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 $\ell$, 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 $\ell$. 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 $\Theta( \ell / n )$ 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 $\ell$ and uses only $O\left(\frac{\ell \log n}{n}\right)$ 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.

中文翻译：

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带有间隔的火车轨道：将概率方法应用于火车

我们确定了火车车轮数与必须安装的轨道数量之间的折衷曲线，以确保火车始终受到轨道的支撑。目标是建造一条高架轨道，覆盖一定距离，但主要由间隙组成，因此实际安装的火车轨道的总脚数仅占其很小的一部分。为了使火车轨道能够在所有点上支撑火车，要求是当火车横穿轨道行驶时，从后四分之一起至少一组车轮，从前四分之一起至少一组车轮。火车必须始终触摸轨道。我们表明，如果火车车厢的后轮平均间隔设置为$ n $，而前轮的间隔均匀设置为$ n $，那么就可以构建一条支持火车但仅使用$ \ Theta（\ ell / n）$英尺轨道的火车轨道。然后，我们考虑如果火车上的车轮间距不均匀（甚至可能在对抗状态下配置）会发生什么情况。我们表明，对于任何配置的火车车厢，在其前后各有$ n $个轮子的情况下，可以建立一条轨道来支撑汽车，使其行驶距离$ \ ell $，并且仅使用$ O \ left（\ frac {\ ell \ log n} {n} \ right）$英尺。另外，我们表明，存在火车的配置，该配置的折衷曲线是渐近最优的。上限和下限均通过应用概率方法来实现。然后，我们考虑如果火车上的车轮间距不均匀（甚至可能在对抗状态下配置）会发生什么情况。我们表明，对于任何配置的火车车厢，在其前后各有$ n $个轮子的情况下，可以建立一条轨道来支撑汽车，使其行驶距离$ \ ell $，并且仅使用$ O \ left（\ frac {\ ell \ log n} {n} \ right）$英尺。另外，我们表明，存在火车的配置，该配置的折衷曲线是渐近最优的。上限和下限均通过应用概率方法来实现。然后，我们考虑如果火车上的车轮间距不均匀（甚至可能在对抗状态下配置）会发生什么情况。我们表明，对于任何配置的火车车厢，在其前后各有$ n $个轮子的情况下，可以建立一条轨道来支撑汽车，使其行驶距离$ \ ell $，并且仅使用$ O \ left（\ frac {\ ell \ log n} {n} \ right）$英尺。另外，我们表明，存在火车的配置，该配置的折衷曲线是渐近最优的。上限和下限均通过应用概率方法来实现。可以建立一条支持汽车距离$ \ ell $且仅使用$ O \ left（\ frac {\ ell \ log n} {n} \ right）$英尺的轨道。另外，我们表明，存在火车的配置，该配置的折衷曲线是渐近最优的。上限和下限均通过应用概率方法来实现。可以建立一条支持汽车距离$ \ ell $且仅使用$ O \ left（\ frac {\ ell \ log n} {n} \ right）$英尺的轨道。另外，我们表明，存在火车的配置，该配置的折衷曲线是渐近最优的。上限和下限均通过应用概率方法来实现。