Abstract

We give improved lower bounds for Shellsort based on a new and relatively simple proof idea. The lower bounds obtained are both stronger and more general than the previously known bounds. In particular, they hold for nonmonotone increment sequences and adaptive Shellsort algorithms, as well as for some recently proposed variations of Shellsort. 1 Introduction Shellsort is a classical sorting algorithm introduced by Shell in 1959 [15]. The algorithm is based on a sequence H = h 0 ; : : : ; hm\Gamma1 of positive integers called an increment sequence. An input file A = A[0]; : : : ; A[n \Gamma 1] of elements is sorted by performing an h j -sort for every increment h j in H, starting with hm\Gamma1 and going down to h 0 . Every h j -sort partitions the positions of the input array into congruence classes modulo h j , and then performs Insertion Sort on each of these classes. It is not difficult to see that at least one of the h j 's must be equal to 1 in order for the algorithm to sort ...

Citations