The problem of merging sorted lists in the least number of pairwise comparisons has been solved completely only for a few special cases. Graham and Karp (Sorting Search 3:197–207, 1999) independently discovered that the tape merge algorithm is optimal in the worst case when the two lists have the same size. In their seminal papers, Stockmeyer and Yao (SIAM J Comput 9(1):85–90, 1980), Murphy and Paull (Inf Control 42(1):87–96, 1979), and Christen (On the optimality of the straight merging algorithm, 1978) independently showed when the lists to be merged are of size m and n satisfying $$m\le n\le \lfloor \frac{3}{2}m\rfloor +1$$ m ≤ n ≤ ⌊ 3 2 m ⌋ + 1 , the tape merge algorithm is optimal in the worst case. This paper extends this result by showing that the tape merge algorithm is optimal in the worst case whenever the size of one list is no larger than 1.52 times the size of the other. The main tool we use to prove the lower bound is Knuth’s (1999) adversary methods. In addition, we show that the lower bound cannot be improved to 1.8 via Knuth’s adversary methods. Moreover, we design a simple procedure, and by invoking this procedure recursively until the remaining subproblem can be solved efficiently by another known algorithm, we achieve constant improvement of the upper bound for $$2m-2\le n\le 3m $$ 2 m - 2 ≤ n ≤ 3 m .

中文翻译：

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关于大小相似的两个列表的磁带合并的最优性

以最少的成对比较次数合并排序列表的问题仅在少数特殊情况下完全解决。Graham 和 Karp（Sorting Search 3:197–207, 1999）独立发现，当两个列表具有相同的大小时，磁带合并算法在最坏的情况下是最佳的。在他们的开创性论文中，Stockmeyer 和 Yao (SIAM J Comput 9(1):85–90, 1980)、Murphy 和 Paull (Inf Control 42(1):87–96, 1979) 和 Christen (On the optimization of the直接合并算法，1978) 独立地显示了当要合并的列表的大小为 m 和 n 时满足 $$m\le n\le \lfloor \frac{3}{2}m\rfloor +1$$ m ≤ n ≤ ⌊ 3 2 m ⌋ + 1 ，磁带合并算法在最坏情况下是最优的。本文扩展了这一结果，表明当一个列表的大小不大于另一个列表大小的 1.52 倍时，磁带合并算法在最坏的情况下是最佳的。我们用来证明下界的主要工具是 Knuth (1999) 的对抗方法。此外，我们表明无法通过 Knuth 的对抗方法将下限提高到 1.8。此外，我们设计了一个简单的过程，通过递归调用这个过程，直到剩下的子问题可以被另一个已知的算法有效地解决，我们实现了 $$2m-2\le n\le 3m $$2 的上限的不断改进m - 2 ≤ n ≤ 3 m 。