Fast Sorting of Weyl Sequences Using Comparisons

An algorithm is given which makes only $O(\log n)$ comparisons, and which will determine the ordering of the uniformly distributed (pseudo random) Weyl sequences given by ${ (k\alpha )\bmod 1:1 \leqq k \leqq n} $, where $\alpha $ is an unspecified irrational number. This result is shown to be best possible in the sense that no algorithm can perform the same task with fewer than $ \Omega (\log n)$ comparisons.