Pattern Matching and Consensus Problems on Weighted Sequences and Profiles

Abstract

We study pattern matching problems on two major representations of uncertain sequences used in molecular biology: weighted sequences (also known as position weight matrices, PWM) and profiles (scoring matrices). In the simple version, in which only the pattern or only the text is uncertain, we obtain efficient algorithms with theoretically-provable running times using a variation of the lookahead scoring technique. We also consider a general variant of the pattern matching problems in which both the pattern and the text are uncertain. Central to our solution is a special case where the sequences have equal length, called the consensus problem. We propose algorithms for the consensus problem parameterised by the number of strings that match one of the sequences. As our basic approach, a careful adaptation of the classic meet-in-the-middle algorithm for the knapsack problem is used. On the lower bound side, we prove that our dependence on the parameter is optimal up to lower-order terms conditioned on the optimality of the original algorithm for the knapsack problem. Therefore, we make an effort to keep the lower order terms of the complexities of our algorithms as small as possible.

Weighted Shortest Common Supersequence Problem Revisited

“…A linear-time solution to this problem is the same as for the problem in the meet-in-the-middle solution for Knapsack. However, for completeness we prove the following lemma (see also [21,Lemma 5.6]):…”

“…By d H (S, T ) we denote the Hamming distance of strings S and T . Let us recall an observation from [21].…”

“…The second improvement is to apply a meet-in-the-middle approach, which is possible due to following observation resembling Observation 6.6 in [21].…”

“…Furthermore, it was shown in [21] that the Weighted Consensus problem cannot be solved in O * (z 0.5−ε ) time for any ε > 0 unless there is an O * (2 (0.5−ε)n )-time algorithm for the Subset Sum problem. Let us recall that the Subset Sum problem, for a set of n integers, asks whether there is a subset summing up to a given integer.…”

Approximate pattern matching on elastic-degenerate text

“…In the literature, many different (compressed) representations and thus algorithms have been considered for pattern matching on a set of similar texts [4][5][6][7][8][9][10]. A natural representation of pan-genomes, or fragments of them, that we consider here are elastic-degenerate texts [11].…”