Preprint (full version):

Sam Buss and Alexander Knop
"Strategies for Stable Merge Sorting"
Submitted for publication, February 2019

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 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 α-merge sort, when φ<α<2, where φ is the golden ratio.