Strategies for stable merge sorting

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 · n log m and c · 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 ± o(1))n log m. We state similar asymptotically matching upper and lower bounds for α-merge sort, when ϕ < α < 2, where ϕ 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 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.