Convergence of local communication chain strategies via linear transformations

Kling, Peter; Heide, Friedhelm Meyer auf der

doi:10.1145/1989493.1989517

Cited by 19 publications

(11 citation statements)

References 18 publications

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“…For the Fsync scheduler, a runtime of O(n 2 • log(n/ε)) rounds has been proven. Later on, an almost matching lower bound (for the algorithm) of Ω(n 2 • log(1/ε)) has been derived [16]. Algorithms with stronger assumptions, e.g., the LUMI model, are able to achieve better runtimes [13,17].…”

Section: Related Workmentioning

confidence: 99%

The Max-Line-Formation Problem

Castenow¹,

Götte²,

Knollmann³

et al. 2021

Preprint

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We consider n robots with limited visibility: each robot can observe other robots only up to a constant distance denoted as the viewing range. The robots operate in discrete rounds that are either fully synchronous (Fsync) or semi-synchronized (Ssync). Most previously studied formation problems in this setting seek to bring the robots closer together (e.g., Gathering or Chain-Formation). In this work, we introduce the Max-Line-Formation problem, which has a contrary goal: to arrange the robots on a straight line of maximal length.First, we prove that the problem is impossible to solve by robots with a constant sized circular viewing range. The impossibility holds under comparably strong assumptions: robots that agree on both axes of their local coordinate systems in Fsync. On the positive side, we show that the problem is solvable by robots with a constant square viewing range, i.e., the robots can observe other robots that lie within a constant-sized square centered at their position. In this case, the robots need to agree on only one axis of their local coordinate systems. We derive two algorithms: the first algorithm considers oblivious robots (OBLOT ) and converges to the optimal configuration in time O(n 2 • log(n/ε)) under the Ssync scheduler (ε is a convergence parameter). The other algorithm makes use of locally visible lights (LUMI). It is designed for the Fsync scheduler and can solve the problem exactly in optimal time Θ(n). We also argue how a combination of the two algorithms can solve the Max-Line-Formation exactly in time O(n 2 ) under the Ssync scheduler with the help of the LUMI model.Afterward, we show that both the algorithmic and the analysis techniques can also be applied to the Gathering and Chain-Formation problem: we introduce an algorithm with a reduced viewing range for Gathering and give new and improved runtime bounds for the Chain-Formation problem.

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Section: Related Workmentioning

confidence: 99%

The Max-Line-Formation Problem

Castenow¹,

Götte²,

Knollmann³

et al. 2021

Preprint

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“…the length) of the straight chain between the base stations. [17] gave an almost matching lower bound of Ω n 2 • log(1/ ) and generalized these bounds to a class of (linear) strategies related to GtM. Note that while there are some discrete Chain-Formation strategies, specifically [19], that achieve a better (linear) asymptotic runtime, such strategies are known only for relaxed models and goals (e.g., reaching only a Θ(1)-approximation).…”

Section: Related Workmentioning

confidence: 99%

A Discrete and Continuous Study of the Max-Chain-Formation Problem: Slow Down to Speed up

Castenow

Kling

Knollmann

et al. 2020

Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures

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Most existing robot formation problems seek a target formation of a certain minimal and, thus, efficient structure. Examples include the Gathering and the Chain-Formation problem. In this work, we study formation problems that try to reach a maximal structure, supporting for example an efficient coverage in exploration scenarios. A recent example is the NASA Shapeshifter project [24], which describes how the robots form a relay chain along which gathered data from extraterrestrial cave explorations may be sent to a home base. As a first step towards understanding such maximization tasks, we introduce and study the Max-Chain-Formation problem, where n robots are ordered along a winding, potentially selfintersecting chain and must form a connected, straight line of maximal length connecting its two endpoints. We propose and analyze strategies in a discrete and in a continuous time model. In the discrete case, we give a complete analysis if all robots are initially collinear, showing that the worst-case time to reach an ε-approximation is upper bounded by O(n 2 • log(n/ε)) and lower bounded by Ω(n 2 • log(1/ε)). If one endpoint of the chain remains stationary, this result can be extended to the non-collinear case. If both endpoints move, we identify a family of instances whose runtime is unbounded. For the continuous model, we give a strategy with an optimal runtime bound of Θ(n). Avoiding an unbounded runtime similar to the discrete case relies crucially on a counter-intuitive aspect of the strategy: slowing down the endpoints while all other robots move at full speed. Surprisingly, we can show that a similar trick does not work in the discrete model.

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“…They follow a variety of goals, as for example graph exploration (e.g., [27]), gathering problems (e.g., [2,16]) , and shape formation problem (e.g., [28]). Surveys of recent results in swarm robotics can be found in [32,37]; other samples of representative work can be found in e.g., [25,7,18,19,17,31,30,43,26,39,34,5,40]. Besides work on how to set up robot swarms in order to solve certain tasks, a significant amount of work has also been invested in order to understand the global effects of local behavior in natural swarms like social insects, birds, or fish (see e.g., [13,9]).…”

Section: Related Workmentioning

confidence: 99%

Infinite Object Coating in the Amoebot Model

Derakhshandeh,

Gmyr,

Richa

et al. 2014

Preprint

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The term programmable matter refers to matter which has the ability to change its physical properties (shape, density, moduli, conductivity, optical properties, etc.) in a programmable fashion, based upon user input or autonomous sensing. This has many applications like smart materials, autonomous monitoring and repair, and minimal invasive surgery. While programmable matter might have been considered pure science fiction more than two decades ago, in recent years a large amount of research has been conducted in this field. Often programmable matter is envisioned as a very large number of small locally interacting computational particles. We propose the Amoebot model, a new model which builds upon this vision of programmable matter. Inspired by the behavior of amoeba, the Amoebot model offers a versatile framework to model self-organizing particles and facilitates rigorous algorithmic research in the area of programmable matter. We present an algorithm for the problem of coating an infinite object under this model, and prove the correctness of the algorithm and that it is work-optimal.

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Convergence of local communication chain strategies via linear transformations

Cited by 19 publications

References 18 publications

The Max-Line-Formation Problem

The Max-Line-Formation Problem

A Discrete and Continuous Study of the Max-Chain-Formation Problem: Slow Down to Speed up

Infinite Object Coating in the Amoebot Model

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