A space-efficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. We describe four space-efficient algorithms for computing the convex hull of a planar point set.

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

It is shown that an array of n elements can be sorted using O(1) extra space, O(n log n= log log n) element moves, and n log 2 n+O(n log log n) comparisons. This is the first in-place sorting algorithm requiring o(n log n) moves in the worst case while guaranteeing O(n log n) comparisons but, due to the constant factors involved, the algorithm is predominantly of theoretical interest. Key words: In-place algorithms, sorting, merging, mergesort, multiway merge 1