Strategies for Stable Merge Sorting

We introduce new stable natural merge sort algorithms, called $2$-merge sort and $\alpha$-merge sort. We prove upper and lower bounds for several merge sort algorithms, including Timsort, Shivers' sort, $\alpha$-stack sorts, and our new $2$-merge and $\alpha$-merge sorts. The upper and lower bounds have the forms $c \cdot n \log m$ and $c \cdot n \log n$ for inputs of length~$n$ comprising $m$~monotone runs. For Timsort, we prove a lower bound of $(1.5 - o(1)) n \log n$. For $2$-merge sort, we prove optimal upper and lower bounds of approximately $(1.089 \pm o(1))n \log m$. We prove similar asymptotically matching upper and lower bounds for $\alpha$-merge sort, when $\varphi < \alpha < 2$, where $\varphi$ is the golden ratio. Our bounds are in terms of merge cost; this upper bounds the number of comparisons and accurately models runtime. The merge strategies can be used for any stable merge sort, not just natural merge sorts. The new $2$-merge and $\alpha$-merge sorts have better worst-case merge cost upper bounds and are slightly simpler to implement than the widely-used Timsort; they also perform better in experiments. We report also experimental comparisons with algorithms developed by Munro-Wild and Jug\'e subsequently to the results of the present paper.