On the Optimality of Tape Merge of Two Lists with Similar Size

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 \cite{taocp} independently discovered that the tape merge algorithm is optimal in the worst case when the two lists have the same size. In the seminal papers, Stockmeyer and Yao\cite{yao}, Murphy and Paull\cite{3k3}, and Christen\cite{christen1978optimality} independently showed when the lists to be merged are of size $m$ and $n$ satisfying $m\leq n\leq\lfloor\frac{3}{2}m\rfloor+1$, 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 used to prove lower bounds is Knuth's adversary methods \cite{taocp}. In addition, we show that the lower bound cannot be improved to 1.8 via Knuth's adversary methods. We also develop a new inequality about Knuth's adversary methods, which might be interesting in its own right. Moreover, we design a simple procedure to achieve constant improvement of the upper bounds for $2m-2\leq n\leq 3m $.