Minimal Suffix and Rotation of a Substring in Optimal Time

For a text given in advance, the substring minimal suffix queries ask to determine the lexicographically minimal non-empty suffix of a substring specified by the location of its occurrence in the text. We develop a data structure answering such queries optimally: in constant time after linear-time preprocessing. This improves upon the results of Babenko et al. (CPM 2014), whose trade-off solution is characterized by $\Theta(n\log n)$ product of these time complexities. Next, we extend our queries to support concatenations of $O(1)$ substrings, for which the construction and query time is preserved. We apply these generalized queries to compute lexicographically minimal and maximal rotations of a given substring in constant time after linear-time preprocessing. Our data structures mainly rely on properties of Lyndon words and Lyndon factorizations. We combine them with further algorithmic and combinatorial tools, such as fusion trees and the notion of order isomorphism of strings

Approximate String Matching With Dynamic Programming and Suffix Trees

The importance and the contribution of string matching algorithms to the modern society cannot be overstated. From basic search algorithms such as spell checking and data querying, to advanced algorithms such as DNA sequencing, trend analysis and signal processing, string matching algorithms form the foundation of many aspects in computing that have been pivotal in technological advancement. In general, string matching algorithms can be divided into the categories of exact string matching and approximate string matching. We study each area and examine some of the well known algorithms. We probe into one of the most intriguing data structure in string algorithms, the suffix tree. The lowest common ancestor extension of the suffix tree is the key to many advanced string matching algorithms. With these tools, we are able to solve string problems that were, until recently, thought intractable by many. Another interesting and relatively new data structure in string algorithms is the suffix array, which has significant breakthroughs in its linear time construction in recent years. Primarily, this thesis focuses on approximate string matching using dynamic programming and hybrid dynamic programming with suffix tree. We study both approaches in detail and see how the merger of exact string matching and approximate string matching algorithms can yield synergistic results in our experiments