Sam Buss and Alexander Knop
    "Strategies for Stable Merge Sorting"
    Preprint, December 2017.

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Abstract: We introduce new stable, natural merge sort algorithms, called 2-merge sort and α-merge sort. We prove upper and lower bounds for several merge sort algorithms, including Timsort, Shiver's sort, α-stack sorts, and our new 2-merge and α-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 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 α-merge sort, when φ<α<2, where φ is the golden ratio. These merge strategies can be used for any stable merge sort, not just natural merge sorts.
The new 2-merge and α-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.

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