Shellsort with three increments

A perturbation technique can be used to simplify and sharpen A. C. Yao's theorems about the behavior of shellsort with increments (h; g; 1). In particular, when h = (n 7=15) and g = (h 1=5), the average running time is O(n 23=15). The proof involves interesting properties of the inversions in random permutations that have been h-sorted and g-sorted. Shellsort, also known as the \diminishing increment sort" 7, Algorithm 5.2.1D], puts the elements of an array (X 0 ; : : : ; X n?1) into order by successively performing a straight insertion sort on larger and larger subarrays of equally spaced elements. The algorithm consists of t passes deened by increments (h t?1 ; : : : ; h 1 ; h 0), where h 0 = 1; the jth pass makes X k X l whenever l ? k = h t?j. A. C. Yao 11] has analyzed the average behavior of shellsort in the general three-pass case when the increments are (h; g; 1). The most interesting part of his analysis dealt with the third pass, where the running time is O(n) plus a term proportional to the average number of inversions that remain after a random permutation has been h-sorted and g-sorted. Yao proved that if g and h are relatively prime, the average number of inversions remaining is where the constant implied by b O depends on g and h. He gave a complicated triple sum for (h; g), which is too diicult to explain here; we will show that Moreover, we will prove that the average number of inversions after such h-sorting and g-sorting is The main technique used in proving (0.3) is to consider a stochastic algorithm A whose output has the same distribution as the inversions of the third pass of shellsort. Then by slightly perturbing the probabilities that deene A, we will obtain an algorithm A whose output has the expected value (h; g)n exactly. Finally we will prove that the perturbations cause the expected value to change by at most O(g 3 h 2). Section 1 introduces basic techniques for inversion counting, and section 2 adapts those techniques to a random input model. Section 3 proves that the crucial random variables needed for inversion counting are nearly uniform; then section 4 shows that the leading term (h; g)n in (0.3) would be exact if those variables were perfectly uniform. …