Efficiently computing runs on a trie

Abstract 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 ( n log ⁡ log ⁡ n ) time and O ( n ) space algorithm for finding all runs.

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