Comparative modelling of interregional transport flows: Applications to multimodal European freight transport This paper aims to compare the descriptive and predictive power of two classes of statistical estimation models for multimodal network flows, viz. the logit model and the neural network model. The application concerns a large data set on inter-regional European freight flows for two commodity categories (food and chemicals). After an exposition of policy issues, methodological and modelling questions and the database, a variety of experiments is carried out. The results show that in general the predictive potential of neural network models is higher than that of logit analysis. The statistical results are also used to investigate the implications of various road tax systems (e.g., e&axes) on various segments of the European road network After the completion of the European market and with the widening of Europe towards easterly direction, mobility in general has shown a steady increase in Europe. In particular, cross-border transport has been at a rising edge with annual growth rates exceeding sometimes 10 percent, a process reinforced by the current globalisation trends. The integration of former segmented markets -and the related liberalisation in the European space -has led to drastic changes in goods and passenger transport. It is expected that the demand for all forms of transport in the EU will continue to rise up to the year 2010, so that a further increase in goods transport of some 70 percent seems to be likely. This trend is explained by a number of factors (see ECMT, 1995 and Banister, et al., 1993) : -an upturn in economic and industrial activity; -enlargement of the EU and co-operation with the European Free Trade Association (EFTA); more transit flows from the South of the EU and the developments in Eastern Europe will lead to more trade, both intra-European and international; -increased globalisation and intemationalisation of economic activity.. Europe will hence face a series of transportation problems, which may hamper its competitive position. This calls for a Europe-wide policy. Co-ordination at the European level regarding the transport sector is necessary in order to cope with this growth, viz. a common transport policy (Cl'?). One of the elements of the CTP is the development of Trans-European Networks (TENS), which will contribute to a more efficiently running transport system (e.g., removing bottlenecks) and will induce more cohesion in the EU; goods, people and information will then travel more freely. As a result, there is an increasing interest in the issue of intermodal competition and complementarity. For surface transport in Europe, especially the competitive position of rail vis-a-vis road is at stake. This holds increasingly also for commodity transport, It needs to be added however, that the analysis of freight transport in Europe is fraught with many difficulties, as freight is not a homogeneous commodity, but is composed of an extremely diversified set of goods with specific haulage requirements and logistic needs. This means that a commodity-specific approach is necessary to analyse in depth the implications of changes in network configurations. To offer a background scene, we will first concisely describe the rail and road network in Europe.. In this section we will present the results of our NN approach. Reggiani and Tritapepe (1997) describe a methodological structure for the main NN modelling steps consisting of three stages: -definition of network architecture; -learning phase; -forecasting phase.. As previously mentioned, in parallel with the logit approach the NN modelling approach has also been used to analyse the same modal split problem with the same data set. Particularly, two NN models have been adopted, viz. model (NN (A) ) for the food sector and model (NN (B)) for the chemical products. The design of the network architecture will be given in Section 5.2. In Subsection 5.3 the NN models will be trained and evaluated. Then the spatial forecasting analysis of the two NN models will be performed and evaluated in Subsection 5.4. The EU acknowledges the need for a TEN and is planning to construct this by connecting the existing national transport networks. This TEN should improve the accessibility of European regions, while paying attention to environmental restrictions. The issue of central and peripheral areas is also important: investments are mainly undertaken in the central areas, because they contribute to the economic growth more than investments made in peripheral areas. On the other hand, social cohesion is necessary for economic growth and investment is needed to connect the peripheral areas. Clearly, the TENS face interoperability problems: technical, social and fiscal harmonisation of the differences between EU member states. A possible way of monitoring the emerging EU flow patterns is the use of forecast / sensitivity scenarios based on specific transport models. The paper explores this research direction by offering a comparative empirical study of the performance of two classes of models (viz. logit models and neural neural network models) in the light of their applicability in the European freight transport sector. The paper is then organised as follows: Section 2 illustrates the analytical approaches adopted for the empirical analysis, while Section 3 describes the European database on freight flows. Section 4 and 5 describe the results emerging from the calibration / training process for the logit and neural network model, respectively. Section 6 offers a comparison of these results, by implementing also a sensitivity analysis based on different tax policy scenarios. Finally, Section 7 will conclude the paper with some methodological reflections.. Although the NN approach has been mainly applied in operational systems for pattern recognition, i.e. optical character recognition, speech recognition, etc., it is becoming an interesting modelling approach to other fields of research, for instance, in economics. The availability of powerful processing systems and easy-to-use software make it widely applicable. The advantage of NN over traditional modelling approaches is that it is not limited by methodological restrictions, e.g. the Independence of Irrelevant Alternatives (IIA) property. This means that an NN model can handle better complex problems than the traditional models. Data, especially transportation data, often contain noise / incorrect information, which can confuse the traditional modelling approaches and prevent them from correct calibration. However, an NN can easily deal with this kind of data 'corruption', since it can recognise the whole patterns from just a part. During learning, an NN is capable of correcting the noise by adjusting the weights, so that the total error term is minimised. A major disadvantage of the NN models is that they are generally a 'black box'. When an NN is trained and performs reasonably, it is not clear if or how a certain variable is correlated with the output variable. Knowledge that an NN has learned during the training process is not stored in some specific region of the system, where it may be retrieved. Instead, this knowledge is somehow represented in the weights of the connections. Because an NN is a parallel distributed processing system, the value of the weights is not an indicator for a possible correlation. Schintler and Olurotimi (1998) showed that there may be some connection between feedforward, backpropagation NNs and logit models. In their work, they demonstrated that a multi-layered feedforward NN can be considered as a binary logit model with non-linearity in the parameters of the utility function. These parameters can be adapted to the data, resulting in an 'adaptive' logit model. By implementing a conceptual layer, where inputs are presented to the NN as weights, adaptive elasticities of demand can be calculated from the trained NN. This research indicates that NNs are perhaps not complete 'black boxes'. Clearly, further research in this direction is needed to make the application of NNs in social sciences more fruitful.. Our findings appear to offer a wealth of new insights. The results indicate that both modelling approaches are somewhat sensitive to changes in the cost attribute, while the logit model is slightly more sensitive overall. The NN model estimates appear to be closer to the real flows in general. However, the above results may be considered to be rather reliable and valid, given the good performance of the calibration / test phase. Moreover, these results may offer a range of plausible values to policy actors aiming to evaluate the impact of cost changes on flows.. It is also noteworthy that on the one hand, the large amount of data at an aggregate level hampers a behavioural perspective inherent in logit models, but that on the other hand, the type of architecture adopted in NN models seems critical for the validity of the results. Consequently, the results of our model may be used as a benchmark for the results of other models. This paper started with a concise review of European freight transport policy. Although there is a policy agreement among the members of the EU, the European Transport Policy is still not effectively and efficiently applied. National policies still have a strong influence, which results in major differences between national transport networks in Europe in both operational and organisational aspects. In order to study the complexity of such networks, a logit modelling approach is a well known and frequently applied procedure. On the other hand, the application of the NN modelling approach to transport economics is rather new and unexplored. An NN model is able to study the input-output relations when it is presented a training data set with all observed inputs and the desired output. Two different learning methods (supervised and unsupervised learning) can be used for this purpose. The cross-validation method can be used to let the NN model know when to stop the learning procedure. The advantages of applying NN models to transport economics are: -NN models are able to map out quite complex problems due to its unique architecture; -overcoming noise in data can be done more properly, because the NN models consist of parallel processing units and hence are better in recognising a pattern in the data set. Some noteworthy disadvantages of NN models are: -it is not certain that a NN model has learned correctly, since it is not known when or whether it reaches a global minimum during the training procedure; -the knowledge, which is stored in its weights after the training is done, is not easily understood. From the sensitivity analysis carried out here, it is clear that the NN models are able to extract more information than the conventional discrete choice models, in the case of different policy scenarios. On the other hand, the results also indicate that the logit modelling approach is slightly more sensitive to changes in the cost attribute, while the NN modelling approach appears to be fairly robust. Based on these findings, it seems likely that the NN modelling approach is more suitable for spatial forecasting of freight flows, while the logit modelling approach is more suitable for temporal forecasting. However, the predictions made by both models should be tested with more recent and accurate real space-time (pooled cross-section time-series) data to evaluate the performance more thoroughly. Clearly, to Cn-ther understand the possibilities and limitations of applying NN modelling approaches to transport policy issues, more studies need to be carried out. The experiments in this paper may thus be seen as an exploratory comparison between the models investigated

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Reduced PCR/PLSR models by subspace projections Latent variables models used in principal component regression (PCR) or partial least squares regression (PLSR) often use a high number of components, and this makes interpretation of score and loading plots difficult. These plots are essential parts of multivariate modeling, and there is therefore a need for a reduction of the number of components without loss of prediction power. In this work it is shown that such reductions of PCR models with a common number of components for all responses, as well as of PLSR (PLS1 and PLS2) models, may be obtained by projection of the X modeling objects onto a subspace containing the estimatorsb i for the different responses y i . The theoretical results are substantiated in three real world data set examples, also showing that the presented model reduction method may work quite well also for PCR models with different numbers of components for different responses, as well as for a set of individual PLSR (PLS1) models. Examples of interpretational advantages of reduced models in process monitoring applications are included. Solutions of ill-posed regression problems using principal component regression (PCR) or partial least squares regression (PLSR) based on latent variables (LV) models, offer the added advantage of scoreloading visualizations. However, for good predictions such models often require so many components that interpretation of the plots is difficult, and this calls for model reduction methods without loss of prediction capability. This is the main results of the orthogonal signal correction (OSC) methods [1] , and as pointed out by Trygg [2] the number of final components need not be higher than the number of independent responses. This fact also follows from an earlier paper by Kvalheim and Karstang [3] . With a single response, one component will thus be enough, but that will obviously not facilitate the score and loading plots that are essential parts of multivariate modeling. This was the motivation behind the 2PLS algorithm [4] , where a single response LV model (PLS1) with many components is reduced to a model with two components only. That is obtained by projection of the objects in the A-dimensional loading weights space onto a two-dimensional subspace containing the vectorb of prediction coefficients (the estimator).. In the present paper the 2PLS theory is extended to show that any PCR or PLSR (PLS1 and PLS2) model with an A-dimensional loading space can be reduced by projection onto a subspace with any dimension m ≤Ã < A, where m is the number of independent responses, and that this can be done while retaining exactly the same predictions. Since the only significant difference between PCR and PLSR (especially PLS1) often is the number of original components, and since this difference disappears after the model reduction, the two methods will often give practically the same final results. The model reduction results may utilized in e.g. operator support systems for process understanding and monitoring [5, 6] .. In the multiresponse case with m > 1 the corresponding reduction of a set of individual PLSR (PLS1) models will not give exactly the same predictions, and the reason for this is that the individual coefficient vectors inB PLS1 are located in different subspaces. The same problem will occur for a set of individual PCR estimators based on different numbers of components. The predictions in these cases may be approximately the same, however, just as a multiresponse PLS2 models may give approximately the same predictions as a set of individual single response PLS1 models.. The theory behind these model reductions is given in Section 2, while comparisons between the different cases using real world data are presented in Section 3. Section 3 also includes illustrations of the interpretational advantages obtained. Conclusions are finally given in Section 4, and a detailed proof in Appendix A. The main idea behind the model reduction results presented above is that models for prediction and interpretation should not necessarily be the same, and the practical result of model reduction is increased interpretability. An example of that is the 2PLS algorithm, where a single response PLSR model is reduced by projection on a plane containingb [4] .. It has been shown that a PCR latent variables model with a common number of components for all responses, as well as a PLSR model for one or several responses (PLS1 or PLS2), can without any change in the predictions be reduced by projection of the X modeling objects onto a subspace containing the estimatorsb i for the different responses y i . The result of this is easier interpretation of the score and loading plots, that are essential parts of multivariate modeling.. The theoretical results are substantiated by use of three real world data set examples, also showing that the presented model reduction method may work quite well also for a set of individual PCR latent variables models with different numbers of components for different responses, as well as for a set of individual PLSR (PLS1) models. Some interpretational advantages are also exemplified, and more details of this are found in the references.

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Natural and extended formulations for the Time-Dependent Traveling Salesman Problem Consider a graph G=(V,A), where V ={1,2,..,n} and A={(i,j): i,j=1,..,n, i ≠ j}. The TimeDependent Travelling Salesman problem (TDTSP) is to find a minimum cost Hamiltonian circuit, starting and ending on node 1, where arc costs depend on its position in the tour.

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Bounding the inefficiency of logit-based stochastic user equilibrium Abstract. Bounding the inefficiency of selfish routing has become an emerging research subject. A central result obtained in the literature is that the inefficiency of deterministic User Equilibrium (UE) is bounded and the bound is independent of network topology. This paper makes a contribution to the literature by bounding the inefficiency of the logit-based Stochastic User Equilibrium (SUE). In a stochastic environment there are two different definitions of system optimization: one is the traditional System Optimum (SO) which minimizes the total actual system travel time, and the other is the Stochastic System Optimum (SSO) which minimizes the total perceived travel time of all users. Thus there are two ways to define the inefficiency of SUE, i.e. to compare SUE with SO in terms of total actual system travel time, or to compare SUE with SSO in terms of total perceived travel time.We establish upper bounds on the inefficiency of SUE in both situations. We have defined the inefficiency of SUE in two different ways, i.e. comparing SUE with SO in terms of total actual system travel time, or comparing SUE with SSO in terms of total perceived system travel time. For both notions, we provided upper bounds on the inefficiency of the logit-based SUE, based on its equivalent VI formulation and the properties of the divergence between two discrete path flow distributions.. When comparing SUE with SO in terms of total actual system travel time, the inefficiency bound of SUE depends on both the class of cost functions and the degree of perception error and the network complexity. Nevertheless, it is found that the effect of network complexity in terms of number of available paths is rather limited. Unlike the price of anarchy of the deterministic UE established in the literature, the inefficiency bounds established for the SUE is generally not tight unless either the users' perception error is zero or the network has constant link travel times.. When comparing SUE with SSO in terms of total perceived travel time (or equivalently net economic benefit), we established an upper bound on the welfare loss of SUE and found that the welfare loss is not larger than a fraction of the total actual system travel time under SUE, and the "fraction" is independent of network topology, but depends solely on the class of link cost functions. We also found that the established inefficiency bound is tight only when link costs are constants (in this case SUE coincides with SSO).

O O B O O B I I O O O B O B I O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O


Minsum Location Extended to Gauges and to Convex Sets Abstract One of the oldest and richest problems from continuous location science is the famous Fermat-Torricelli problem, asking for the unique point in Euclidean space that has minimal distance sum to n given (non-collinear) points. Many natural and interesting generalizations of this problem were investigated, e.g., by extending it to non-Euclidean spaces and modifying the used distance functions, or by generalizing the configuration of participating geometric objects. In the present paper, we extend the Fermat-Torricelli problem in a two-fold way: more general than for normed spaces, the unit balls of our spaces are compact convex sets having the origin as interior point (but without symmetry condition), and the n given objects can be general convex sets (instead of points). We combine these two viewpoints, and the presented sequence of new theorems follows in a comparing sense that of theorems known for normed spaces. Some of these results holding for normed spaces carry over to our more general setting, and others not. In addition, we present analogous results for related questions, like, e.g., for Heron's problem. And finally we derive a collection of results holding particularly for the Euclidean norm. The classical Fermat-Torricelli problem asks for the unique point minimizing the distance sum to finitely many non-collinear points in d-dimensional Euclidean space. It is, regarding the variety of contributions and contributors, one of the richest problems from continuous location science. Going back to the 17th century, it still creates new research problems. At its historically first step, it was connected with famous names like R. Descartes, P. de Fermat, E. Torricelli, V. Viviani, B. Cavalieri, and E. W. von Tschirnhaus. Later on, mathematicians like J. Bertrand, C. F. Gauss, J. Steiner, L. Lindelöf, R. Sturm, J. Hadamard, G. Polya, H. W. Kuhn, P. Erdős and many others added results more related to modern branches of mathematics, like (convex) optimization, approximation theory, functional analysis, algebraic geometry, convex analysis, computational geometry etc. (a comprehensive representation is given in [1, Chapter II]). More recently, deeper generalizations of this problem were added, for example extensions to non-Euclidean spaces of different types, and generalizations of the participating geometric configuration. Extensions to normed spaces (i.e., to real, finite dimensional Banach spaces; see [2] , [3] , and [4] ) and replacements of the participating points by hyperplanes or spheres (cf., e.g., [5] and [6] ) yield especially interesting and geometrically rich approaches and algorithms. Our goal here is to continue this line of research by generalizing the latter two viewpoints at the same time, thus reaching a more general step in the combined sense. Namely, first we extend the basic theory on this problem from normed spaces to generalized Minkowski spaces, having arbitrary convex bodies as unit balls which no longer need to be centrally symmetric; they create general convex distance functions (gauges). And second, we generalize the geometric properties of the participating given objects: they are no longer points or hyperplanes, but arbitrary convex sets. Related topics are also discussed in [7, pp. 146-168] . To do this, we give in our second section also novel extensions of several common notions from Banach space theory to gauges, e.g., by introducing generalized norming functionals. Such notions are fundamental for the geometric description of the solution sets of our location problems. It turns out that for introducing generalized norming functionals, a correspondingly generalized version of the Hahn-Banach theorem is necessary. So we also use a version of the Hahn-Banach theorem extended to gauges.. In the third section we derive results on the Fermat-Torricelli problem for finite point sets with respect to gauges, mainly generalizing the sequence of theorems presented for normed spaces in [4] (see also [2] ). It turns out that some of the results from [4] directly carry over to generalized Minkowski spaces, and some not. Denoting by ft(P ) the solution set of the Fermat-Torricelli problem for a non-collinear finite point set P in a generalized Minkowski space, we clarify the cases when ft(P ) is a singleton or a polytope. Strict convexity of the unit ball is now only a necessary criterion for the property that ft(P ) is a singleton, for every P as above. We present also characterizations of normed spaces within the family of generalized Minkowski spaces via an extension of Menger's notion of d-segments (often needed for metrical problems in normed spaces; see [8] and [9, Chapter II] ). We show that in this more general setting ft(P ) can be represented as intersection of certain cones determined by the boundary structure of the convex unit balls. In this section we also investigate the boundary structure of sublevel sets for the generalized Fermat-Torricelli problem; their geometry again depends on the boundary structure of the unit balls.. In the fourth section we combine the approach via gauges with the replacement of the given finite point set by a given finite family of arbitrary convex sets, even considering finitely many respective gauges. Using generalized norming functionals and a corresponding extension of distance functions to gauges, we derive similar results as they exist for given point sets, and some interesting additional observations are obtained, too. For example, we investigate also Heron's problem in our generalized setting. In the final fifth section, further theorems holding only for the Euclidean norm are derived (the given objects are still arbitrary convex sets). Inspired by various well-known results on the Fermat-Torricelli problem considered in finite dimensional real Banach spaces and by its analogues for searched minimizing hyperplanes or spheres, in the present paper this problem is generalized to arbitrary convex distance functions (gauges), and at the same time the participating geometric configuration is extended to families of arbitrary convex sets. This two-fold generalization of existing results is reached via certain preliminary steps (e.g., by considering gauges, but still finite point sets as geometric configurations) which are interesting for themselves and therefore presented, too. It turns out that many results known from the literature suitably carry over to this more general setting, but some of them do not. For deriving these new results, basic geometric notions from Banach space theory are extended to gauges, directly yielding interesting questions for further generalizations (for instance, to extend the problem to infinite-dimensional vector spaces with gauges). Hence we pose also a couple of new research problems. From our paper one can read off the power of geometric tools and of methods from convex analysis to obtain far-reaching, but natural generalizations of one of the oldest and most famous problems from continuous location science.

O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Optical mapping in plant comparative genomics Optical mapping has been widely used to improve de novo plant genome assemblies, including rice, maize, Medicago, Amborella, tomato and wheat, with more genomes in the pipeline. Optical mapping provides long-range information of the genome and can more easily identify large structural variations. The ability of optical mapping to assay long single DNA molecules nicely complements short-read sequencing which is more suitable for the identification of small and short-range variants. Direct use of optical mapping to study population-level genetic diversity is currently limited to microbial strain typing and human diversity studies. Nonetheless, optical mapping shows great promise in the study of plant trait development, domestication and polyploid evolution. Here we review the current applications and future prospects of optical mapping in the field of plant comparative genomics. Optical mapping is a molecular technique that produces fingerprints of DNA sequences in order to construct genome-wide maps [1] . The sequence markers can be ordered restriction fragments [1] , or specific sequence motifs (nick sites) [2] . The optical mapping procedure first stretches relatively intact (minimally-sheared) linear DNA fragments on a glass surface or in a nanochannel array, and then directly images the locations of the restriction sites or sequence motifs under light microscopes, with the aid of dye or fluorescent labels [1, 2] . Automation of optical measuring and processing devices has led to the development of commercial platforms, such as OpGen Argus [3] and BioNano Genomics Irys systems [4] .. Optical mapping offers several unique advantages over traditional mapping approaches, including single molecule analysis and the ability to assay long DNA molecules (~250Kb to 3 Mb in conventional optical mapping [1] and 20-220Kb in nanochannel arrays [2] ). The ability to assay large DNA molecules has allowed accurate reconstruction of chromosomal pieces during de novo genome assembly and identification of relatively large structural variants in genetic diversity studies. While optical mapping is readily available across a wide range of organisms including bacterial, fungi, plant and mammalian genomes [5] [6] [7] [8] [9] , this review focuses on the applications and of optical mapping in the field of plant comparative genomics. Optical mapping is an important technique that can provide long genomic linkage information in a highthroughput manner, which has substantially improved the assemblies of several important model plant genomes sequenced to date. Direct comparisons of genome structures have so far been lacking in plants, but optical mapping shows great promises at revealing genomic regions that are not easily accessible through conventional sequencing methods. Optical mapping could become an integral part of the mapping tools in the study of plant domestication, polyploid evolution, and trait development.

B I O O B I B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O


About

the

domino

problem

in

the

hyperbolic

plane

from

an

algorithmic

point

of

view

In

this

paper,

we

prove

that

the

general

problem

of

tiling

the

hyperbolic

plane

withà

la

Wang

tiles

is

undecidable.

The

question,

whether

it

is

possible

to

tile

the

plane

with

copies

of

a

fixed

set

of

tiles

was

raised

by

Wang,

in

the

late

of

the

previous

century.

Wang

solved

the

partial

problem

which

consists

in

fixing

an

initial

finite

set

of

tiles:

indeed,

fixing

one

tile

is

enough

to

entail

the

undecidability

of

the

problem.

The

general

case,

when

no

initial

tile

is

fixed,

was

proved

by

Berger

in

Both

Wang's

and

Berger's

proofs

deal

with

the

problem

in

the

Euclidean

plane.

In

Robinson

found

an

alternative

proof

of

the

undecidability

of

the

general

problem

in

the

Euclidean

plane,

see

In

this

paper,

he

raises

the

question

of

the

general

problem

for

the

hyperbolic

plane.

Seven

years

later,

in

he

proved

that

in

the

hyperbolic

plane,

the

partial

problem

is

undecidable,

see

Up

to

now,

and

as

far

as

I

know,

the

problem

remained

open..

In

this

paper,

we

give

a

proof

that

the

general

problem

is

also

undecidable

in

the

case

of

the

hyperbolic

plane.

It

is

important

to

stress

that

in

its

spirit,

our

proof

is

a

bit

different

from

the

proofs

of

Berger

and

Robinson.

We

consider

the

tiling

process

from

the

point

of

view

of

computer

science.

This

means

that

it

must

be

an

algorithmic

process

which

evolves

in

time..

We

start

from

the

initial

idea

of

Berger's

proof.

It

consists

in

simulating

infinitely

many

computations

of

a

Turing

machine,

indeed

of

the

same

Turing

machine

starting

from

an

empty

tape

from

which

the

set

of

prototiles

is

derived.

The

computations

go

as

far

as

they

can.

If

the

Turing

machine

halts,

the

computations

of

the

tiling

arrive

until

the

halting

state

is

called

and

at

this

time,

the

tiling

process

is

interrupted

and

it

is

no

more

possible

to

tile.

If

the

Turing

machine

does

not

halt,

the

simulations

go

on

endlessly

or

may

be

stopped

by

the

limitations

imposed

on

the

tiling.

But

Berger,

and

after

him

Robinson,

proved

that

among

this

infinity

of

computations

in

the

case

of

a

non-halting

situation,

either

at

least

one

of

them

can

be

performed

without

being

interrupted,

or

there

are

arbitrary

long

computations.

Accordingly

the

plane

can

be

tiled.

And

so,

tiling

the

plane

is

possible

if

and

only

if

the

Turing

machine

does

not

stop.

And

this

proves

the

undecidability

of

the

problem..

Analysing

the

proof,

we

can

see

that

the

restricted

amount

of

space

in

the

Euclidean

plane

forces

to

find

a

way

to

generate

infinitely

many

bounded

domains

whose

size

is

exponentially

increasing.

The

exponential

size

is

needed

to

overcome

the

meeting

problem.

It

is

also

needed

from

Wang's

remarks

of

the

if

tilings

of

the

plane

are

necessarily

periodic,

the

general

tiling

problem

is

decidable.

And,

historically,

Berger's

proof

had,

as

a

side-product,

the

construction

of

a

non-periodic

tiling.

This

was

not

the

goal

of

Berger's

paper,

but

this

was

entailed

by

Wang's

remark.

Berger

managed

to

produce

exponential

signals

in

a

rather

easy

way.

Also,

the

exponential

growth

guarantees

the

existence

of

at

least

an

infinite

sequence

of

increasingly

longer

and

longer

computations.

It

is

important

to

focus

on

the

fact

that

the

constructions

performed

both

by

Berger

and

by

Robinson

in

their

respective

proofs

make

a

huge

use

of

similarity.

In

his

paper,

speaking

of

the

problem

in

the

hyperbolic

plane,

Robinson

And

it

can

be

noticed

that

the

non-periodicity

problem

for

tilings

of

the

hyperbolic

plane

has

drawn

more

attention

than

the

undecidability

problem,

see

for

instance

for

works

in

this

direction..

From

the

tools

I

devised

to

locate

cells

of

a

cellular

automaton

implemented

in

the

hyperbolic

plane,

see

for

more

complete

references,

it

came

to

me

that

perhaps,

this

could

provide

us

with

a

new

angle

to

tackle

the

problem..

Due

to

the

very

different

context,

an

important

point

of

the

discussion

is

the

exact

meaning

of

the

general

problem.

It

cannot

be

the

exclusion

of

any

constraints

on

the

initial

tiling

set

and

on

the

construction.

Indeed,

if

the

first

tile

can

be

chosen

at

random,

it

can

bear

the

computation

signs

of

a

Turing

machine.

In

this

case,

it

may

turn

out

that

this

tile

will

never

be

used,

and

it

should

not

be

placed.

But

the

reachability

problem

is

undecidable

for

Turing

machines.

Consequently,

the

organisation

of

the

tiling

must

force

any

computation

to

be

started

at

the

very

beginning

of

the

Turing

computation.

And

so,

the

first

tile

cannot

be

taken

at

random.

In

his

proof,

Berger

defines

a

subset

of

the

initial

set

of

tiles,

the

skeleton,

from

which

the

first

tile

is

chosen.

Here

also,

we

define

such

a

skeleton

and,

as

in

Berger's

proof,

it

allows

to

construct

the

general

pattern

in

which

areas

are

delimited

for

the

Turing

computation..

Here,

we

suggest

to

give

a

precise

definition

of

the

partial

case

and

of

the

general

case.

The

partial

problem

means

that

at

least

one

tile

is

fixed

in

the

initial

set

which

must

be

used

but

at

most

finitely

many

times.

In

the

general

problem,

there

is

a

subset

of

the

initial

set,

the

skeleton,

whose

elements

must

all

be

used

infinitely

many

times

in

the

tiling..

Our

solution

is

algorithmic.

The

undecidability

of

the

tiling

problem

has

no

meaning

for

solutions

which

would

not

be

algorithmic.

As

in

Berger's

and

Robinson's

proof,

our

set

of

initial

tiles

has

the

property

that

there

is

an

algorithm

which

produces

any

possible

solution

by

running

the

algorithm

non-deterministically.

In

Berger's

and

Robinson's

proof,

the

initial

tiles

can

be

only

translated

when

they

are

duplicated

to

be

placed

in

the

tiling.

Rotations

and

reflections

are

ruled

out.

Here,

we

rule

out

only

reflections:

from

the

negative

curvature

of

the

hyperbolic

plane,

we

know

that

the

translations

which

leave

the

tiling

invariant

also

generate

the

rotations

which

leave

the

tiling

invariant..

Our

solution

sticks

to

Wang's

convention

on

the

tiles

as

in

Berger's

paper.

Also,

in

the

case

when

the

Turing

machine

does

not

halt,

we

have

infinitely

many

periodic

solutions.

This

confirms

the

quoted

above

prediction

of

Robinson.

We

also

prove

that

are

there

countably

many

periodic

solutions

which,

indeed,

can

be

enumerated..

In

this

abstract,

we

have

no

room

to

remind

the

reader

of

what

is

needed

from

hyperbolic

geometry.

It

is

also

impossible

to

redefine

the

tools

I

introduced

for

the

location

of

tiles

in

the

hyperbolic

plane.

The

reader

is

invited

to

look

at

the

technical

report,

on

which

the

paper

is

based

and

which

is

available

at

the

following

address:.

dominoes.ps.gzip.

In

the

second

section,

the

paper

constructs

the

tiling

defined

by

the

skeleton,

the

mantilla.

In

the

third

section,

it

presents

the

partial

solution,

within

the

frame

of

the

harp.

In

the

fourth

section,

it

presents

the

general

solution.

O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A Fast Algorithm for Computing Geodesic Distances in Tree Space Comparing and computing distances between phylogenetic trees are important biological problems, especially for models where edge lengths play an important role. The geodesic distance measure between two phylogenetic trees with edge lengths is the length of the shortest path between them in the continuous tree space introduced by Billera, Holmes, and Vogtmann. This tree space provides a powerful tool for studying and comparing phylogenetic trees, both in exhibiting a natural distance measure and in providing a euclidean-like structure for solving optimization problems on trees. An important open problem is to find a polynomial time algorithm for finding geodesics in tree space. This paper gives such an algorithm, which starts with a simple initial path and moves through a series of successively shorter paths until the geodesic is attained. A phylogenetic tree describes the evolutionary history of a set of organisms, with the leaf vertices representing the organisms and the interior vertices representing points at which the evolutionary history branches. Researchers use different criteria and methods for constructing phylogenetic trees from available data about the set of organisms, which can result in several possible trees or a distribution of trees describing the phylogenetic history. For example, reconstructing the most likely tree for different genes may yield different trees [19] ; different reconstruction methods can also produce different trees on the same set of organisms [11] . Thus, a way is needed to quantitatively compare different phylogenetic trees, by computing some metric describing the differences between them. Many such distance measures have been proposed, including the Robinson-Foulds or partition distance [18] , the Nearest Neighbor Interchange (NNI) distance [17] , the Subtree-Prune-and-Regraft (SPR) distance [7] , and the Tree Bisection and Reconnection (TBR) distance [2] . These measures tend to emphasize the differences in topologies between the trees, and often do not account for edge lengths. If the edge lengths represent such information as number of mutations between speciation events, we lose important information by ignoring them. Worst yet, most of these measures cannot be computed efficiently and so are of little use when applied to large trees.. To address this issue, Billera et al. [4] propose the concept of a continuous tree space, and its associated geodesic distance metric, as a natural way to embed and compare phylogenetic trees. This tree space consists of a set of euclidean regions, called orthants, one for each tree topology. Orthants are joined together whenever one tree topology can be made into another by exchanging edges between the trees. Within an orthant, the coordinates of each point represent the edge lengths for a particular tree with the topology associated with that orthant. The geodesic between two trees is the unique shortest path connecting the two associated points in this space. Thus, traversing the geodesic corresponds to continuously transforming one tree into the other. In contrast to previous measures, geodesic distance incorporates in a natural way edge lengths as well as the tree topology. Furthermore, the uniqueness of the geodesic between any pair of trees and the continuity of the tree space suggest this framework has useful properties with respect to optimization over trees and to formulating statistical measures associated with trees [9] and [10] . Other versions of tree space with different metrics or no metric have been investigated in phylogenetics contexts ( [8] , [6] , and [13] for example) and in combinatorial ones [21] and [16] .. Two algorithms have been previously proposed for computing the geodesic distance: GEOMETREE [12] and GEODEMAPS [15] . Both these algorithms search through an exponential number of candidate paths to find the geodesic, so their runtime is exponential in the size of the trees. Currently, there are no known polynomial time algorithms to find tree space geodesics, although a polynomial time ffiffi ffi 2 p -approximation of the geodesic distance was given by Amenta et al. [3] . Some combinatorial and geometric properties of the space of phylogenetic trees were also presented in [15] . This paper presents the first polynomial time method for computing geodesic distances-and the associated geodesics-between trees in tree space. The algorithm uses a different approach from the previous papers, by starting with a simple path between the trees and transforming it into successively shorter paths until the geodesic is obtained. At each step, the algorithm identifies one new orthant that intersects the geodesic, and transforms the current path so that it passes through this new orthant in an optimal manner. By restricting consideration to the orthants intersecting the geodesic, the algorithm makes only a polynomial number of path transformations. Each new orthant is identified by finding a weighted vertex cover in a specially constructed bipartite graph, which also is a polynomial time problem. Section 2 describes the tree space in which we define the geodesic distance, along with some important geometric and combinatorial properties relevant to finding the geodesic. Section 3 gives the geodesic path algorithm between trees with disjoint edge sets, and establishes its correctness and complexity. Section 4 explains how to efficiently use the geodesic path algorithm when the trees have common edges. The final section outlines some interesting problems that extend the scope of this algorithm and the associated structures. This paper presents the first polynomial time algorithm for finding geodesics between phylogenetic trees in tree space, as well as further characterizing properties of geodesics. This significantly increases the usefulness of the geodesic distance as a modeling tool, since the previous exponential algorithms essentially restricted the geodesic distance measure to trees with fewer than 50 leaves.. We first note that the technique presented here also solves GTP in the case where there is a specific right-left ordering on the nonroot leaves of the tree, or equivalently, where the tree must be planar with respect to a given clockwise ordering of the leaves. An example of such trees are binary search trees. This condition simply adds to the definition of a tree that the splits of the tree must be noncrossing, that is, for any split X e jX e there are no pairs v 1 ; v 2 2 X e and v 3 ; v 4 2 X e which appear in clockwise order v 1 ; v 3 ; v 2 ; v 4 . Since if T and T 0 both satisfy the noncrossing property, and all of the splits in the intermediate trees on the geodesic between T and T 0 are made up of the splits of T and T 0 , then these trees must also satisfy the noncrossing property, and so the geodesic for this case is the same as that for the unrestricted case.. The properties and techniques given here potentially apply to the much wider range of problems and measures on trees that make use of the intrinsic euclidean nature of tree space. For example, Nye [14] compares and groups trees through the idea of "medial trees" which serve as representatives for topological types within data sets of trees. Hillis et al. [8] also investigate tree sets using distance as the distinguishing feature to find statistical groupings and common features. Using a more euclidean-related measure for dissimilarity could allow more powerful statistical techniques to be employed in these situations.. Billera et al. [4] look at the concept of medial trees in their paper by defining the centroid of a set of points in tree space. Their definition involves an iterative process that is based on finding a converging sequence of midpoints of geodesics between trees. The implementation of this would require a fast method of computing geodesics. Another way of thinking about a centroid in standard euclidean space, though, is as the point of minimum sum squared distance to the trees. The framework for finding geodesics here naturally lends itself to finding centroids in this alternate sense as well, and could yield a more direct and efficient way of computing centroids.. A further extension of the idea of centroid comes up in the development of object-oriented data analysis (OODA) as it has been applied to trees [22] . This involves fitting a "line" to a set of trees in such a way as to minimize least-squares distances. The set of nearest points on this line can then be analyzed to yield statistical discriminators that can, in turn, isolate significant properties of the underlying objects. This can be done a second time, as a result gaining "secondorder" information about the set of trees, and so on. Two problems with OODA have been 1) the difficulty in determining the right concept of "line" and "least-square distances" when the objects are not euclidean in nature and 2) the computational challenge in actually finding these "least-fit" objects. The path space concept presents a compelling model for facilitating both these kinds of analyses, with the CAT(0) property providing the framework for efficient iterative improvement methods to extract useful statistical information in this context.

O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O B O O O O O O O B O O O O O B O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Efficient parametric analysis of the chemical master equation through model order reduction Background: Stochastic biochemical reaction networks are commonly modelled by the chemical master In this paper, we have introduced the application of parametric model reduction methods to finite-state approximations of the chemical master equation. We have also presented two case studies where these methods are applied to CME models of different networks in order to make parametric analysis tasks This contribution is just a first step in the application of parametric model reduction methods to the CME.. One particularly important aspect that we have not discussed here is the computation of error estimates for certifying that the simulation output of the reduced model is within some tolerance of the corresponding simulation output of the original model. To maintain computational efficiency, the error estimation should be done without actually simulating the original model. Error estimation methods have been developed for parametric model reduction of generic models [9] , but tighter estimates could likely be obtained by taking into account the special structure of the CME models. Recent work for example refined the previous generic error bounds for stable models [25] . The chemical master equation (CME) is the most basic mathematical description of stochastic biomolecular reaction networks [1, 2] . The CME is a generally infinite-dimensional linear differential equation. It characterizes the temporal development of the probabilities that the network is in any of its possible configurations, where the different configurations are characterized by the molecular copy numbers of the network's chemical species.. Due to its infinite dimension, the CME is usually not directly solvable, not even with numerical methods.. A recent breakthrough in the numerical treatment of the CME was the establishment of the finite state projection (FSP) method by Munsky and Khammash [3] . They showed that it is possible to compute a good approximation to the real solution by projecting the CME to a suitable finite subdomain of the network's state space, and solving the resulting finite-dimensional linear differential equation on that domain. Nevertheless, the FSP approach still yields very high-dimensional models which are computationally expensive to simulate, even for small biochemical networks. The efficient simulation of the CME is an area of active research, and recently other simulation methods have been developed that can also be used for larger networks [4, 5] .. Despite this progress, the direct simulation of the CME remains a computational bottleneck for common model analysis tasks in systems biology. It is especially problematic for tasks which require the repeated simulation of the model using different parameter values, for example identifiability analysis, parameter estimation, or model sensitivity analysis. Thereby, while a single or a few evaluations of a CME model with the FSP or other approaches may still be computationally feasible, the necessity of many repeated simulations will quickly render higher-level analysis tasks infeasible.. Mathematical methods that approximate the behaviour of a high-dimensional original model through a low-dimensional reduced model are a common way to deal with complex models. Especially for linear differential equations, model order reduction is a well established field and several methods to compute reduced order models are available [6] . Note that the step of generating a reduced model is usually computationally more expensive than a single or even a few simulations of the original high-dimensional model. But the simulation of the resulting reduced models is frequently orders of magnitude faster than the solution of the original model. So, model reduction is worth the effort if many repeated simulations are to be expected. Unfortunately, for analysis tasks which require the repeated model simulation with different parameters, classical model reduction methods are not helpful. With these methods, the reduced model depends on specific parameter values in the original model, and the reduction needs to be redone for different parameter values. Thus, for the mentioned analysis tasks, the model reduction process would have to be repeated for each new parameter value, and no gain in computational efficiency would typically be possible. While classical model reduction techniques have been applied to the CME in the past [7] , they are not so suitable for parametric analysis tasks.. Fortunately, model reduction methods where parameters from the original model are retained as adjustable parameters also in the reduced model are now being developed. These methods allow to compute a reduced model which uses the same parameters as the original model, and where the reduced model can directly be simulated with any choice of parameter values [8] [9] [10] [11] .. The purpose of this paper is to introduce the application of these parametric model reduction methods to finite-state approximations of the chemical master equation, and to show possible usage scenarios of such an approach. The structure is as follows. In the following section, we introduce some background and notation concerning the modelling of chemical reaction networks and parametric model order reduction.. We also show how the parametric model order reduction methods can in fact be applied to the CME.. Afterwards, we apply the reduction technique on two reaction network models and corresponding parametric analysis tasks.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


AutoGrow 3.0: An improved algorithm for chemically tractable, semi-automated protein inhibitor design We here present an improved version of AutoGrow (version 3.0), an evolutionary algorithm that works in conjunction with existing open-source software to automatically optimize candidate ligands for predicted binding affinity and other druglike properties. Though no substitute for the medicinal chemist, AutoGrow 3.0, unlike its predecessors, attempts to introduce some chemical intuition into the automated optimization process. AutoGrow 3.0 uses the rules of click chemistry to guide optimization, greatly enhancing synthesizability. Additionally, the program discards any growing ligand whose physical and chemical properties are not druglike. By carefully crafting chemically feasible druglike molecules, we hope that AutoGrow 3.0 will help supplement the chemist's efforts.To demonstrate the utility of the program, we use AutoGrow 3.0 to generate predicted inhibitors of three important drug targets: Trypanosoma brucei RNA editing ligase 1, peroxisome proliferator-activated receptor ␥, and dihydrofolate reductase. In all cases, AutoGrow generates druglike molecules with high predicted binding affinities.

B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O


An improved relaxed complex scheme for receptor flexibility in computer-aided drug design Abstract The interactions among associating (macro)molecules are dynamic, which adds to the complexity of molecular recognition. While ligand flexibility is well accounted for in computational drug design, the effective inclusion of receptor flexibility remains an important challenge. The relaxed complex scheme (RCS) is a promising computational methodology that combines the advantages of docking algorithms with dynamic structural information provided by molecular dynamics (MD) simulations, therefore explicitly accounting for the flexibility of both the receptor and the docked ligands. Here, we briefly review the RCS and discuss new extensions and improvements of this methodology in the context of ligand binding to two example targets: kinetoplastid RNA editing ligase 1 and the W191G cavity mutant of cytochrome c peroxidase. The RCS improvements include its extension to virtual screening, more rigorous characterization of local and global binding effects, and methods to improve its computational efficiency by reducing the receptor ensemble to a representative set of configurations. The choice of receptor ensemble, its influence on the predictive power of RCS, and the current limitations for an accurate treatment of the solvent contributions are also briefly discussed. Finally, we outline potential methodological improvements that we anticipate will assist future development. Accounting for receptor flexibility in computer-aided drug design is still a major challenge. Recent examples illustrate the importance of predicting and including induced-fit effects upon receptor-ligand binding. MD simulations of receptors in complex with known and potential inhibitors provide relevant biochemical insights, which are otherwise not accessible through standard experimental techniques. Despite this, a general and highly transferable procedure that reliably and efficiently accommodates receptor flexibility is still lacking. The extensions and methodological improvements to the RCS presented here take important steps toward offering such a streamlined procedure. Our examples indicate that alternative choices of receptor ensembles can significantly alter the predictive power of RCS calculations, and that it is possible to reduce the receptor ensemble to a non-redundant set of configurations by various techniques without losing relevant binding information. Furthermore, both example systems indicate that the role of explicit water molecules in molecular association remains one of the key components of computer-aided drug design methods to be further investigated. A summary of the crucial points that we anticipate will help drive the future development of the RCS was presented and what emerges is that a clear and pressing challenge, closely coupled to receptor flexibility, is the development of methods to better estimate ligand and receptor entropy. Their subsequent application to molecular association thermodynamics will allow an increased accuracy in the description of enthalpy-entropy compensation effects during the ligand binding process.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Energy efficiency in wireless sensor networks: A top-down survey The design of sustainable wireless sensor networks (WSNs) is a very challenging issue. On the one hand, energyconstrained sensors are expected to run autonomously for long periods. However, it may be cost-prohibitive to replace exhausted batteries or even impossible in hostile environments. On the other hand, unlike other networks, WSNs are designed for specific applications which range from small-size healthcare surveillance systems to large-scale environmental monitoring. Thus, any WSN deployment has to satisfy a set of requirements that differs from one application to another. In this context, a host of research work has been conducted in order to propose a wide range of solutions to the energysaving problem. This research covers several areas going from physical layer optimization to network layer solutions. Therefore, it is not easy for the WSN designer to select the efficient solutions that should be considered in the design of application-specific WSN architecture.We present a top-down survey of the trade-offs between application requirements and lifetime extension that arise when designing wireless sensor networks. We first identify the main categories of applications and their specific requirements. Then we present a new classification of energy-conservation schemes found in the recent literature, followed by a systematic discussion as to how these schemes conflict with the specific requirements. Finally, we survey the techniques applied in WSNs to achieve trade-off between multiple requirements, such as multi-objective optimisation. There is abundant literature relating to energy-saving in WSNs as numerous methods have been proposed in the last few years, and there is still much ongoing research on how to optimise power usage in battery-limited sensor networks. However, none of the proposed solutions is universally applicable. For example, if safety applications require fast and timely responsiveness, this is not the case for other applications, such as in agriculture where the delay property is not as important. We believe that WSN energy-saving problems should be tackled by taking into consideration application requirements in a more systematic manner.. In [1] , Yick et al. provide a general survey of wireless sensor networks. This study reviews sensor platforms and operating systems, network services issues and communication protocol challenges, but it does not addresses the energy issues. In [2] , Anastasi et al. present a valuable taxonomy of energy-conservation schemes. However, the authors mainly focus on duty cycling and data-reduction approaches. There also exist several technique-specific sur-Email addresses: tifenn.rault@utc.fr (Tifenn Rault), abdelmadjid.bouabdallah@utc.fr (Abdelmadjid Bouabdallah), yacine.challal@utc.fr (Yacine Challal) veys that concentrate on only one energy-efficient mechanism (like energy-efficient routing protocols, data aggregation techniques, energy harvesting approaches [3] [4] [5] ) since every category of solution often represents a whole research area in itself.. Our aim is to provide WSN designers with a top-down survey that offers a holistic view of energy-saving solutions while taking into consideration the specific requirements of the applications. In this paper, we propose a new classification of energy-efficient mechanisms which integrates the most recent techniques and up-to-date references. Moreover, we give particular attention to the design of energy-efficient sensor networks that satisfy application requirements. Our study is original in that we focus on the trade-offs between meeting specifications and sustainability that necessarily arise when designing a WSN. We thus discuss mechanisms that enable a satisfactory trade-off between multiple requirements to be achieved. To the best of our knowledge, this is the first time that this approach has been taken.. The rest of this paper is organised as follows. In the next section, we present the main categories of applications we have identified and their respective requirements. Then, in Section 3, we discuss existing standards for low-power wireless sensor networks and show that current standards cannot respond to all application needs. In Section 4, we give an overview of the major energy-saving mechanisms developed so far and discuss their advantages and shortcomings regarding the set of identified requirements. In Section 5, we review techniques proposed in the literature to achieve a trade-off between multiple requirements, including network lifetime maximisation. Finally, Section 6 concludes this paper. In the last decade, we have witnessed a proliferation of potential application domains of wireless sensor networks. These applications include, but are not limited to, life-critical healthcare surveillance, large-scale precision agriculture, security-oriented industrial process monitoring and nation-wide smart grid systems. In this paper, we surveyed the recent advances in the development of energy-efficient solutions for WSNs while taking into consideration the other application requirements. We first categorized the different WSN applications and we identified their specific requirements. Then we introduced a new taxonomy of energy conservation schemes and we provided the reader with a comprehensive analysis of how these techniques can affect performance of applications. We finally reviewed some existing methods that allow trade-offs between multiple requirements to be achieved for efficient and sustainable sensor networks. The main WSN requirements that we identified in the different applications are scalability, coverage, latency, QoS, security, mobility and robustness. In Table 1 , we summarise the importance of these requirements for every class of application considered in this section. In these applications, sensors are expected to operate autonomously for a long period of time, ranging from weeks to months. However, every application is constrained in terms of energy due to the scarce battery resources of the sensors, which limits the network lifetime. Indeed, it may not always be possible to manually replenish the motes because of their number, the maintenance cost or the inaccessibility of monitored regions. This is the case of structural health monitoring applications, precision agriculture and environment monitoring, transportation systems. Furthermore, some applications such as healthcare applications can tolerate battery replacement, but we believe that the rapid depletion of the battery prevent their wide adoption. Indeed, efforts are still made to propose energy-efficient solutions for body area networks to foster the acceptance of these technologies by the patients. This is why the design of WSNs requires, in both cases, the development of energy-efficient solutions that meet a specific set of requirements.. In order to achieve energy efficiency, we first present in the next section existing standards developed for lowpower wireless sensor networks.. Bluetooth low energy and IEEE 802.15.4-based standards have been specifically developed for batteryoperated devices. They enable energy-saving through duty cycling and include optional modes that can be disabled for further network lifetime optimisation. In Table 2 we compare these two WSN-specific standards with other wellknown wireless standards (Wi-Fi, WiMax, WiMedia, Bluetooth) regarding data rate, transmission range, scalability and applications.. In terms of applications, existing healthcare platforms often interface with Bluetooth due to the suitability of this technology for body area networks that demand short communication ranges and high data rates. However, Bluetooth technology may quickly deplete a nodes' energy. In this case, BLE or IEEE 802.15.6 may be considered as alternatives. ZigBee technology is suitable for a large number of applications thanks to its scalability and energyefficiency. For example, in smart home automation, Zig-Bee data rate and radio range are sufficient for room supervision. Nevertheless, in a more complex monitoring system, both ambient sensor networks and body sensor networks may be integrated together and further connected to the Internet via 6LowPan. In large-scale outdoor deployment, the Zigbee 100-meter achievable radio range may quickly become limiting. In this case, we envision that WiMax-enabled gateways will be able to mesh the topology to connect the network to the Internet. Thus, the integration of different technologies and standards is necessary to respond to the needs of emerging and challenging applications such as Smart grids, Intelligent Transportation Systems and Healthcare Information Systems.. Standardisation is a key issue for the success of WSN markets. Although application-specific standards are emerging, such as WirelessHART and ISA100.11a for industry, and IEEE 802.15.6 for body sensor networks, they can still be improved in regard to application requirements. For instance, some research studies propose to optimise standard parameters such as packet size, slot length, contention window length or even introduce alternative protocols. Moreover, the performance of recent standards (e.g., MQTT, IEEE 802.15.6) need further investigation, because there is a lack of evaluation concerning these solutions and a lack of comparisons with well-established protocols. It also appears that current standards cannot respond to all application needs, notably regarding hard real-time requirements and security issues. In parallel to ongoing standardisation efforts, many solutions have been developed which strongly consider energy-saving.. It is clear that many efforts have been made to enhance the lifetime of WSNs through a variety of energyefficient mechanisms. It also appears that energy efficiency and other applications requirements are strongly dependent, so that various performance metrics have to be optimized jointly. Indeed, energy-efficient routing protocols and sleep/wakeup schemes directly influence network latency. Similarly, radio optimisation trades off signal quality for battery conservation, and data reduction approaches can affect the accuracy of the collected information. Additionally, if sensor recharging techniques are promising, energy-saving mechanisms remain essential. In Table 4 , we summarise the different energy-saving mechanisms and the WSN requirements they directly influence. For example, we can see that energy-efficient routing solutions can improve the robustness by using multipath routing protocols that provide alternative paths in case of a node failure. Furthermore, in Table 5 , we link Tables 1 (which represents the applications requirements) and Table 4 (which represents the interdependence between energy-efficient mechanisms and other requirements) in order to show how some energy-efficient techniques can be used in specific applications to jointly optimize multiple criteria. Here is an example to explain how to read Table 5 : in agricultural applications, sleep/wakeup schemes can influence the coverage by using topology control mechanisms. The justification of these statements can be found in Table 4 , and more generally in the discussions in Section 4.. In this section, a classification is provided of solutions proposed in the literature to satisfy multiple objectives. We can distinguish the aforementioned techniques based on the flexibility of the obtained trade-off over time. By flexibility we mean that the trade-off may change over the time depending on the network conditions. For instance, multi-metric protocols usually specify a desired trade-off between the requirements at the conception phase when setting the parameters. Preference is given to a requirement by the designer. For this reason, even if the requirements are highly dependent, the trade-off is decided once and for all. In contrast, MOO solutions explore a variety of candidate solutions at run-time that represent different trade-off points of the design space. Generation after generation, behaviour policy evolves and is adjusted to the network dynamics. In-between, cross-layer approaches are expected to improve network performances by exploiting the interactions between layers. The downside is the complexity of the method that necessitates a good understanding of the interplay of various variables.

O O O B I I O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Space-Frequency Quantization for Image Compression With Directionlets Abstract-The standard separable 2-D wavelet transform (WT) has recently achieved a great success in image processing because it provides a sparse representation of smooth images. However, it fails to efficiently capture 1-D discontinuities, like edges or contours. These features, being elongated and characterized by geometrical regularity along different directions, intersect and generate many large magnitude wavelet coefficients. Since contours are very important elements in the visual perception of images, to provide a good visual quality of compressed images, it is fundamental to preserve good reconstruction of these directional features. In our previous work, we proposed a construction of critically sampled perfect reconstruction transforms with directional vanishing moments imposed in the corresponding basis functions along different directions, called directionlets. In this paper, we show how to design and implement a novel efficient space-frequency quantization (SFQ) compression algorithm using directionlets. Our new compression method outperforms the standard SFQ in a rate-distortion sense, both in terms of mean-square error and visual quality, especially in the low-rate compression regime. We also show that our compression method, does not increase the order of computational complexity as compared to the standard SFQ algorithm. We have proposed a novel adaptive image compression algorithm that combines the SFQ method proposed in [20] and directionlets. In our algorithm, image is segmented using the quadtree segmentation method and transform directions are adapted to dominant directions in each segment. The spatial segmentation and the choice of transform directions are optimized in a R-D sense using Lagrangian optimization.. We showed in our previous work that directionlets are capable of providing a more efficient representation of elongated oriented features in images, like edges or contours, than the standard 2-D WT [2] . For that reason, our compression method outperforms the standard SFQ in terms of both the numerical and visual quality of compressed images at the same bit rate. The gain is especially significant at low operational bit rates, as we have demonstrated in Section IV.. Since we use spatial segmentation in our method, the reconstructed images are affected by blocking effect. Hence, we apply a postprocessing deblocking method that successfully removes the sharp edges along segment boundaries, as we have explained in Section III-C. We have shown that the improvement of the visual quality of reconstructed images is significant, although the difference in the numerical quality is negligible.. We have also analyzed the total computational complexity of our method. We have shown in Section III-D that the order of complexity of our method is of same as the order of complexity of the standard SFQ algorithm. In this section, we apply our compression algorithm using directionlets to several standard natural test images of the size 512 512. The results are compared to the results of the standard SFQ in terms of both numerical and visual quality.. First, we compare the behavior of the standard WT and the adaptive S-AWT choosing best directions per segment in NLA of images. Then, we show the numerical results of the compression methods and reconstructed images for compression at different bit rates.

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