Train Tracks with Gaps
We identify a tradeoff curve between the number of wheels on a train car, and the amount of track that must be installed in order to ensure that the train car is supported by the track at all times. The goal is to build an elevated track that covers some large distance đ, but that consists primarily of gaps, so that the total amount of feet of train track that is actually installed is only a small fraction of đ. In order so that the train track can support the train at all points, the requirement is that as the train drives across the track, at least one set of wheels from the rear quarter and at least one set of wheels from the front quarter of the train must be touching the track at all times.
We show that, if a train car has n sets of wheels evenly spaced apart in its rear and n sets of wheels evenly spaced apart in its front, then it is possible to build a train track that supports the train car but uses only Î(đ / n) feet of track. We then consider what happens if the wheels on the train car are not evenly spaced (and may even be configured adversarially). We show that for any configuration of the train car, with n wheels in each of the front and rear quarters of the car, it is possible to build a track that supports the car for distance đ and uses only O((đ log n)/n) feet of track. Additionally, we show that there exist configurations of the train car for which this tradeoff curve is asymptotically optimal. Both the upper and lower bounds are achieved via applications of the probabilistic method.
The algorithms and lower bounds in this paper provide simple illustrative examples of many of the core techniques in probabilistic combinatorics and randomized algorithms. These include the probabilistic method with alterations, the use of McDiarmidâs inequality within the probabilistic method, the algorithmic LovĂĄsz Local Lemma, the min-hash technique, and the method of conditional probabilities.
probabilistic method
algorithms
trains
LovĂĄsz Local Lemma
McDiarmidâs Inequality
Theory of computation
19:1-19:11
Regular Paper
Funded by a Fannie and John Hertz Fellowship and an NSF GRFP fellowship. Research also was sponsored by the United States Air Force Research Laboratory and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
The author would like to thank Michael A. Bender, Bradley C. Kuszmaul, and Charles E. Leiserson for several useful conversations about train tracks. The author would also like to thank Jake Hillard for offering his engineering expertise, and observing that train tracks with an asymptotically small number of pillars would likely encounter practical difficulties in the real world.
William
Kuszmaul
William Kuszmaul
Massachusetts Institute of Technology, CSAIL, Cambrige, MA, USA
10.4230/LIPIcs.FUN.2021.19
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William Kuszmaul
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