A Provably, Linear Time, In-place and Stable Merge Algorithm via the Perfect Shuffle

We reconsider a previously published (Dalkilic et al.) algorithm for merging lists by way of the perfect shuffle permutation. The original publication gave only experimental results which, although consistent with linear execution time on the samples tested, provided no analysis. Here we prove that the time complexity, in the average case, is indeed linear, although there is a Θ(n) worst case. This is then the first provably linear time merge algorithm based on the use of the perfect shuffle. We provide a proof of correctness, extend the algorithm to the general case where the lists are of unequal length and show how it can be made stable, all aspects not included in the original presentation and we give a much more concise definition of the algorithm. keywords: algorithms, perfect shuffle, merging, sorting, stability. ∗This work was supported in part by the Natural Sciences and Engineering Research Council of Canada