MergeInsertion, also known as the Ford-Johnson algorithm, is a sorting algorithm which, up to today, for many input sizes achieves the best known upper bound on the number of comparisons. Indeed, it gets extremely close to the information-theoretic lower bound. While the worst-case behavior is well understood, only little is known about the average case. This work takes a closer look at the average case behavior. In particular, we establish an upper bound of \(n \log n - 1.4005n + o(n)\) comparisons. We also give an exact description of the probability distribution of the length of the chain a given element is inserted into and use it to approximate the average number of comparisons numerically. Moreover, we compute the exact average number of comparisons for n up to 148. Furthermore, we experimentally explore the impact of different decision trees for binary insertion. To conclude, we conduct experiments showing that a slightly different insertion order leads to a better average case and we compare the algorithm to Manacher’s combination of merging and MergeInsertion as well as to the recent combined algorithm with (1,2)-Insertionsort by Iwama and Teruyama.

MergeInsertion，也称为Ford-Johnson算法，是一种排序算法，到今天，对于许多输入大小，它都达到了比较次数最著名的上限。确实，它非常接近信息理论的下限。虽然最坏情况的行为已广为人知，但对一般情况知之甚少。这项工作将仔细研究平均案例行为。特别是，我们建立了\（n \ log n-1.4005n + o（n）\）比较的上限。我们还给出了插入给定元素的链的长度的概率分布的精确描述，并使用它来近似地计算比较的平均次数。此外，我们计算出n的精确平均比较数最多148。此外，我们实验性地探索了不同决策树对二进制插入的影响。总而言之，我们进行的实验表明，插入顺序略有不同会导致更好的平均情况，我们将算法与Manacher合并和MergeInsertion的组合以及最近由Iwama和（1）提出的（1,2）-Insertionsort的组合算法进行了比较照山