Train tracks with gaps: Applying the probabilistic method to trains

We identify a tradeoff curve between the number of wheels on train car, and amount track that must be installed in order to ensure car is supported by at all times. The goal build an elevated covers some large distance ?, but consists primarily gaps, so total feet actually only small fraction ?. In can support points, requirement as drives across track, least one set from rear quarter front touching show that, if has n sets evenly spaced apart its front, then it possible supports uses ?(?/n) track. consider what happens are not (and may even configured adversarially). for any configuration with each quarters ? O(?log?nn) Additionally, we there exist configurations which this asymptotically optimal. Both upper lower bounds achieved via applications probabilistic method. algorithms paper provide simple illustrative examples many core techniques combinatorics randomized algorithms. These include method alterations, use McDiarmid's inequality within method, algorithmic Lovász Local Lemma, min-hash technique, conditional probabilities.