A Lower Bound on the Average-Case Complexity of Shellsort

TL;DR

This paper establishes a fundamental lower bound on the average-case complexity of Shellsort, demonstrating that it requires at least on the order of p times n to the power of 1 plus 1 over p data movements, for any number of passes p.

Contribution

It introduces a novel incompressibility-based proof technique to derive lower bounds on Shellsort's average-case complexity, applicable to various sorting algorithms.

Findings

Shellsort's average-case complexity is at least Ω(p n^{1 + 1/p}) for any p.

The proof employs Kolmogorov complexity and incompressibility arguments.

Similar techniques are used to analyze other sorting algorithms.

Abstract

We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a $p$-pass Shellsort for any incremental sequence is $\Omega (pn^{1 + 1/p})$ for every $p$. The proof method is an incompressibility argument based on Kolmogorov complexity. Using similar techniques, the average-case complexity of several other sorting algorithms is analyzed.