A tight lower bound for the train reversal problem

Abstract

In a 1987 Scientific American Computer Recreations article, A.K. Dewdney posed the problem of reversing an n-car train on a track with a one-car spur using the minimum amount of work. In that article, Dewdney indicated an algorithm for reversing the train that uses O( n 3) work. Shortly thereafter, Amato, Blum, Irani and Rubinfeld (Reversing Trains: A Turn of the Century Sorting Problem, J. Algorithms, Vol. 10, 1989, pp. 413-428) discovered a simple recursive algorithm that requires O( n 2log n) work to reverse a train. In this paper, we prove that Amato et al.'s algorithm is optimal up to a constant factor, i.e., we prove that any algorithm for reversing an n-car train in the Dewdney model requires Ω(n 2 log n) work.