Lyndon Arrays Simplified

A Lyndon word is a string that is lexicographically smaller than all of its proper suffixes (e.g., "airbus" is a Lyndon word; "amtrak" is not a Lyndon word because its suffix "ak" is lexicographically smaller than "amtrak"). The Lyndon array (sometimes called Lyndon table) identifies the longest Lyndon prefix of each suffix of a string. It is well known that the Lyndon array of a length-n string can be computed in O(n) time. However, most of the existing algorithms require the suffix array, which has theoretical and practical disadvantages. The only known algorithms that compute the Lyndon array in O(n) time without the suffix array (or similar data structures) do so in a particularly space efficient way (Bille et al., ICALP 2020), or in an online manner (Badkobeh et al., CPM 2022). Due to the additional goals of space efficiency and online computation, these algorithms are complicated in technical detail. Using the main ideas of the aforementioned algorithms, we provide a simpler and easier to understand algorithm that computes the Lyndon array in O(n) time

Space Efficient Construction of Lyndon Arrays in Linear Time

Given a string S of length n, its Lyndon array identifies for each suffix S[i..n] the next lexicographically smaller suffix S[j..n], i.e. the minimal index j > i with S[i..n] ? S[j..n]. Apart from its plain (n log? n)-bit array representation, the Lyndon array can also be encoded as a succinct parentheses sequence that requires only 2n bits of space. While linear time construction algorithms for both representations exist, it has previously been unknown if the same time bound can be achieved with less than ?(n lg n) bits of additional working space. We show that, in fact, o(n) additional bits are sufficient to compute the succinct 2n-bit version of the Lyndon array in linear time. For the plain (n log? n)-bit version, we only need ?(1) additional words to achieve linear time. Our space efficient construction algorithm makes the Lyndon array more accessible as a fundamental data structure in applications like full-text indexing

Algorithms and Lower Bounds for Ordering Problems on Strings

This dissertation presents novel algorithms and conditional lower bounds for a collection of string and text-compression-related problems. These results are unified under the theme of ordering constraint satisfaction. Utilizing the connections to ordering constraint satisfaction, we provide hardness results and algorithms for the following: recognizing a type of labeled graph amenable to text-indexing known as Wheeler graphs, minimizing the number of maximal unary substrings occurring in the Burrows-Wheeler Transformation of a text, minimizing the number of factors occurring in the Lyndon factorization of a text, and finding an optimal reference string for relative Lempel-Ziv encoding

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum