Abstract Evolutionary trees describing the relationship for a set of species are central in evolutionarybiology, and quantifying differences between evolutionary trees is therefore an important task. The quartet distance is a distance measure between trees previously proposed by Estabrook,McMorris and Meacham. The quartet distance between two unrooted evolutionary trees is the number of quartet topology differences between the two trees, where a quartet topologyis the topological subtree induced by four species. In this paper, we present an algorithm for computing the quartet distance between two unrooted evolutionary trees of n species, whereall internal nodes have degree three, in time O(n log n). The previous best algorithm for theproblem uses time O(n2).

The string statistics problem consists of preprocessing a string of length n such that given a query pattern of length m, the maximum number of non-overlapping occurrences of the query pattern in the string can be reported efficiently...

A seed in a word is a relaxed version of a period. We show a linear time algorithm computing a compact representation of all the seeds of a word, in particular, the shortest seed. Thus, we solve an open problem stated in the survey by Smyth (2000) and improve upon a previous over 15-year old O(n log n) algorithm by Iliopoulos, Moore and Park (1996). Our approach is based on combinatorial relations between seeds and a variant of the LZ-factorization (used here for the first time in context of seeds). 1