Efficiently computing runs on a trie

A maximal repetition, or run, in a string, is a maximal periodic 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 each edge is labeled with a single symbol, and 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 asymptotic lower bound on the maximum density of runs in tries: lim_n→∞⁡ρ_T(n)/n>0.9932348 where ρ_T(n) is the maximum number of runs in a trie with n edges. Furthermore, we also show an O(nlog⁡log⁡n) time and O(n) space algorithm for finding all runs.