizes in O(n) steps after the token circulation stabilizes, where n is the number of processors in the network. The second protocol is designed on an underlying spanning tree protocol and stabilizes in O(h) time, after the spanning tree is constructed, where h is the height of the spanning tree. Although the second protocol assumes the existence of a spanning tree of the rooted network, it orients all edges--both tree and non-tree edges--of the network

Communication Efficient Self-Stabilizing Leader Election

This paper presents a randomized self-stabilizing algorithm that elects a leader r in a general n-node undirected graph and constructs a spanning tree T rooted at r. The algorithm works under the synchronous message passing network model, assuming that the nodes know a linear upper bound on n and that each edge has a unique ID known to both its endpoints (or, alternatively, assuming the KT? model). The highlight of this algorithm is its superior communication efficiency: It is guaranteed to send a total of O? (n) messages, each of constant size, till stabilization, while stabilizing in O? (n) rounds, in expectation and with high probability. After stabilization, the algorithm sends at most one constant size message per round while communicating only over the (n - 1) edges of T. In all these aspects, the communication overhead of the new algorithm is far smaller than that of the existing (mostly deterministic) self-stabilizing leader election algorithms. The algorithm is relatively simple and relies mostly on known modules that are common in the fault free leader election literature; these modules are enhanced in various subtle ways in order to assemble them into a communication efficient self-stabilizing algorithm

Self-Stabilization in the Distributed Systems of Finite State Machines

The notion of self-stabilization was first proposed by Dijkstra in 1974 in his classic paper. The paper defines a system as self-stabilizing if, starting at any, possibly illegitimate, state the system can automatically adjust itself to eventually converge to a legitimate state in finite amount of time and once in a legitimate state it will remain so unless it incurs a subsequent transient fault. Dijkstra limited his attention to a ring of finite-state machines and provided its solution for self-stabilization. In the years following his introduction, very few papers were published in this area. Once his proposal was recognized as a milestone in work on fault tolerance, the notion propagated among the researchers rapidly and many researchers in the distributed systems diverted their attention to it. The investigation and use of self-stabilization as an approach to fault-tolerant behavior under a model of transient failures for distributed systems is now undergoing a renaissance. A good number of works pertaining to self-stabilization in the distributed systems were proposed in the yesteryears most of which are very recent. This report surveys all previous works available in the literature of self-stabilizing systems

Fast and compact self-stabilizing verification, computation, and fault detection of an MST

This paper demonstrates the usefulness of distributed local verification of proofs, as a tool for the design of self-stabilizing algorithms.In particular, it introduces a somewhat generalized notion of distributed local proofs, and utilizes it for improving the time complexity significantly, while maintaining space optimality. As a result, we show that optimizing the memory size carries at most a small cost in terms of time, in the context of Minimum Spanning Tree (MST). That is, we present algorithms that are both time and space efficient for both constructing an MST and for verifying it.This involves several parts that may be considered contributions in themselves.First, we generalize the notion of local proofs, trading off the time complexity for memory efficiency. This adds a dimension to the study of distributed local proofs, which has been gaining attention recently. Specifically, we design a (self-stabilizing) proof labeling scheme which is memory optimal (i.e., $O(\log n)$ bits per node), and whose time complexity is $O(\log ^2 n)$ in synchronous networks, or $O(\Delta \log ^3 n)$ time in asynchronous ones, where $\Delta$ is the maximum degree of nodes. This answers an open problem posed by Awerbuch and Varghese (FOCS 1991). We also show that $\Omega(\log n)$ time is necessary, even in synchronous networks. Another property is that if $f$ faults occurred, then, within the requireddetection time above, they are detected by some node in the $O(f\log n)$ locality of each of the faults.Second, we show how to enhance a known transformer that makes input/output algorithms self-stabilizing. It now takes as input an efficient construction algorithm and an efficient self-stabilizing proof labeling scheme, and produces an efficient self-stabilizing algorithm. When used for MST, the transformer produces a memory optimal self-stabilizing algorithm, whose time complexity, namely, $O(n)$, is significantly better even than that of previous algorithms. (The time complexity of previous MST algorithms that used $\Omega(\log^2 n)$ memory bits per node was $O(n^2)$, and the time for optimal space algorithms was $O(n|E|)$.) Inherited from our proof labelling scheme, our self-stabilising MST construction algorithm also has the following two properties: (1) if faults occur after the construction ended, then they are detected by some nodes within $O(\log ^2 n)$ time in synchronous networks, or within $O(\Delta \log ^3 n)$ time in asynchronous ones, and (2) if $f$ faults occurred, then, within the required detection time above, they are detected within the $O(f\log n)$ locality of each of the faults. We also show how to improve the above two properties, at the expense of some increase in the memory

Self-stabilizing interval routing algorithm with low stretch factor

A compact routing scheme is a routing strategy which suggests routing tables that are space efficient compared to traditional all-pairs shortest path routing algorithms. An Interval Routing algorithm is a compact routing algorithm which uses a routing table at every node in which a set of destination addresses that use the same output port are grouped into intervals of consecutive addresses. Self-stabilization is a property by which a system is guaranteed to reach a legitimate state in a finite number of steps starting from any arbitrary state. A self-stabilizing Pivot Interval Routing (PIR) algorithm is proposed in this work. The PIR strategy allows routing along paths whose stretch factor is at most five, and whose average stretch factor is at most three with routing tables of size O(n3/2log 23/2n) bits in total, where n is the number of nodes in the network. Stretch factor is the maximum ratio taken over all source-destination pairs between the length of the paths computed by the routing algorithm and the distance between the source and the destination. PIR is also an Interval Routing Scheme (IRS) using at most 2n( 1+lnn)1/2 intervals per link for the weighted graphs and 3n(1+ lnn)1/2 intervals per link for the unweighted graphs. The preprocessing stage of the PIR algorithm consists of nodelabeling and arc-labeling functions. The nodelabeling function re-labels the nodes with unique integers so as to facilitate fewer number of intervals per arc. The arc-labeling is done in such a fashion that the message delivery protocol takes an optimal path if both the source and the destination are located within a particular range from each other and takes a near-optimal path if they are farther from each other

Self-stabilizing routing protocols

In systems made up of processors and links connecting the processors, the global state of the system is defined by the local variables of the individual processors. The set of global states can be defined as being either legal or illegal. A self-stabilizing system is one that forces a system from an illegal state to a global legal state without external interference, using a finite number of steps. This thesis will concentrate on application of self-stabilization to routing problems, in particular path identification, connectivity and methods involved in destinational routing. Traditional methods for creation of rooted paths to multiple destinations in a computer network involve the creation of spanning trees, and broadcasting information on the tree to be picked up by the individual nodes on the tree. The information for the creation of the tree are all sourced at the root, and the individual nodes update information from the centralized source. The self-stabilization model for networks allows the decision for a creation of a tree and message checking to occur automatically, locally, and more important, in contrast to traditional networks, asynchronously. The creation, message passing occur with a node and its immediate neighbor, and the tree, path is created based on this communicated data. In addition, the self-stabilization model eliminates the requisite initialization of traditional networks, i.e. given any arbitrary initial state the system (a given network) is guaranteed to stabilize to a legal global state, in the case of a broadcast network, a minimal spanning tree rooted at a source