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 $h=\Theta(n^{7/15})$ and $g=\Theta(h^{1/5})$, 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

Janson, Svante

Knuth, Donald E.

English

arXiv

. A perturbation technique can be used to simplify and sharpen A. C. Yao&apos;s theorems about the behavior of shellsort with increments (h; g; 1). In particular, when  h = \Theta(n  7=15  ) and g = \Theta(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 &quot;diminishing increment sort&quot; [7, Algorithm 5.2.1D], puts the elements of an array (X 0 ; : : : ; X n\Gamma1 ) into order by successively performing a straight insertion sort on larger and larger subarrays of equally spaced elements. The algorithm consists of t passes defined by increments (h t\Gamma1 ; : : : ; h 1 ; h 0 ), where h 0 = 1; the jth pass makes X k  X l whenever l \Gamma k = h t\Gammaj . 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 ..

Shellsort with three increments

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