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Register a new userLeader Election in Arbitrarily Connected Networks with Process Crashes and Weak Channel Reliability
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Added by
Karla Vargas
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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A channel from a process p to a process q satisfies the ADD property if there are constants K and D, unknown to the processes, such that in any sequence of K consecutive messages sent by p to q, at least one of them is delivered to q at most D time units after it has been sent. This paper studies implementations of an eventual leader, namely, an {Omega} failure detector, in an arbitrarily connected network of eventual ADD channels, where processes may fail by crashing. It first presents an algorithm that assumes that processes initially know n, the total number of processes, sending messages of size O( log n). Then, it presents a second algorithm that does not assume the processes know n. Eventually the size of the messages sent by this algorithm is also O( log n). These are the first implementations of leader election in the ADD model. In this model, only eventually perfect failure detectors were considered, sending messages of size O(n log n).
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In this paper, we look at the problem of randomized leader election in synchronous distributed networks with a special focus on the message complexity. We provide an algorithm that solves the implicit version of leader election (where non-leader nodes need not be aware of the identity of the leader) in any general network with $O(sqrt{n} log^{7/2} n cdot t_{mix})$ messages and in $O(t_{mix}log^2 n)$ time, where $n$ is the number of nodes and $t_{mix}$ refers to the mixing time of a random walk in the network graph $G$. For several classes of well-connected networks (that have a large conductance or alternatively small mixing times e.g. expanders, hypercubes, etc), the above result implies extremely efficient (sublinear running time and messages) leader election algorithms. Correspondingly, we show that any substantial improvement is not possible over our algorithm, by presenting an almost matching lower bound for randomized leader election. We show that $Omega(sqrt{n}/phi^{3/4})$ messages are needed for any leader election algorithm that succeeds with probability at least $1-o(1)$, where $phi$ refers to the conductance of a graph. To the best of our knowledge, this is the first work that shows a dependence between the time and message complexity to solve leader election and the connectivity of the graph $G$, which is often characterized by the graphs conductance $phi$. Apart from the $Omega(m)$ bound in [Kutten et al., J.ACM 2015] (where $m$ denotes the number of edges of the graph), this work also provides one of the first non-trivial lower bounds for leader election in general networks.
It was suggested that a programmable matter system (composed of multiple computationally weak mobile particles) should remain connected at all times since otherwise, reconnection is difficult and may be impossible. At the same time, it was not clear that allowing the system to disconnect carried a significant advantage in terms of time complexity. We demonstrate for a fundamental task, that of leader election, an algorithm where the system disconnects and then reconnects automatically in a non-trivial way (particles can move far away from their former neighbors and later reconnect to others). Moreover, the runtime of the temporarily disconnecting deterministic leader election algorithm is linear in the diameter. Hence, the disconnecting -- reconnecting algorithm is as fast as previous randomized algorithms. When comparing to previous deterministic algorithms, we note that some of the previous work assumed weaker schedulers. Still, the runtime of all the previous deterministic algorithms that did not assume special shapes of the particle system (shapes with no holes) was at least quadratic in $n$, where $n$ is the number of particles in the system. (Moreover, the new algorithm is even faster in some parameters than the deterministic algorithms that did assume special initial shapes.) Since leader election is an important module in algorithms for various other tasks, the presented algorithm can be useful for speeding up other algorithms under the assumption of a strong scheduler. This leaves open the question: can a deterministic algorithm be as fast as the randomized ones also under weaker schedulers?
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.
Ad-hoc networks, a promising trend in wireless technology, fail to work properly in a global setting. In most cases, self-organization and cost-free local communication cannot compensate the need for being connected, gathering urgent information just-in-time. Equipping mobile devices additionally with GSM or UMTS adapters in order to communicate with arbitrary remote devices or even a fixed network infrastructure provides an opportunity. Devices that operate as intermediate nodes between the ad-hoc network and a reliable backbone network are potential injection points. They allow disseminating received information within the local neighborhood. The effectiveness of different devices to serve as injection point differs substantially. For practical reasons the determination of injection points should be done locally, within the ad-hoc network partitions. We analyze different localized algorithms using at most 2-hop neighboring information. Results show that devices selected this way spread information more efficiently through the ad-hoc network. Our results can also be applied in order to support the election process for clusterheads in the field of clustering mechanisms.
In this paper, we consider the Byzantine reliable broadcast problem on authenticated and partially connected networks. The state-of-the-art method to solve this problem consists in combining two algorithms from the literature. Handling asynchrony and faulty senders is typically done thanks to Gabriel Brachas authenticated double-echo broadcast protocol, which assumes an asynchronous fully connected network. Danny Dolevs algorithm can then be used to provide reliable communications between processes in the global fault model, where up to f processes among N can be faulty in a communication network that is at least 2f+1-connected. Following recent works that showed that Dolevs protocol can be made more practical thanks to several optimizations, we show that the state-of-the-art methods to solve our problem can be optimized thanks to layer-specific and cross-layer optimizations. Our simulations with the Omnet++ network simulator show that these optimizations can be efficiently combined to decrease the total amount of information transmitted or the protocols latency (e.g., respectively, -25% and -50% with a 16B payload, N=31 and f=4) compared to the state-of-the-art combination of Brachas and Dolevs protocols.
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