Message Delivery in the Plane by Robots with Different Speeds
Abstract
We study a fundamental cooperative messagedelivery problem on the plane. Assume $n$ robots which can move in any direction, are placed arbitrarily on the plane. Robots each have their own maximum speed and can communicate with each other facetoface (i.e., when they are at the same location at the same time). There are also two designated points on the plane, $S$ (the source) and $D$ (the destination). The robots are required to transmit the message from the source to the destination as quickly as possible by facetoface message passing. We consider both the offline setting where all information (the locations and maximum speeds of the robots) are known in advance and the online setting where each robot knows only its own position and speed along with the positions of $S$ and $D$. In the offline case, we discover an important connection between the problem for tworobot systems and the wellknown Apollonius circle which we employ to design an optimal algorithm. We also propose a $\sqrt 2$ approximation algorithm for systems with any number of robots. In the online setting, we provide an algorithm with competitive ratio $\frac 17 \left( 5+ 4 \sqrt{2} \right)$ for tworobot systems and show that the same algorithm has a competitive ratio less than $2$ for systems with any number of robots. We also show these results are tight for the given algorithm. Finally, we give two lower bounds (employing different arguments) on the competitive ratio of any online algorithm, one of $1.0391$ and the other of $1.0405$.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.12185
 Bibcode:
 2021arXiv210912185C
 Keywords:

 Computer Science  Distributed;
 Parallel;
 and Cluster Computing;
 Computer Science  Discrete Mathematics