Parameterized Strings: Algorithms and Applications

The parameterized string (p-string), a generalization of the traditional string, is composed of constant and parameter symbols. A parameterized match (p-match) exists between two p-strings if the constants match exactly and there exists a bijection between the parameter symbols. Historically, p-strings have been employed in source code cloning, plagiarism detection, and structural similarity between biological sequences. By handling the intricacies of the parameterized suffix, we can efficiently address complex applications with data structures also reusable in traditional matching scenarios. In this dissertation, we extend data structures for p-strings (and variants) to address sophisticated string computations.;We introduce a taxonomy of classes for longest factor problems. Using this taxonomy, we show an interesting connection between the parameterized longest previous factor (pLPF) and familiar data structures in string theory, including the border array, prefix array, longest common prefix array, and analogous p-string data structures. Exploiting this connection, we construct a multitude of data structures using the same general pLPF framework.;Before this dissertation, the p-match was defined predominately by the matching between uncompressed p-strings. Here, we introduce the compressed parameterized pattern match to find all p-matches between a pattern and a text, using only the pattern and a compressed form of the text. We present parameterized compression (p-compression) as a new way to losslessly compress data to support p-matching. Experimentally, it is shown that p-compression is competitive with standard compression schemes. Using p-compression, we address the compressed p-match independent of the underlying compression routine.;Currently, p-string theory lacks the capability to support indeterminate symbols, a staple essential for applications involving inexact matching such as in music analysis. In this work, we propose and efficiently address two new types of p-matching with indeterminate symbols. (1) We introduce the indeterminate parameterized match (ip-match) to permit matching with indeterminate holes in a p-string. We support the ip-match by introducing data structures that extend the prefix array. (2) From a different perspective, the equivalence parameterized match (e-match) evolves the p-match to consider intra-alphabet symbol classes as equivalence classes. We propose a method to perform the e-match using the p-string suffix array framework, i.e. the parameterized suffix array (pSA) and parameterized longest common prefix array (pLCP). Historically, direct constructions of the pSA and pLCP have suffered from quadratic time bounds in the worst-case. Here, we introduce new p-string theory to efficiently construct the pSA/pLCP and break the theoretical worst-case time barrier.;Biological applications have become a classical use of p-string theory. Here, we introduce the structural border array to provide a lightweight solution to the biologically-oriented variant of the p-match, i.e. the structural match (s-match) on structural strings (s-strings). Following the s-match, we show how to use s-string suffix structures to support various pattern matching problems involving RNA secondary structures. Finally, we propose/construct the forward stem matrix (FSM), a data structure to access RNA stem structures, and we apply the FSM to the detection of hairpins and pseudoknots in an RNA sequence.;This dissertation advances the state-of-the-art in p-string theory by developing data structures for p-strings/s-strings and using p-string/s-string theory in new and old contexts to address various applications. Due to the flexibility of the p-string/s-string, the data structures and algorithms in this work are also applicable to the myriad of problems in the string community that involve traditional strings.