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Register a new userEfficient Deterministic Leader Election for Programmable Matter
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Added by
Fabien Dufoulon
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Fabien Dufoulon
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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?
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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.
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.
We show that the $(degree+1)$-list coloring problem can be solved deterministically in $O(D cdot log n cdotlog^2Delta)$ rounds in the CONGEST model, where $D$ is the diameter of the graph, $n$ the number of nodes, and $Delta$ the maximum degree. Using the recent polylogarithmic-time deterministic network decomposition algorithm by Rozhov{n} and Ghaffari [STOC 2020], this implies the first efficient (i.e., $polylog n$-time) deterministic CONGEST algorithm for the $(Delta+1)$-coloring and the $(mathit{degree}+1)$-list coloring problem. Previously the best known algorithm required $2^{O(sqrt{log n})}$ rounds and was not based on network decompositions. Our techniques also lead to deterministic $(mathit{degree}+1)$-list coloring algorithms for the congested clique and the massively parallel computation (MPC) model. For the congested clique, we obtain an algorithm with time complexity $O(logDeltacdotloglogDelta)$, for the MPC model, we obtain algorithms with round complexity $O(log^2Delta)$ for the linear-memory regime and $O(log^2Delta + log n)$ for the sublinear memory regime.
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).
This paper provides an algorithmic framework for obtaining fast distributed algorithms for a highly-dynamic setting, in which *arbitrarily many* edge changes may occur in each round. Our algorithm significantly improves upon prior work in its combination of (1) having an $O(1)$ amortized time complexity, (2) using only $O(log{n})$-bit messages, (3) not posing any restrictions on the dynamic behavior of the environment, (4) being deterministic, (5) having strong guarantees for intermediate solutions, and (6) being applicable for a wide family of tasks. The tasks for which we deduce such an algorithm are maximal matching, $(degree+1)$-coloring, 2-approximation for minimum weight vertex cover, and maximal independent set (which is the most subtle case). For some of these tasks, node insertions can also be among the allowed topology changes, and for some of them also abrupt node deletions.
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