String Matching and 1d Lattice Gases

We calculate the probability distributions for the number of occurrences $n$ of a given $l$ letter word in a random string of $k$ letters. Analytical expressions for the distribution are known for the asymptotic regimes (i) $k \gg r^l \gg 1$ (Gaussian) and $k,l \to \infty$ such that $k/r^l$ is finite (Compound Poisson). However, it is known that these distributions do now work well in the intermediate regime $k \gtrsim r^l \gtrsim 1$. We show that the problem of calculating the string matching probability can be cast into a determining the configurational partition function of a 1d lattice gas with interacting particles so that the matching probability becomes the grand-partition sum of the lattice gas, with the number of particles corresponding to the number of matches. We perform a virial expansion of the effective equation of state and obtain the probability distribution. Our result reproduces the behavior of the distribution in all regimes. We are also able to show analytically how the limiting distributions arise. Our analysis builds on the fact that the effective interactions between the particles consist of a relatively strong core of size $l$, the word length, followed by a weak, exponentially decaying tail. We find that the asymptotic regimes correspond to the case where the tail of the interactions can be neglected, while in the intermediate regime they need to be kept in the analysis. Our results are readily generalized to the case where the random strings are generated by more complicated stochastic processes such as a non-uniform letter probability distribution or Markov chains. We show that in these cases the tails of the effective interactions can be made even more dominant rendering thus the asymptotic approximations less accurate in such a regime.Comment: 44 pages and 8 figures. Major revision of previous version. The lattice gas analogy has been worked out in full, including virial expansion and equation of state. This constitutes the main part of the paper now. Connections with existing work is made and references should be up to date now. To be submitted for publicatio

On the effects of the fix geometric constraint in 2D profiles on the reusability of parametric 3D CAD models

[EN] In order to be reusable, history-based feature-based parametric CAD models must reliably allow for modifications while maintaining their original design intent. In this paper, we demonstrate that relations that fix the location of geometric entities relative to the reference system produce inflexible profiles that reduce model reusability. We present the results of an experiment where novice students and expert CAD users performed a series of modifications in different versions of the same 2D profile, each defined with an increasingly higher number of fix geometric constraints. Results show that the amount of fix constraints in a 2D profile correlates with the time required to complete reusability tasks, i.e., the higher the number of fix constraints in a 2D profile, the less flexible and adaptable the profile becomes to changes. In addition, a pilot software tool to automatically track this type of constraints was developed and tested. Results suggest that the detection of fix constraint overuse may result in a new metric to assess poor quality models with low reusability. The tool provides immediate feedback for preventing high semantic level quality errors, and assistance to CAD users. Finally, suggestions are introduced on how to convert fix constraints in 2D profiles into a negative metric of 3D model quality.The authors would like to thank Raquel Plumed for her support in the statistical analysis. This work has been partially funded by Grant UJI-A02017-15 (Universitat Jaume I) and DPI201784526-R (MINECO/AEI/FEDER, UE), project CAL-MBE. The authors also wish to thank the editor and reviewers for their valuable comments and suggestions that helped us improve the quality of the paper.González-Lluch, C.; Company, P.; Contero, M.; Pérez Lopez, DC.; Camba, JD. (2019). On the effects of the fix geometric constraint in 2D profiles on the reusability of parametric 3D CAD models. 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(Colección Sapientia, Núm. 86). http://cad3dconsolidworks.uji.es .Contero, M., Company, P., Vila, C., & Aleixos, N. (2002). Product data quality and collaborative engineering. IEEE Computer Graphics Applications, 22(3), 32–42. https://doi.org/10.1109/MCG.2002.999786 .Dixon, B. M., & Dannenhoffer, J. F., III. (2014). Geometric sketch constraint solving with user feedback. Journal of Aerospace Information Systems, 11(5), 316–325. https://doi.org/10.2514/1.I010110 .Fudos, I., & Hoffmann, C. M. (1997). A graph-constructive approach to solving systems of geometric constraints. ACM Transactions on Graphics, 16(2), 179–216. https://doi.org/10.1145/248210.248223 .Ge, J. X., Chou, S. C., & Gao, X. S. (1999). Geometric constraint satisfaction using optimization methods. Computer-Aided Design, 31(14), 867–879. https://doi.org/10.1016/S0010-4485(99)00074-3 .González-Lluch, C., Company, P., Contero, M., Camba, J. D., & Colom, J. (2017a). 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Difficulties with the novices’ comprehension of the computer-aided design (CAD) interface: Understanding visual representations of CAD tools. Journal of Engineering Design, 14(2), 169–185. https://doi.org/10.1080/0954482031000091491

Editable Representations for 2D Geometric Design

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii 1. INTRODUCTION AND RELATED WORK : : : : : : : : : : : : : : : : 1 1.1 Trends in two dimensional sketching : : : : : : : : : : : : : : : : : : : 2 1.1.1 The descriptive approach : : : : : : : : : : : : : : : : : : : : : 2 1.1.2 The constructive approach : : : : : : : : : : : : : : : : : : : : 2 1.1.3 The declarative approach : : : : : : : : : : : : : : : : : : : : : 3 1.2 Constraint solving methods : : : : : : : : : : : : : : : : : : : : : : : 5 1.2.1 Numerical constraint solvers : : : : : : : : : : : : : : : : : : : 5 1.2.2 Constructive constraint solvers : : : : : : : : : : : : : : : : : : 6 1.2.3 Propagation methods : : : : : : : : : : : : : : : : : : : : : : : 7 1.2.4 Symbolic constraint solvers : : : : : : : : : : : : : : : : : : : : 9 1.2.5 Solvers using hybrid methods : : : : : : : : : : : : : : : : : : 9 1.2.6 Other methods : : : : : : : : : : : : : : : : : : : : : : : : : : 10 1.3 The repertoire of con..

A Constraint Programming Approach for Solving Rigid Geometric Systems

This paper introduces a new rigidification method--using interval constraint programming techniques--to solve geometric constraint systems. Standard rigidification techniques are graph-constructive methods exploiting the degrees of freedom of geometric objects. They work in two steps: a planning phase which identifies rigid clusters, and a solving phase which computes the coordinates of the geometric objects in every cluster. We propose here a new heuristic for the planning algorithm that yields in general small systems of equations. We also show that interval constraint techniques can be used not only to efficiently implement the solving phase, but also generalize former ad-hoc solving techniques. First experimental results show that this approach is more efficient than systems based on equational decomposition techniques