Sorting in-place with a Worst Case complexity of n log n-1.3n + O(log n) comparisons and ε n log n + O(1) transports

Abstract

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 enlogn 4-O(1) for any ε > 0.