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.
翻译:我们确定火车车轮数与必须安装的轨迹数量之间的折中曲线,以确保火车车轮在任何时候都得到轨道的支持。目标是建立一个高轨,覆盖一定的远距离$\ell$,但主要是缺口,这样实际安装的火车轨数的总数只有小部分美元。为了让火车轨迹在所有点都能支持火车,需要的是,火车轨轨迹跨轨道,至少从后方转来一组轮子,至少从火车前方转来一组离车轮得到轨道支持。目标是建立一个高轨道,覆盖一定的轨道,覆盖一定的轨道,如果火车车轮的轮子在车尾间平平平平平平平,那么火车的轨轨迹就有可能建造一个支持火车车轮子,但只能用$\ell\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\