Straightforward Entropy-Sensitive Mergesort

In this paper, we present a stable mergesort variant, "directional mergesort", that to sort an array of $n$ elements makes no more than $nH+3n$ comparisons and $1.5nH+O(n)$ moves where $H$ is the run-based entropy of the input sequence, matching the best existing algorithms in run-adaptive sorting. However, our algorithm is surprisingly minimalistic: it leverages only skip checks, i.e., bypassing the merge step when both halves are already in order, and dynamically changing the direction of merging based on the state of the subarrays. As dynamic run scanning is avoided, all merge steps remain static and derivable from $n$, enabling a reduction to $O(1)$ words of stack space (i.e. space usage excluding the merge buffer) in "directional mergesort$^{++}$", thus improving over the predecessors' $O(\lg n)$ words. Importantly, as directional mergesort adapts only to non-decreasing runs, directional mergesort$^{++}$ also applies a parallel set of rules for (strictly) decreasing runs, allowing the algorithm to also adapt to decreasing runs whilst retaining the original comparison bounds.