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

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

Given a boolean predicate ? on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for ? 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 ?. 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 ?(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

Self-Stabilizing Depth-First Token Circulation In Arbitrary Rooted Networks

We present a deterministic distributed depth-first token passing protocol on a rooted network. This protocol uses neither the processor identifiers nor the size of the network, but assumes the existence of a distinguises hed processor, called the root of the network. The protocol is self-stabilizing, meaning that starting from an arbitrary state (in response to an arbitrary perturbation modifying the memory state), it is guaranteed to reach a state with no more than one token in the network. Our protocol implements a fair token circulation scheme, i.e., in every round, every processor obtains the token at least once. The proposed protocol has extremely small state requirement---only 3(\Delta +1) states per processor, i.e., O(log\Delta) bits per processor, where \Delta is the degree of the network. The protocol can be used to implement a fair distributed mutual exclusion in any rooted network. This protocol can also be used to construct a DFS spanning tree

Self-stabilizing depth-first token circulation in arbitrary rooted networks

Abstract: We present a deterministic distributed depth- rst token passing protocol on a rooted network. This protocol uses neither the processor identi ers nor the size of the network, but assumes the existence of a distinguises hed processor, called the root of the network. The protocol is self-stabilizing, meaning that starting from an arbitrary state (in response to an arbitrary perturbation modifying the memory state), it is guaranteed to reach a state with no more than one token in the network. Our protocol implements a fair token circulation scheme, i.e., in every round, every processor obtains the token at least once. The proposed protocol has extremely small state requirement|only 3 ( +1) states per processor, i.e., O(log) bits per processor, where is the degree of the network. The protocol can be used to implement a fair distributed mutual exclusion in any rooted network. This protocol can also be used to construct a DFS spanning tree

Maintaining Balanced Trees for Structured Distributed Streaming Systems

International audienceIn this paper, we propose and analyze a simple local algorithm to balance a tree. The motivation comes from live distributed streaming systems in which a source diffuses a content to peers via a tree, a node forwarding the data to its children. Such systems are subject to a high churn, peers frequently joining and leaving the system. It is thus crucial to be able to repair the diffusion tree to allow an efficient data distribution. In particular, due to bandwidth limitations, an efficient diffusion tree must ensure that node degrees are bounded. Moreover, to minimize the delay of the streaming, the depth of the diffusion tree must also be controlled. We propose here a simple distributed repair algorithm in which each node carries out local operations based on its degree and on the subtree sizes of its children. In a synchronous setting, we first prove that starting from any n-node tree our process converges to a balanced binary tree in O(n 2) rounds. We then describe a more restrictive model, adding a small extra information to each node, under which we adapt our algorithm to converge in Θ(n log n) rounds