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Galloping in Fast-Growth Natural Merge Sorts

Authors Elahe Ghasemi, Vincent Jugé , Ghazal Khalighinejad

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Elahe Ghasemi
  • Sharif University of Technology, Teheran, Iran
  • Univ Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France
Vincent Jugé
  • Univ Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France
Ghazal Khalighinejad
  • Duke University, Durham, NC, USA
  • Sharif University of Technology, Teheran, Iran

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Elahe Ghasemi, Vincent Jugé, and Ghazal Khalighinejad. Galloping in Fast-Growth Natural Merge Sorts. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 68:1-68:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.68

Abstract

We study the impact of sub-array merging routines on merge-based sorting algorithms. More precisely, we focus on the galloping sub-routine that TimSort uses to merge monotonic (non-decreasing) sub-arrays, hereafter called runs, and on the impact on the number of element comparisons performed if one uses this sub-routine instead of a naive merging routine.
The efficiency of TimSort and of similar sorting algorithms has often been explained by using the notion of runs and the associated run-length entropy. Here, we focus on the related notion of dual runs, which was introduced in the 1990s, and the associated dual run-length entropy. We prove, for this complexity measure, results that are similar to those already known when considering standard run-induced measures: in particular, TimSort requires only 𝒪(n + n log(σ)) element comparisons to sort arrays of length n with σ distinct values.
In order to do so, we introduce new notions of fast- and middle-growth for natural merge sorts (i.e., algorithms based on merging runs). By using these notions, we prove that several merge sorting algorithms, provided that they use TimSort’s galloping sub-routine for merging runs, are as efficient as TimSort at sorting arrays with low run-induced or dual-run-induced complexities.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sorting and searching
Keywords
  • Sorting algorithms
  • Merge sorting algorithms
  • Analysis of algorithms

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