Abstract
Two mobile robots are initially placed at the same point on an infinite line. Each robot may move on the line in either direction not exceeding its maximal speed. The robots need to find a stationary target placed at an unknown location on the line. The search is completed when both robots arrive at the target point. The target is discovered at the moment when either robot arrives at its position. The robot knowing the placement of the target may communicate it to the other robot. We look for the algorithm with the shortest possible search time (i.e. the worst-case time at which both robots meet at the target) measured as a function of the target distance from the origin (i.e. the time required to travel directly from the starting point to the target at unit velocity).
We consider two standard models of communication between the robots, namely wireless communication and communication by meeting. In the case of communication by meeting, a robot learns about the target while sharing the same location with the robot possessing this knowledge. We propose here an optimal search strategy for two robots including the respective lower bound argument, for the full spectrum of their maximal speeds. This extends the main result of Chrobak et al. (SOFSEM 2015) referring to the exact complexity of the problem for the case when the speed of the slower robot is at least one third of the faster one. In addition, we consider also the wireless communication model, in which a message sent by one robot is instantly received by the other robot, regardless of their current positions on the line. In this model, we design an optimal strategy whenever the faster robot is at most 6 times faster than the slower one.
J. Czyzowicz—Partially funded by NSERC. Part of this work was done while Jurek Czyzowicz was visiting the LaBRI as a guest professor of the University of Bordeaux.
L. Gąsieniec—Sponsored in part by the University of Liverpool initiative Networks Systems and Technologies NeST.
D. Ilcinkas—Partially funded by the ANR project MACARON (ANR-13-JS02-002). This study has been carried out in the frame of the “Investments for the future” Programme IdEx Bordeaux – CPU (ANR-10-IDEX-03-02).
R. Klasing—Partially funded by the ANR project DISPLEXITY (ANR-11-BS02-014).
D. Pająk—Partially funded by the National Science Centre, Poland - grant number 2015/17/B/ST6/01897.
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Notes
- 1.
Note that the sequence \((p_k)_{k\in \mathbb {Z}}\) is understood as prescribing infinitesimally small moves for the robot in the two directions around the origin at the beginning of the execution (when time is in the neighborhood of 0, i.e., at the beginning of the execution, the robot visits points \(p_k\) for k in the neighborhood of \(-\infty \), hence it makes infinitesimal moves). Algorithm \(\mathcal {A^*}\), described below, has similar behavior. In order to avoid this, we could start the sequence \(p_k\) from any finite k (instead of \(-\infty \)). This would result in small constant additive terms appearing throughout the calculations, but the asymptotic behavior of the algorithm and in particular the efficiency measure \(\tau (\mathcal {A})\) would be unaffected.
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Bampas, E. et al. (2016). Linear Search by a Pair of Distinct-Speed Robots. In: Suomela, J. (eds) Structural Information and Communication Complexity. SIROCCO 2016. Lecture Notes in Computer Science(), vol 9988. Springer, Cham. https://doi.org/10.1007/978-3-319-48314-6_13
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