On the pivot strategy of Quicksort

Quicksort is probably the most widely used sorting method in computer science. Although it is well known that its worst-case running time is Θ(N²), it is not clear that what kind of data will cause such slow behavior. In this paper, we prove that a sorted array followed by one or more small elements will do. We also give examples for the pivot method of the middle element, since this is another popular pivot technique in practice. Our examples are pessimistic in the sense that they reach the worst case of Quicksort and cost the maximum number of comparisons for Quicksort, that they are extremely simple for they differ from sorted arrays only by one or two elements, and that they are well constructed so that one may tell their running time even at the first glance. Furthermore, we consider a general non-random pivoting scheme that is based on a constant number of messages about the array elements. We provide a general guideline to construct examples of running time Θ(N²) to this general pivoting scheme