Abstract
Consider k identical robots traversing the edges of a geometric tree. The robots have to patrol the tree by perpetually moving along edges, but without exceeding their maximum unit speed. The robots can change direction and speed anywhere on vertices or interiors of edges. The quality of patrolling is measured by idleness, which is defined as the longest time period during which any point on the tree is not visited by any of the robots. Goal is to provide algorithms describing the movement of the robots along the tree so as to minimize the idleness.
Our main contribution is to show that there is an off-line schedule, where placing k robots at specific initial positions on a geometric tree T and making them move at unit speed, permits to achieve the optimal idle time. We extend this to a graph tree model (where the robots can change direction only on vertices). We also consider on-line schedules, working for collections of simple, identical, memoryless robots, walking with constant speed, which behave according to so-called rotor-router model. We conclude with a discussion of experimental work indicating that in a random setting the rotor router is efficient on tree graphs.
J. Czyzowicz and E. Kranakis—Research supported in part by NSERC Discovery grant.
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Acknowledgements
Many thanks to Leszek Gasieniec for useful conversations in the early stages of the research.
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Czyzowicz, J., Kosowski, A., Kranakis, E., Taleb, N. (2017). Patrolling Trees with Mobile Robots. In: Cuppens, F., Wang, L., Cuppens-Boulahia, N., Tawbi, N., Garcia-Alfaro, J. (eds) Foundations and Practice of Security. FPS 2016. Lecture Notes in Computer Science(), vol 10128. Springer, Cham. https://doi.org/10.1007/978-3-319-51966-1_22
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