On the Average Density and Selectivity of Nodes in Multi-Digit Tries

We introduce and study two parameters of nodes in tries with multi-digit branching. The first parameter, which we call a density of a multi-digit node, is a ratio of the number of non-empty pointers (i.e. pointers to the attached non-empty sub-tries) to the total number of pointers in this node. The second parameter, which we call a selectivity of a node, is a ratio of the number of pointers to external nodes (containing uniquely identified strings) to the total number of strings processed by this node. We show, that in a memoryless model, the average density and the average selectivity of an r-digit node over n binary strings both yield asymptotic expressions in the form Φ (r − r∗) / σ∗√log n , where, however, their central values r∗ and factors σ∗ are different if the source is asymmetric. We use our findings to explain several interesting facts in the average behaviour of multidigit tries, and complement our presentation with a number of experimental results.