Introduction and Survey; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems --- Sorting and Searching; E.5 [Data]: Files --- Sorting/searching; G.3 [Mathematics of Computing]: Probability and Statistics --- Probabilistic algorithms; E.2 [Data Storage Representation]: Composite structures, linked representations. General Terms: Algorithms, Theory. Additional Key Words and Phrases: Adaptive sorting algorithms, Comparison trees, Measures of disorder, Nearly sorted sequences, Randomized algorithms. A Survey of Adaptive Sorting Algorithms 2 CONTENTS INTRODUCTION I.1 Optimal adaptivity I.2 Measures of disorder I.3 Organization of the paper 1.WORST-CASE ADAPTIVE (INTERNAL) SORTING ALGORITHMS 1.1 Generic Sort 1.2 Cook--Kim division 1.3 Partition Sort 1.4 Exponential Search 1.5 Adaptive Merging 2.EXPECTED-CASE ADAPTIV

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

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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.

Two in-place variants of the classical mergesort algorithm are analysed in detail. The first, straightforward variant performs at most N log 2 N + O(N ) comparisons and 3N log 2 N + O(N ) moves to sort N elements. The second, more advanced variant requires at most N log 2 N + O(N ) comparisons and "N log 2 N moves, for any fixed " ? 0 and any N ? N ("). In theory, the second one is superior to advanced versions of heapsort. In practice, due to the overhead in the index manipulation, our fastest in-place mergesort behaves still about 50 per cent slower than the bottom-up heapsort. However, our implementations are practical compared to mergesort algorithms based on in-place merging. Key words: sorting, mergesort, in-place algorithms CR Classification: F.2.2 1.

First we present a new variant of Merge-sort, which needs only 1.25n space, because it uses space again, which becomes available within the current stage. It does not need more comparisons than classical Merge-sort. The main result is an easy to implement method of iterating the procedure in-place starting to sort 4/5 of the elements. Hereby we can keep the additional transport costs linear and only very few comparisons get lost, so that n log n − 0.8n comparisons are needed. We show that we can improve the number of comparisons if we sort blocks of constant length with Merge-Insertion, before starting the algorithm. Another improvement is to start the iteration with a better version, which needs only (1+ε)n space and again additional O(n) transports. The result is, that we can improve this theoretically up to n log n − 1.3289n comparisons in the worst case. This is close to the theoretical lower bound of n log n − 1.443n. The total number of transports in all these versions can be reduced to ε n log n+O(1) for any ε> 0. 1

An in-place 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. In this paper we describe three in-place algorithms for computing the convex hull of a planar point set. All three algorithms are optimal, some more so than others...

A parallel sorting algorithm is presented for general purpose internal sorting on MIMD machines. The algorithm initially sorts the elements within each node using a serial sorting algorithm, then proceeds with a two-phase parallel merge.

Abstract. We introduce a new stable minimum storage algorithm for merging that needs O(m log ( n + 1)) element comparisons, where m and m n are the sizes of the input sequences with m ≤ n. According to the lower bound for merging, our algorithm is asymptotically optimal regarding the number of comparisons. The presented algorithm rearranges the elements to be merged by rotations, where the areas to be rotated are determined by a simple principle of symmetric comparisons. This style of minimum storage merging is novel and looks promising. Our algorithm has a short and transparent definition. Experimental work has shown that it is very efficient and so might be of high practical interest. 1