Computing runs on a trie

A maximal repetition, or run, in a string, is a periodically maximal substring whose smallest period is at most half the length of the substring. In this paper, we consider runs that correspond to a path on a trie, or in other words, on a rooted edge-labeled tree where the endpoints of the path must be a descendant/ancestor of the other. For a trie with $n$ edges, we show that the number of runs is less than $n$. We also show an $O(n\sqrt{\log n}\log \log n)$ time and $O(n)$ space algorithm for counting and finding the shallower endpoint of all runs. We further show an $O(n\sqrt{\log n}\log^2\log n)$ time and $O(n)$ space algorithm for finding both endpoints of all runs.