Serial and parallel subsequence finding algorithms

Abstract

The first part of this work is on the longest increasing subsequence problem (LIS) and its variants (a subsequence can be obtained from a sequence by removing zero or more symbols). The problem has applications in bioinformatics, e.g., in sequence alignment, searching new genes. The first variant of the LIS problem, which is considered in this work, is a problem of longest increasing subsequences that are extremal from some point of view. Next variant is a slope-constrained longest increasing subsequence problem. The last two discussed variants of the LIS problem are a longest increasing cyclic subsequence problem (LICS) and a longest increasing subsequence in a sliding window problem (LISW). The algorithms for the recent two problems use cover representation of a sequence. Original algorithms for cover merging are crucial to the proposed algorithms for the LICS and LISW problems. The second part of this work is on the longest common subsequence problem (LCS) and its variants. The applications of these problems are numerous and concentrate mainly on the sequence comparison. For the transposition-invariant LCS problem (LCTS), a few sequential algorithms were proposed. Experiments show that they are much faster than the existing algorithms. For the constrained LCS problem (CLCS), a few sequential algorithms were also proposed. They are faster than the known algorithms. Moreover, for the CLCS problem, the first bit-parallel algorithm was invented. For the merged LCS problem (MerLCS), a bit parallel algorithm, tens times faster than the existing algorithms was proposed. For the LCS, LCTS, CLCS problems also algorithms for graphical processors were invented. All the proposed algorithms were analysed and their time and space complexities in the worst case were determined. For some algorithms the average case was also analysed. Obtained time complexities allow to show that the proposed algorithms are usually faster than the existing algorithms also in an asymptotic sense.