Smoothed heights of tries and patricia tries

Tries and patricia tries are two popular data structures for storing strings. Let HnHn denote the height of the trie (the patricia trie, respectively) on a set of n   strings. Under the uniform distribution model on the strings, it is well known that Hn/log⁡n→2Hn/log⁡n→2 for tries and Hn/log⁡n→1Hn/log⁡n→1 for patricia tries, when n   approaches infinity. Nevertheless, in the worst case, the height of a trie can be unbounded and the height of a patricia trie is in Θ(n)Θ(n). To better understand the practical performance of both tries and patricia tries, we investigate these two classical data structures in a smoothed analysis model. Given a set S={s1,s2,…,sn}S={s1,s2,…,sn} of n binary strings, we perturb the set by adding an i.i.d.   Bernoulli random noise to each bit of every string. We show that the resulting smoothed heights of the trie and the patricia trie are both in Θ(log⁡n)Θ(log⁡n).