Replacement Paths via Row Minima of Concise Matrices

Matrix $M$ is {\em $k$-concise} if the finite entries of each column of $M$ consist of $k$ or less intervals of identical numbers. We give an $O(n+m)$-time algorithm to compute the row minima of any $O(1)$-concise $n\times m$ matrix. Our algorithm yields the first $O(n+m)$-time reductions from the replacement-paths problem on an $n$-node $m$-edge undirected graph (respectively, directed acyclic graph) to the single-source shortest-paths problem on an $O(n)$-node $O(m)$-edge undirected graph (respectively, directed acyclic graph). That is, we prove that the replacement-paths problem is no harder than the single-source shortest-paths problem on undirected graphs and directed acyclic graphs. Moreover, our linear-time reductions lead to the first $O(n+m)$-time algorithms for the replacement-paths problem on the following classes of $n$-node $m$-edge graphs (1) undirected graphs in the word-RAM model of computation, (2) undirected planar graphs, (3) undirected minor-closed graphs, and (4) directed acyclic graphs.Comment: 23 pages, 1 table, 9 figures, accepted to SIAM Journal on Discrete Mathematic

Approximating the Permanent with Fractional Belief Propagation

We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze the Belief Propagation (BP) approach and its Fractional Belief Propagation (FBP) generalization for computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. The Fractional Free Energy (FFE) functional is parameterized by a scalar parameter $\gamma\in[-1;1]$, where $\gamma=-1$ corresponds to the BP limit and $\gamma=1$ corresponds to the exclusion principle (but ignoring perfect matching constraints) Mean-Field (MF) limit. FFE shows monotonicity and continuity with respect to $\gamma$. For every non-negative matrix, we define its special value $\gamma_*\in[-1;0]$ to be the $\gamma$ for which the minimum of the $\gamma$-parameterized FFE functional is equal to the permanent of the matrix, where the lower and upper bounds of the $\gamma$-interval corresponds to respective bounds for the permanent. Our experimental analysis suggests that the distribution of $\gamma_*$ varies for different ensembles but $\gamma_*$ always lies within the $[-1;-1/2]$ interval. Moreover, for all ensembles considered the behavior of $\gamma_*$ is highly distinctive, offering an emprirical practical guidance for estimating permanents of non-negative matrices via the FFE approach.Comment: 42 pages, 14 figure

AutoGraff: towards a computational understanding of graffiti writing and related art forms.

The aim of this thesis is to develop a system that generates letters and pictures with a style that is immediately recognizable as graffiti art or calligraphy. The proposed system can be used similarly to, and in tight integration with, conventional computer-aided geometric design tools and can be used to generate synthetic graffiti content for urban environments in games and in movies, and to guide robotic or fabrication systems that can materialise the output of the system with physical drawing media. The thesis is divided into two main parts. The first part describes a set of stroke primitives, building blocks that can be combined to generate different designs that resemble graffiti or calligraphy. These primitives mimic the process typically used to design graffiti letters and exploit well known principles of motor control to model the way in which an artist moves when incrementally tracing stylised letter forms. The second part demonstrates how these stroke primitives can be automatically recovered from input geometry defined in vector form, such as the digitised traces of writing made by a user, or the glyph outlines in a font. This procedure converts the input geometry into a seed that can be transformed into a variety of calligraphic and graffiti stylisations, which depend on parametric variations of the strokes

Accurate Prediction of Core Properties for Chiral Molecules using Pseudo Potentials

Pseudo potentials (PPs) constitute perhaps the most common way to treat relativity, often in a formally non-relativistic framework, and reduce the electronic structure to the chemically relevant part. The drawback is that orbitals obtained in this picture (called pseudo orbitals (POs)) show a reduced nodal structure and altered amplitude in the vicinity of the nucleus, when compared to the corresponding molecular orbitals (MOs). Thus expectation values of operators localized in the spatial core region that are calculated with POs, deviate significantly from the same expectation values calculated with all-electron (AE) MOs. This study describes the reconstruction of AE MOs from POs, with a focus on POs generated by energy consistent pseudo Hamiltonians. The method reintroduces the nodal structure into the POs, thus providing an inexpensive and easily implementable method that allows to use nonrelativistic, efficiently calculated POs for good estimates of expectation values of core-like properties. The discussion of the method proceeds in two parts: Firstly, the reconstruction scheme is developed for atomic cases. Secondly, the scheme is discussed in the context of MO reconstruction and successfully applied to numerous numerical examples. Starting from the equations of the state-averaged multi-configuration self- consistent field method, used for the generation of energy consistent pseudo potentials, the electronic spectrum of the many-electron Hamiltonian is linked to the spectrum of the effective one-electron Fock operator by means of various models systems. This relation and the Topp–Hopfield–Kramers theorem, are used to show the shape-consistency of energy-consistent POs for atomic systems. Shape-consistency describes POs that follow distinct AOs exactly outside a core-radius r_core . In the cases presented here, shape-consistency holds to a high degree and it follows that in atomic systems every PO has one distinct partner in the set of AOs. The overlap integral between these two orbitals is close to one, as it is determined mainly by the spatial orbital parts outside r_core . Expanding, e.g., a 5s PO in occupied AOs, the 5s AOs will have the highest contribution. The POs itself contains contributions from high-energy unoccupied AOs as well (e.g. 15s), which damp the nodal structure of the POs near the nucleus. Consequently, neglecting contributions from unoccupied orbitals in a projection of the POs reintroduces the nodal structure. This approach is not directly suitable for the reconstruction of MOs, as they often need to be expanded in a full set of AOs at each atomic center, including all unoccupied orbitals, to properly account for the electron density distribution in the molecule. However, it is shown that the occupied MOs are well described by occupied and low-energy unoccupied AOs only and a mapping of the POs onto a basis containing only these orbitals reconstructs the nodal structure of the MO. The approach uses only standard integrals available in most quantum chemistry programs. The computational cost of these integrals scales with N^2 , where N is the number of basis functions. The most time consuming step is a Gram-Schmidt orthogonalization, which scales in this implementation with MN^2 , M being the number of reconstructed orbitals. The reconstruction method is subsequently tested: Valence orbitals of atomic, closed-shell systems were reconstructed numerically exactly. The influence of numerical parameters is investigated using the molecule BaF . It is shown that the method is basis set dependent: One has to ensure that the PO basis can be expanded exactly in the basis of AOs. Violating this rule of thumb may degrade the quality of reconstructed orbitals. Additionally, the representation of MOs by a linear combination of occupied and unoccupied AOs is investigated. For the exemplary systems, the shells included in the fitting procedure of the PP were sufficient. Reconstruction of the alkaline earth monofluorides showed that periodic trends can be reconstructed as well. Scaling of hyperfine structure parameters with increasing atomic number is discussed. For hydrogenic atoms, the scaling should be linear, whereas small deviations from the linear behavior were observed for molecules. The scaling laws computed from reconstructed and reference orbitals were almost identical. In this context, the failure of commonly used relativistic enhancement factors beyond atomic number 100 is discussed. Applicability of the method is also tested on parity violating properties for which the main contribution is generated by the valence orbitals near the nucleus. Symmetry-independence of the method is shown by successful reconstruction of orbitals of the tetrahedral PbCl_4 and chiral NWHClF. The reliable reconstruction of chemical trends is shown with the help of the NWHClF derivatives NWHBrF and NWHFI. The study of chiral compounds as, e.g., NWHClF and its group 17 derivatives, which have been proposed as paradigm for the detection of parity-violation in chiral molecules, remains of great importance. Especially the direct determination of absolute configuration of chiral centers is still non-trivial. The author contributed to this field with a self-written molecular dynamics (MD) program to simulate Coulomb explosions and thus to provide an insight especially into the early explosion stages directly after an instantaneous multi-ionization of the molecule CHBrClF, comparable to experiments using the Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) technique. An algorithm for the determination of the investigated molecule’s absolute configuration from time-of-flight data and detection locations of molecular fragments is included in the program. The program was used to generate experiment-equivalent data which allowed for the first time the investigation of non-racemic mixtures by the analysis routines of the experiment. The MD program includes harmonic and anharmonic bond potentials. A charge-exchange model can model partial charges in early phases of the Coulomb explosion. Furthermore, Born–Oppenheimer MD simulations and statistical models are used to explain the relative abundance of products belonging to competing reaction channels, as obtained by photoion coincidence measurements. Additionally, qualitative statements about reaction branching ratios are made by comparing the partition functions of involved degrees of freedom. Analytic equations for partition functions of simple models are used to provide a simple formula allowing fast estimates of reaction branching ratios