On unary nodes in tries

Abstract

International audience The difference between ordinary tries and Patricia tries lies in the fact that all unary nodes are removed in the latter. Their average number is thus easily determined from earlier results on the size of tries/Patricia tries. In a well-known contention resolution algorithm, whose probabilistic model is essentially equivalent to tries, unary nodes correspond to repetitions, i.e., steps in the algorithm that do not resolve anything at all. In this paper, we take an individual's view on such repetitions: we consider the distribution of the number of repetitions a certain contender encounters in the course of the algorithm―-which is equivalent to the number of unary nodes on the path from the root to a random string in a trie. We encounter an example of a sequence of distributions that does not actually converge to a limit distribution, but rather oscillates around a (discrete) limit distribution.

An analytic approach to the asymptotic variance of trie statistics and related structures

“…When removing all such nodes, the resulting tree has n − 1 internal nodes (for n external nodes). See [104] for an analysis connected to unary nodes of random tries, and [5,15,62,69,95] for other linear shape measures.…”