Adaptive Shivers Sort: An Alternative Sorting Algorithm

We present a new sorting algorithm, called adaptive ShiversSort, that exploits the existence of monotonic runs for sorting efficiently partially sorted data. This algorithm is a variant of the wellknown algorithm TimSort, which is the sorting algorithm used in standard libraries of programming languages such as Python or Java (for non-primitive types). More precisely, adaptive ShiversSort is a so-called k-aware merge-sort algorithm, a class that was introduced by Buss and Knop that captures "TimSort-like" algorithms.

In this article, we prove that, although adaptive ShiversSort is simple to implement and differs only slightly from TimSort, its computational cost, in number of comparisons performed, is optimal within the class of natural merge-sort algorithms, up to a small additive linear term: this makes adaptive ShiversSort the first k-aware algorithm to benefit from this property, which is also a 33% improvement over TimSort's worst-case. This suggests that adaptive ShiversSort could be a strong contender for being used instead of TimSort.

Then, we investigate the optimality of k-aware algorithms: we give lower and upper bounds on the best approximation factors of such algorithms, compared to optimal stable natural merge-sort algorithms. In particular, we design generalisations of adaptive ShiversSort whose computational costs are optimal up to arbitrarily small multiplicative factors.