Two linear-time algorithms for in-place merging are presented. Both algorithms perform at most m(t+1)+n=2 t +o(m) comparisons, where m and n are the sizes of the input sequences, m n, and t = blog 2 (n=m)c. The first algorithm is for unstable merging and it carries out no more than 3(n+m)+o(m) element moves. The second algorithm is for stable merging and it accomplishes at most 5n+12m+o(m) moves. Key words: In-place algorithms, merging, sorting ? A preliminary and weaker version of this work appeared in Proceedings of the 20th Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 969, Springer-Verlag, Berlin/Heidelberg (1995), 211--220. 1 Supported by the Slovak Grant Agency for Science under contract 1/4376/97 (Project "Combinational Structures and Complexity of Algorithms"). 2 Partially supported by the Danish Natural Science Research Council under contracts 9400952 (Project "Computational Algorithmics") and 9701414 (Project "Experimental Algorithmics"). Preprint submitted to Elsevier Preprint December 19, 1995 1

In an earlier research paper [HL1], we presented a novel, yet straightforward linear-time algorithm for merging two sorted lists in a fixed amount of additional space. Constant of proportionality estimates and empirical testing reveal that this procedure is reasonably competitive with merge routines free to squander unbounded additional memory, making it particularly attractive whenever space is a critical resource. In this paper, we devise a relatively simple strategy by which this efficient merge can be made stable, and extend our results in a nontrivial way to the problem of stable sorting by merging. We also derive upper bounds on our algorithms' constants of proportionality, suggesting that in some environments (most notably external file processing) their modest run-time premiums may be more than offset by the dramatic space savings achieved.