Analysis of Algorithms for Permutations Biased by Their Number of Records

TL;DR

This paper studies the complexity of algorithms on permutations considering data bias towards records, introducing a new Ewens-like distribution model that accounts for presortedness and analyzing expected algorithm performance.

Contribution

It introduces a novel Ewens-like distribution model for permutations based on records, linking data bias to algorithm complexity analysis.

Findings

Expected values of permutation statistics under the new model

Expected running times of Insertion Sort and Min-Max variants

Model's relevance to presortedness and data bias

Abstract

The topic of the article is the parametric study of the complexity of algorithms on arrays of pairwise distinct integers. We introduce a model that takes into account the non-uniformness of data, which we call the Ewens-like distribution of parameter $\theta$ for records on permutations: the weight $\theta^r$ of a permutation depends on its number $r$ of records. We show that this model is meaningful for the notion of presortedness, while still being mathematically tractable. Our results describe the expected value of several classical permutation statistics in this model, and give the expected running time of three algorithms: the Insertion Sort, and two variants of the Min-Max search.