Repetition Detection in a Dynamic String

A string UU for a non-empty string U is called a square. Squares have been well-studied both from a combinatorial and an algorithmic perspective. In this paper, we are the first to consider the problem of maintaining a representation of the squares in a dynamic string S of length at most n. We present an algorithm that updates this representation in n^o(1) time. This representation allows us to report a longest square-substring of S in O(1) time and all square-substrings of S in O(output) time. We achieve this by introducing a novel tool - maintaining prefix-suffix matches of two dynamic strings. We extend the above result to address the problem of maintaining a representation of all runs (maximal repetitions) of the string. Runs are known to capture the periodic structure of a string, and, as an application, we show that our representation of runs allows us to efficiently answer periodicity queries for substrings of a dynamic string. These queries have proven useful in static pattern matching problems and our techniques have the potential of offering solutions to these problems in a dynamic text setting

Dynamic Dictionary Matching in External Memory

AbstractIn thedynamic dictionary matchingproblem, a dictionaryDcontains a set of patterns that can change over time under insertion and deletion of individual patterns. Given an arbitrary textT, we must efficiently list all the dictionary patterns that occur at each text position. We investigate the I/O complexity of this problem for a large dictionary that must be stored in external storage devices. By following a completely new approach, we devise an efficient solution which is based upon the SB-tree data structure (P. Ferragina and R. Grossi, 1995,in“Proc. ACM Symposium on Theory of Computing,” pp. 693–702), and a novel notion of certificate for the dictionary matching problem. Our data structure can be adapted to efficiently work in main memory and to solve other problems, thus providing a new insight into the nature of the dictionary matching problem