The Ubiquitous Digital Tree

The digital tree also known as trie made its first appearance as a general-purpose data structure in the late 1950’s. Its principle is a recursive partitioning based on successive bits or digits of data items. Under various guises, it has then surfaced in the management of very large data bases, in the design of efficient communication protocols, in quantitative data mining, in the leader election problem of distributed computing, in data compression, as well as in some corners of computational geometry. The algorithms are invariably very simple, easy to implement, and in a number of cases surprisingly efficient. The corresponding quantitative analyses pose challenging mathematical problems and have triggered a flurry of research works. Generating functions and symbolic methods, singularity analysis, the saddle-point method, transfer operators of dynamical systems theory, and the Mellin transform have all been found to have a bearing on the probabilistic behaviour of trie algorithms. We offer here a perspective on the rich algorithmic, analytic, and probabilistic aspects of tries, culminating with a connection between a sorting problem and the Riemann hypothesis. Invited lecture, STACS06, Marseille, February 2006. Proceedings in Lecture Notes in Computer Science. While, in the course of the 1980s and 1990s, a large portion of the theoretical computer science community was massively engaged in worst-case design and analysis issues, the discovery of efficient algorithms continued to make tangible progress. Such algorithms are often based on simple and elegant ideas, and, accordingly, their study is likely to reveal structures of great mathematical interest. Also, efficiency is much better served by probabilistic analyses than by the teratological constructions of worst-case theory. I propose to illustrate this point of view by discussing a fundamental process shared by algorithmics, combinatorics, and discrete probability theory—the digital tree process. Because of space-time limitations, this text, an invited lecture at STACS’06, cannot be but a brief guide to a rich subject whose proper development would require a book of full length. 1. The basic structure Consider first as domain of our data items the set of all infinitely long binary strings, B = {0, 1}∞. The goal is to devise a data structure in which elements of B can be stored and easily retrieved. Given a finite set ω ⊂ B like ω = { 110100 · · · , 01011 · · · , 01101 · · · } , a natural idea is to form a tree in which the left subtree will contain all the elements starting with 0, all elements starting with 1 going to the right subtree. (On the example, the last two strings would then go to the left subtree, the first one to the To be mitigated by common sense and a good feel for algorithmic engineering, of course!