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Background: The longest common subsequence (LCS) problem is a classical problem in computer science, and forms the basis of the current best-performing reference-based compression schemes for genome resequencing data. Methods: First, we present a new algorithm for the LCS problem. Using the generalized suffix tree, we identify the common substrings shared between the two input sequences. Using the maximal common substrings, we construct a directed acyclic graph (DAG), based on which we determine the LCS as the longest path in the DAG. Then, we introduce an LCS-motivated reference-based compression scheme using the components of the LCS, rather than the LCS itself. Results: Our basic scheme compressed the Homo sapiens genome (with an original size of 3,080,436,051 bytes) to 15,460,478 bytes. An improvement on the basic method further reduced this to 8,556,708 bytes, or an overall compression ratio of 360. This can be compared to the previous state-of-the-art compression ratios of 157 (Wang and Zhang, 2011) and 171 (Pinho, Pratas, and Garcia, 2011). Conclusion: We propose a new algorithm to address the longest common subsequence problem. Motivated by our LCS algorithm, we introduce a new reference-based compression scheme for genome resequencing data. Comparative results against state-of-the-art reference-based compression algorithms demonstrate the performance of the proposed method.

The challenge of direct parameterized suffix sorting (p-suffix sorting) for a parameterized string (p-string), say T of length-n, is the dynamic nature of the n parameterized suffixes (p-suffixes) of T . In this work, we propose transformative approaches to direct p-suffix sorting by generating and sorting lexicographically numeric fingerprints and arithmetic codes that correspond to individual p-suffixes. Our algorithm to p-suffix sort via fingerprints is the first theoretical linear time algorithm for p-suffix sorting for nonbinary parameter alphabets, which assumes that, in practice, all codes are within the range of an integral data type. We eliminate the key problems of fingerprints by introducing an algorithm that exploits the ordering of arithmetic codes to sort p-suffixes in linear time on average. The arithmetic coding approach is further extended to handle p-strings in the worst case. This algorithm is the first direct p-suffix sorting approach in theory to execute in o(n 2 ) time in the worst case, which improves on the best known theoretical result on this problem that sorts p-suffixes based on p-suffix classifications in O (n 2 ) time. We show that, based on the algorithmic parameters and the input data, our algorithm does indeed execute in linear time in various cases, which is confirmed with experimental results.

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The parameterized longest previous factor (pLPF) problem as defined for parameterized strings (p-strings) adds a level of parameterization to the longest previous factor (LPF) problem originally defined for traditional strings. In this work, we consider the construction of the pLPF data structure and identify the strong relationship between the pLPF linear time construction and several variations of the problem. Initially, we propose a taxonomy of classes for longest factor problems. Using this taxonomy, we show an interesting connection between the pLPF and popular data structures. It is shown that a subset of longest factor problems may be created with the pLPF construction. More specifically, the pLPF problem is used as a foundation to achieve the linear time construction of popular data structures such as the LCP, parameterized-LCP (pLCP), parameterized-border (p-border) array, and border array. We further generalize the permuted-LCP for p-strings and provide a linear time construction. A number of new variations of the pLPF problem are proposed and addressed in linear time for both p-strings and traditional strings, including the longest not-equal factor (LneF), longest reverse factor (LrF), and longest factor (LF). The framework of the pLPF construction is exploited to efficiently address a multitude of data structures with prospects in various applications. Finally, we implement our algorithms and perform various experiments to confirm theoretical results.