Strategies for Stable Merge Sorting

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 · n logm and c · n logn for inputs of length n comprising m runs. For Timsort, we prove a lower bound of (1.5−o(1))n logn. For 2-merge sort, we prove optimal upper and lower bounds of approximately (1.089±o(1))n logm. 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.