Given two strings of lengths m and n , with m ≤ n , the longest common subsequence problem consists of computing a common subsequence of maximum length by deleting symbols from both strings. While the O ( m n ) algorithm devised in 1974 is optimal in the most general setting, algorithms that depend on parameters other than m and n have been proposed since then. In the word RAM model, let w be the word size, s be the alphabet size, d be the number of dominant symbol matches between the strings, and p be the length of the longest common subsequence. Fast algorithms for this problem have complexities O ( m n / log ⁡ n ) , O ( m n / w ) , O ( n s + min ⁡ ( p ( n − p ) , p m ) ) , O ( n log ⁡ s + d log ⁡ log ⁡ min ⁡ ( d , m n / d ) ) , O ( n s + min ⁡ ( d s , p m ) ) , and O ( n s + s ! 2 s + d log ⁡ s ) . In this work, we present an O ( n ( s + log ⁎ ⁡ n ) + min ⁡ ( d log ⁡ s , p m ) ) algorithm when s ∈ O ( w ) , and also an O ( n ( s + log ⁎ ⁡ n ) + d ) algorithm when s ≤ w which uses bitwise instructions that became recently available in modern processors.

MOTIVATION: Biological network analysis for high-throughput biomedical data interpretation relies heavily on topological characteristics. Networks are commonly composed of nodes representing genes or proteins that are connected by edges when interacting. In this study, we use the rich information available in the Reactome pathway database to build biological networks accounting for small molecules and proteoforms modeled using protein isoforms and post-translational modifications to study the topological changes induced by this refinement of the network representation. RESULTS: We find that improving the interactome modeling increases the number of nodes and interactions, but that isoform and post-translational modification annotation is still limited compared to what can be expected biologically. We also note that small molecule information can distort the topology of the network due to the high connectedness of these molecules, which does not necessarily represent the reality of biology. However, by restricting the connections of small molecules to the context of biochemical reactions, we find that these improve the overall connectedness of the network and reduce the prevalence of isolated components and nodes. Overall, changing the representation of the network alters the prevalence of articulation points and bridges globally but also within and across pathways. Hence, some molecules can gain or lose in biological importance depending on the level of detail of the representation of the biological system, which might in turn impact network-based studies of diseases or druggability. AVAILABILITY AND IMPLEMENTATION: Networks are constructed based on data publicly available in the Reactome Pathway knowledgebase: reactome.org.

Abstract The plowing with precedence problem is a variant of the windy postman problem, where a plow is required to clean streets after a heavy snowfall with traversing costs depending on the direction of traversal as well as whether a street has been previously plowed or not. We prove that this problem can be solved in polynomial time under natural cost structures. We also propose heuristics for this problem, which compare favorably with the state of the art.