Relations Between Average-Case and Worst-Case Complexity

2005 ◽  
pp. 422-432
Author(s):  
A. Pavan ◽  
N. V. Vinodchandran
Keyword(s):  
Worst Case ◽  
Average Case ◽  
Case Complexity ◽  
Worst Case Complexity

scholarly journals Asymptotic and Exact Results on the Complexity of the Novelli-Pak-Stoyanovskii Algorithm

10.37236/6354 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Carsten Schneider ◽  
Robin Sulzgruber
Keyword(s):  
Exact Formula ◽  
Sorting Algorithm ◽  
Limit Curve ◽  
Hook Length ◽  
Asymptotic Results ◽  
Worst Case ◽  
Average Case ◽  
Case Complexity ◽  
Average Case Complexity ◽  
Worst Case Complexity

The Novelli-Pak-Stoyanovskii algorithm is a sorting algorithm for Young tableaux of a fixed shape that was originally devised to give a bijective proof of the hook-length formula. We obtain new asymptotic results on the average case and worst case complexity of this algorithm as the underlying shape tends to a fixed limit curve. Furthermore, using the summation package Sigma we prove an exact formula for the average case complexity when the underlying shape consists of only two rows. We thereby answer questions posed by Krattenthaler and Müller.

scholarly journals Computing Maximal Lyndon Substrings of a String

2020 ◽  
Author(s):  
Frantisek Franek ◽  
Michael Liut
Keyword(s):  
Theoretical Analysis ◽  
Suffix Array ◽  
Input String ◽  
Sorting Algorithm ◽  
Worst Case ◽  
Average Case ◽  
Linear Algorithm ◽  
Case Complexity ◽  
Worst Case Complexity ◽  
Novel Algorithm

There are two reasons to have an efficient algorithm for identifying all maximal Lyndon substrings of a string: firstly, Bannai et al. introduced in 2015 a linear algorithm to compute all runs of a string that relies on knowing all maximal Lyndon substrings of the input string, and secondly, Franek et al. showed in 2017 a linear equivalence of sorting suffixes and sorting maximal Lyndon substrings of a string, inspired by a novel suffix sorting algorithm of Baier. In 2016, Franek et al. presented a brief overview of algorithms for computing the Lyndon array that encodes the knowledge of maximal Lyndon substrings of the input string. Among the presented were two well-known algorithms for computing the Lyndon array: a quadratic in-place algorithm based on iterated Duval's algorithm for Lyndon factorization, and a linear algorithmic scheme based on linear suffix sorting, computing inverse suffix array, and applying to it the Next Smaller Value algorithm. Duval's algorithm works for strings over any ordered alphabet, while for linear suffix sorting, a constant or an integer alphabet is required. The authors at that time were not aware of Baier's algorithm. In 2017, our research group proposed a novel algorithm for the Lyndon array. Though the proposed algorithm is linear in the average case and has O(n log(n)) worst-case complexity, it is interesting as it emulates the fast Fourier algorithm's recursive approach and introduces tau-reduction that might be of independent interest. In 2018, we presented a linear algorithm to compute the Lyndon array of a string inspired by Phase I of Baier's algorithm for suffix sorting. This paper presents theoretical analysis of these two algorithms and provides empirical comparisons of both their C++ implementations with respect to iterated Duval's algorithm.

Relations between Average-Case and Worst-Case Complexity

2007 ◽  
Vol 42 (4) ◽  
pp. 596-607
Author(s):  
A. Pavan ◽  
N. V. Vinodchandran
Keyword(s):  
Worst Case ◽  
Average Case ◽  
Case Complexity ◽  
Worst Case Complexity

scholarly journals Computing Maximal Lyndon Substrings of a String

Algorithms ◽  
2020 ◽  
Vol 13 (11) ◽  
pp. 294
Author(s):  
Frantisek Franek ◽  
Michael Liut
Keyword(s):  
Theoretical Analysis ◽  
Suffix Array ◽  
Input String ◽  
Sorting Algorithm ◽  
Worst Case ◽  
Average Case ◽  
Linear Algorithm ◽  
Case Complexity ◽  
Worst Case Complexity ◽  
Novel Algorithm

There are two reasons to have an efficient algorithm for identifying all right-maximal Lyndon substrings of a string: firstly, Bannai et al. introduced in 2015 a linear algorithm to compute all runs of a string that relies on knowing all right-maximal Lyndon substrings of the input string, and secondly, Franek et al. showed in 2017 a linear equivalence of sorting suffixes and sorting right-maximal Lyndon substrings of a string, inspired by a novel suffix sorting algorithm of Baier. In 2016, Franek et al. presented a brief overview of algorithms for computing the Lyndon array that encodes the knowledge of right-maximal Lyndon substrings of the input string. Among those presented were two well-known algorithms for computing the Lyndon array: a quadratic in-place algorithm based on the iterated Duval algorithm for Lyndon factorization and a linear algorithmic scheme based on linear suffix sorting, computing the inverse suffix array, and applying to it the next smaller value algorithm. Duval’s algorithm works for strings over any ordered alphabet, while for linear suffix sorting, a constant or an integer alphabet is required. The authors at that time were not aware of Baier’s algorithm. In 2017, our research group proposed a novel algorithm for the Lyndon array. Though the proposed algorithm is linear in the average case and has O(nlog(n)) worst-case complexity, it is interesting as it emulates the fast Fourier algorithm’s recursive approach and introduces τ-reduction, which might be of independent interest. In 2018, we presented a linear algorithm to compute the Lyndon array of a string inspired by Phase I of Baier’s algorithm for suffix sorting. This paper presents the theoretical analysis of these two algorithms and provides empirical comparisons of both of their C++ implementations with respect to the iterated Duval algorithm.

scholarly journals On the optimal order of worst case complexity of direct search

2015 ◽  
Vol 10 (4) ◽  
pp. 699-708 ◽  
Author(s):  
M. Dodangeh ◽  
L. N. Vicente ◽  
Z. Zhang
Keyword(s):  
Direct Search ◽  
Optimal Order ◽  
Worst Case ◽  
Case Complexity ◽  
Worst Case Complexity
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