Smoothed Analysis of Trie Height

Tries are very simple general purpose data structures for information retrieval. A crucial parameter of a trie is its height. In the worst case the height is unbounded when the trie is built over a set of n strings. Analytical investigations have shown that the average height under many random sources is logarithmic in n. Experimental studies of trie height suggest that this holds for non-random data. In order to give an analytical explanation to these findings we perform a smoothed analysis of trie height. Smoothed analysis combines elements from both, worstcase and average-case analysis: the paradigm assumes that inputs are chosen by an adversary and that the costs are expected costs over slight random pertubations of the chosen inputs. We consider a special class of string perturbation functions which can be modelled by probabilistic finite automata (PFAs). Those perturbation functions constitute an extension of a very natural class of random edit perturbations. We observe that for the case of random deletions the smoothed trie height is unbounded. We introduce read-deterministic star-like perturbation functions, which include random substitutions and insertion as a special case, and give a necessary and sufficient condition for the smoothed trie height under read-deterministic star-like perturbation functions being logarithmic. This condition is particularly appealing since it can easily be checked by looking at the transition probabilities of the corresponding probabilistic automaton. Corresponding author. Email address: [email protected] Email address: [email protected] Email address: [email protected] 2