Convergence of local communication chain strategies via linear transformations: or how to trade locality for speed

Consider two far apart base stations connected by an arbitrarily winding chain of n relay robots to transfer messages between them. Each relay acts autonomously, has a limited communication range, and knows only a small, local part of its environment. We seek a strategy for the relays to minimize the chain's length. We describe a large strategy class in form of linear transformations of the spatial vectors connecting neighboring robots. This yields surprising correlations between several strategy properties and characteristics of these transformations (e.g., "reasonable" strategies correspond to transformations given by doubly stochastic matrices). Based on these results, we give almost tight bounds on the strategies' convergence speed by applying and extending results about the mixing time of Markov chains. Eventually, our framework enables us to define strategies where each relay bases its decision where to move only on the positions of its k next left and right neighbors, and to prove a convergence speed of Θ(n2/k2 log n) for these strategies. This not only closes a gap between upper and lower runtime bounds of a known strategy (Go-To-The-Middle), but also allows for a trade-off between convergence properties and locality.

[1]  V. Climenhaga Markov chains and mixing times , 2013 .

[2]  Friedhelm Meyer auf der Heide,et al.  A Continuous, Local Strategy for Constructing a Short Chain of Mobile Robots , 2010, SIROCCO.

[3]  Patrick Doherty,et al.  Positioning unmanned aerial vehicles as communication relays for surveillance tasks , 2009, Robotics: Science and Systems.

[4]  L. Saloff-Coste,et al.  Convergence of some time inhomogeneous Markov chains via spectral techniques , 2007 .

[5]  Michael P. Knapp Sines and Cosines of Angles in Arithmetic Progression , 2009 .

[6]  R. Oldenburger,et al.  Infinite powers of matrices and characteristic roots , 1940 .

[7]  Franck Petit,et al.  Self-stabilizing Deterministic Gathering , 2009, ALGOSENSORS.

[8]  Friedhelm Meyer auf der Heide,et al.  Optimal strategies for maintaining a chain of relays between an explorer and a base camp , 2009, Theor. Comput. Sci..

[9]  Patrick Doherty,et al.  Relay Positioning for Unmanned Aerial Vehicle Surveillance* , 2010, Int. J. Robotics Res..

[10]  Ichiro Suzuki,et al.  Distributed motion coordination of multiple mobile robots , 1990, Proceedings. 5th IEEE International Symposium on Intelligent Control 1990.

[11]  Friedhelm Meyer auf der Heide,et al.  Local strategies for maintaining a chain of relay stations between an explorer and a base station , 2007, SPAA '07.

[12]  Friedhelm Meyer auf der Heide,et al.  Maintaining Communication Between an Explorer and a Base Station , 2006, BICC.

[13]  Friedhelm Meyer auf der Heide,et al.  A local O(n2) gathering algorithm , 2010, SPAA '10.

[14]  Albrecht Böttcher,et al.  Spectral properties of banded Toeplitz matrices , 1987 .

[15]  Friedhelm Meyer auf der Heide,et al.  A tight runtime bound for synchronous gathering of autonomous robots with limited visibility , 2011, SPAA '11.

[16]  Sayaka Kamei,et al.  Randomized Gathering of Mobile Robots with Local-Multiplicity Detection , 2009, SSS.

[17]  Jaroslaw Kutylowski Using mobile relays for ensuring connectivity in sparse networks , 2007 .

[18]  L. Breuer Introduction to Stochastic Processes , 2022, Statistical Methods for Climate Scientists.

[19]  Masafumi Yamashita,et al.  Distributed Anonymous Mobile Robots: Formation of Geometric Patterns , 1999, SIAM J. Comput..

[20]  Anoop Gupta,et al.  Maintaining Communication Link for a Robot Operating in a Hazardous Environment , 2004 .