Dynamic Z-Fast Tries

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 variablelength 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{vertical bar x vertical bar, vertical bar x(+)vertical bar, vertical bar x(-)vertical bar}) (x(+/-) is the successor/predecessor of x in S) for strings of length linear in the machineword size w. Prefix queries are answered in time O(log vertical bar x vertical bar + k), and range queries in time O(log max{vertical bar x vertical bar, vertical bar y vertical bar, vertical bar x(-)vertical bar, vertical bar y(+)vertical bar} 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(vertical bar x vertical bar/w + log max{vertical bar x vertical bar, vertical bar x(+)vertical bar, vertical bar x(-)vertical bar}) time, (and O(vertical bar x vertical bar/B + log max{vertical bar x vertical bar, vertical bar x(+)vertical bar,vertical bar x(-)vertical bar}) 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 (w) bits each.