Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is ${\sf NP}$-hard in general graphs. Confining to trees renders polynomial time algorithms possible and is of fundamental importance for approaches on more general graph classes. Various variants of this problem in trees have been intensively studied. We consider the general case, where trees are neither rooted nor ordered and the isomorphism is maximum w.r.t. a weight function on the mapped vertices and edges. For trees of order $n$ and maximum degree $\Delta$ our algorithm achieves a running time of $\mathcal{O}(n^2\Delta)$ by exploiting the structure of the matching instances arising as subproblems. Thus our algorithm outperforms the best previously known approaches. No faster algorithm is possible for trees of bounded degree and for trees of unbounded degree we show that a further reduction of the running time would directly improve the best known approach to the assignment problem. Combining a polynomial-delay algorithm for the enumeration of all maximum common subtree isomorphisms with central ideas of our new algorithm leads to an improvement of its running time from $\mathcal{O}(n^6+Tn^2)$ to $\mathcal{O}(n^3+Tn\Delta)$, where $n$ is the order of the larger tree, $T$ is the number of different solutions, and $\Delta$ is the minimum of the maximum degrees of the input trees. Our theoretical results are supplemented by an experimental evaluation on synthetic and real-world instances

Computational Molecular Biology

Computational Biology is a fairly new subject that arose in response to the computational problems posed by the analysis and the processing of biomolecular sequence and structure data. The field was initiated in the late 60's and early 70's largely by pioneers working in the life sciences. Physicists and mathematicians entered the field in the 70's and 80's, while Computer Science became involved with the new biological problems in the late 1980's. Computational problems have gained further importance in molecular biology through the various genome projects which produce enormous amounts of data. For this bibliography we focus on those areas of computational molecular biology that involve discrete algorithms or discrete optimization. We thus neglect several other areas of computational molecular biology, like most of the literature on the protein folding problem, as well as databases for molecular and genetic data, and genetic mapping algorithms. Due to the availability of review papers and a bibliography this bibliography

A note on computing a maximal planar subgraph using PQ-trees

The problem of computing a maximal planar subgraph of a non planar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQ-trees. The latest attempt has been reported by Jayakumar et al. [10]. In this paper we show that the algorithm presented by Jayakumar et al. is not correct. We show that it does not necessarily compute a maximal planar subgraph and we note that the same holds for a modified version of the algorithm presented by Kant [12]. Our conclusions most likely suggest not to use PQ-trees at all for this specific problem

Largest Weight Common Subtree Embeddings with Distance Penalties

The largest common embeddable subtree problem asks for the largest possible tree embeddable into two input trees and generalizes the classical maximum common subtree problem. Several variants of the problem in labeled and unlabeled rooted trees have been studied, e.g., for the comparison of evolutionary trees. We consider a generalization, where the sought embedding is maximal with regard to a weight function on pairs of labels. We support rooted and unrooted trees with vertex and edge labels as well as distance penalties for skipping vertices. This variant is important for many applications such as the comparison of chemical structures and evolutionary trees. Our algorithm computes the solution from a series of bipartite matching instances, which are solved efficiently by exploiting their structural relation and imbalance. Our analysis shows that our approach improves or matches the running time of the formally best algorithms for several problem variants. Specifically, we obtain a running time of O(|T| |T\u27|Delta) for two rooted or unrooted trees T and T\u27, where Delta=min{Delta(T),Delta(T\u27)} with Delta(X) the maximum degree of X. If the weights are integral and at most C, we obtain a running time of O(|T| |T\u27|sqrt Delta log (C min{|T|,|T\u27|})) for rooted trees

Characterisation and optimisation of a sample preparation method for the detection and quantification of atmospherically relevant carbonyl compounds in aqueous medium

Carbonyl compounds are ubiquitous in the atmosphere and either emitted primarily from anthropogenic and biogenic sources or they are produced secondarily from the oxidation of volatile organic compounds. Despite a number of studies about the quantification of carbonyl compounds a comprehensive description of optimised methods is scarce for the quantification of atmospherically relevant carbonyl compounds. The method optimisation was conducted for seven atmospherically relevant carbonyl compounds including acrolein, benzaldehyde, glyoxal, methyl glyoxal, methacrolein, methyl vinyl ketone and 2,3-butanedione. O-(2,3,4,5,6-pentafluorobenzyl)hydroxylamine hydrochloride (PFBHA) was used as derivatisation reagent and the formed oximes were detected by gas chromatography/mass spectrometry (GC/MS). With the present method quantification can be carried out for each carbonyl compound originating from fog, cloud and rain or sampled from the gas- and particle phase in water. Detection limits between 0.01 and 0.17 μmol L−1 were found, depending on carbonyl compounds. Furthermore, best results were found for the derivatisation with a PFBHA concentration of 0.43 mg mL−1 for 24 h followed by a subsequent extraction with dichloromethane for 30 min at pH = 1. The optimised method was evaluated in the present study by the OH radical initiated oxidation of 3-methylbutanone in the aqueous phase. Methyl glyoxal and 2,3-butanedione were found to be oxidation products in the samples with a yield of 2% for methyl glyoxal and 14% for 2,3-butanedione after a reaction time of 5 h