Algorithms to compute the lyndon array revisited

The motivation for having an efficient algorithm for identifying all maximal Lyndon substrings of a string comes from the work of Bannai et al. on the Runs Conjecture. In 2015, they resolved the conjecture by considering Lyndon roots of runs and they also presented a unique linear algorithm for computing all runs. The uniqueness of the algorithm lies in the fact that it relies on the knowledge of all maximal Lyndon substrings, while all other linear algorithms for runs rely on Lempel-Ziv factorization of the input string. A Lyndon array is a data structure that encodes the information of all maximal Lyndon substrings of a string. In a 2016 Prague Stringology Conference paper, Franek et al. discussed various known algorithms for computing the Lyndon array. In 2015, in his Masters' thesis, Baier designed a linear algorithm for suffix sorting. Inspired by Phase II of Baier's algorithm, in a 2017 Prague Stringology Conference paper, Franek et al. discussed the linear co-equivalency of sorting suffixes and sorting maximal Lyndon substrings. As noticed by C. Diegelmann, the first phase of Baier's algorithm identifies and sorts all maximal Lyndon substrings of the input string. Based on Phase I of Baier's algorithm, in a 2018 Prague Stringology Conference paper, Franek et. al presented an elementary (in the sense of not relying on a pre-processed global data structure) linear algorithm for identifying and sorting all maximal Lyndon substrings. This paper revisits the subject of algorithms for the Lyndon array and closes off the series of our Prague Stringology Conference contributions on the topic - it provides a simple overview of all currently known algorithms including the new development since 2016. In particular, it presents a detailed analysis of a new algorithm TRLA, and comparative measurements of three of the algorithms.