Dynamic Z-Fast Tries

Abstract

We describe a dynamic version of the z-fast trie, a new data structure inspired by the research started by the van Emde Boas trees [12] and followed by the development of y-fast tries [13]. The dynamic z-fast trie is a very simple, uniform data structure: given a set S of n variable-length strings, it is formed by a standard compacted trie on S (with two additional pointers per node), endowed with a dictionary of size n − 1. With this simple setup, the dynamic z-fast trie provides predecessors/successors in time O(log max{|x|,|x ⁺|,|x ⁻ |}) (x ^± is the successor/predecessor of x in S) for strings of length linear in the machine-word size w. Prefix queries are answered in time O(log|x| + k), and range queries in time O(log max{|x|,|y|,|x ⁻ |,|y ⁺|} + k), where k is the number of elements in the output and x (and y) represent the input of the prefix (range) queries. Updates are performed within the same bounds in expectation (or with high probability using an appropriate dictionary). We then show a simple modification that makes it possible to handle strings of length up to 2^w; in this case, predecessor/successor queries and updates are supported in O(|x|/w + log max{|x|,|x ⁺|,|x ⁻ |}) time, (and O(|x|/B + log max{|x|,|x ⁺|,|x ⁻ |}) I/Os in the cache-oblivious model) with high probability. The space occupied by a dynamic z-fast trie, beside that necessary to store S, is just of 12n pointers, n integers and, in the “long string” case, O(n) signatures of O(w) bits each.