Pattern matching with variables: A multivariate complexity analysis

Abstract

a b s t r a c tThe water quality of Feng-qing Lake, which is a landscape lake supplemented with reclaimed water, was surveyed to investigate the relationship between phytoplankton and environmental variables. A total of 29 water samples were collected to analyze temporal variations of phytoplankton and environmental factors from July 2013 to June 2014. Six phyla and 39 genera of phytoplankton were identified when the lake was supplied with reclaimed water. Among these, Cyanophyta and Chlorophyta account for 38.46% and 30.77% of phytoplankton, respectively. The dominant species in the lake are Pseudanabaena limnetica and Chlorella vulgaris, which are present the entire year. Other leading species include Cosmarium sp. and Raphidiopsis curvata. Principal component analysis (PCA) was conducted to analyze the relationship among environmental factors. Canonical correspondence analysis (CCA) was performed to investigate the relationship between environment factors and dominant species. The PCA result showed that temperature (T), total phosphorus (TP), total nitrogen (TN), transparency, and dissolved oxygen are the main factors that affect the eutrophication level of the lake. The CCA result revealed that TN, PO 4 3− -P, chemical oxygen demand (COD), T, and chlorophyll a exhibit a close relation with dominant species. In particular, TN, salinity, and COD influence the growth of P. limnetica; T and COD influence the growth of R. curvata; and T, PO 4 3− -P, NH 3 -N, and pH influence the growth of C. vulgaris and Cosmarium sp.

On the Parameterised Complexity of String Morphism Problems

“…While questions of learnability as well as language theoretical properties were the main focus of research, results regarding the complexity of their membership problem (except of its general N P-completeness) were sparse and only appeared as by-products (see, e. g., [3,18,20,28]). In the pattern matching community, independent of Angluin's work, the string morphism problem has been investigated in terms of a special kind of pattern matching paradigm, called parameterised pattern matching (see [2,4,8,13]). The string morphism problem can also be seen as the solvability problem for word equations where one side does not contain variables (for more details on word equations, see Mateescu and Salomaa [24]).…”

“…In [25][26][27], several possibilities are presented of how to restrict the structure of the source strings, such that StrMorph can be solved in polynomial time, and in [13], the N P-completeness of a large number of strongly restricted versions of StrMorph is shown.…”

“…In [13], for every variant of StrMorph and for every subset of the above parameters, it is shown whether the chosen parameters can be bounded by constants such that the resulting version of StrMorph can be solved in polynomial time or whether it is still N P-complete. For example, if the cardinality of A is bounded by a constant, then all variants of StrMorph can be solved in polynomial time.…”

“…This trivially follows from the fact that there is a simple brute-force algorithm that runs in time that is exponential only in p 1 . Close inspection reveals that StrMorph can also be solved in time that is exponential only in p 2 (for details, the reader is referred to [18] and also [13] …”

Closure properties of pattern languages

“…Hence, the knowledge on pattern languages is still patchy, despite recent progress mainly regarding decision problems (see, e. g., Freydenberger, Reidenbach [5], Fernau, Schmid [3], Fernau et al [4] and Reidenbach, Schmid [13]) and the relation to the Chomsky hierarchy (see Jain et al [6] and Reidenbach, Schmid [14]). …”