We present a new sorting algorithm, called adaptive Shivers Sort, that exploits the existence of monotonic runs for sorting efficiently partially sorted data. This algorithm is a variant of the well-known 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 Shivers Sort is a so-called k-aware merge-sort algorithm, a class that captures ``TimSort-like'' algorithms and that was introduced by Buss and Knop.

In this article, we prove that, although adaptive Shivers Sort 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 Shivers Sort 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 Shivers Sort 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 Shivers Sort whose computational costs are optimal up to arbitrarily small multiplicative factors.