How to Walk Your Dog in the Mountains with No Magic Leash

We describe a $O(\log n )$-approximation algorithm for computing the homotopic \Frechet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms were known only for curves on the Euclidean plane with polygonal obstacles. A key technical ingredient in our analysis is a $O(\log n)$-approximation algorithm for computing the minimum height of a homotopy between two curves. No algorithms were previously known for approximating this parameter. Surprisingly, it is not even known if computing either the homotopic \Frechet distance, or the minimum height of a homotopy, is in NP

The method of Gaussian weighted trajectories. V. On the 1GB procedure for polyatomic processes

In recent years, many chemical reactions have been studied by means of the quasi-classical trajectory (QCT) method within the Gaussian binning (GB) procedure. The latter consists in "quantizing" the final vibrational actions in Bohr spirit by putting strong emphasis on the trajectories reaching the products with vibrational actions close to integer values. A major drawback of this procedure is that if N is the number of product vibrational modes, the amount of trajectories necessary to converge the calculations is ~ 10^N larger than with the standard QCT method. Applying it to polyatomic processes is thus problematic. In a recent paper, however, Czako and Bowman propose to quantize the total vibrational energy instead of the vibrational actions [G. Czako and J. M. Bowman, J. Chem. Phys., 131, 244302 (2009)], a procedure called 1GB here. The calculations are then only ~ 10 times more time-consuming than with the standard QCT method, allowing thereby for considerable numerical saving. In this paper, we propose some theoretical arguments supporting the 1GB procedure and check its validity on model test cases as well as the prototype four-atom reaction OH+D_2 -> HOD+D

Core percolation in random graphs: a critical phenomena analysis

We study both numerically and analytically what happens to a random graph of average connectivity "alpha" when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated vertices plus an induced subgraph we call the "core". In the thermodynamic limit of an infinite random graph, we compute analytically the dynamics of leaf removal, the number of isolated vertices and the number of vertices and edges in the core. We show that a second order phase transition occurs at "alpha = e = 2.718...": below the transition, the core is small but above the transition, it occupies a finite fraction of the initial graph. The finite size scaling properties are then studied numerically in detail in the critical region, and we propose a consistent set of critical exponents, which does not coincide with the set of standard percolation exponents for this model. We clarify several aspects in combinatorial optimization and spectral properties of the adjacency matrix of random graphs. Key words: random graphs, leaf removal, core percolation, critical exponents, combinatorial optimization, finite size scaling, Monte-Carlo.Comment: 15 pages, 9 figures (color eps) [v2: published text with a new Title and addition of an appendix, a ref. and a fig.

Characterization of complex networks: A survey of measurements

Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements. Important related issues covered in this work comprise the representation of the evolution of complex networks in terms of trajectories in several measurement spaces, the analysis of the correlations between some of the most traditional measurements, perturbation analysis, as well as the use of multivariate statistics for feature selection and network classification. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the proper application and interpretation of measurements.Comment: A working manuscript with 78 pages, 32 figures. Suggestions of measurements for inclusion are welcomed by the author

Matching Triangles and Triangle Collection: Hardness based on a Weak Quantum Conjecture

Classically, for many computational problems one can conclude time lower bounds conditioned on the hardness of one or more of key problems: k-SAT, 3SUM and APSP. More recently, similar results have been derived in the quantum setting conditioned on the hardness of k-SAT and 3SUM. This is done using fine-grained reductions, where the approach is to (1) select a key problem $X$ that, for some function $T$, is conjectured to not be solvable by any $O(T(n)^{1-\epsilon})$ time algorithm for any constant $\epsilon > 0$ (in a fixed model of computation), and (2) reduce $X$ in a fine-grained way to these computational problems, thus giving (mostly) tight conditional time lower bounds for them. Interestingly, for Delta-Matching Triangles and Triangle Collection, classical hardness results have been derived conditioned on hardness of all three mentioned key problems. More precisely, it is proven that an $n^{3-\epsilon}$ time classical algorithm for either of these two graph problems would imply faster classical algorithms for k-SAT, 3SUM and APSP, which makes Delta-Matching Triangles and Triangle Collection worthwhile to study. In this paper, we show that an $n^{1.5-\epsilon}$ time quantum algorithm for either of these two graph problems would imply faster quantum algorithms for k-SAT, 3SUM, and APSP. We first formulate a quantum hardness conjecture for APSP and then present quantum reductions from k-SAT, 3SUM, and APSP to Delta-Matching Triangles and Triangle Collection. Additionally, based on the quantum APSP conjecture, we are also able to prove quantum lower bounds for a matrix problem and many graph problems. The matching upper bounds follow trivially for most of them, except for Delta-Matching Triangles and Triangle Collection for which we present quantum algorithms that require careful use of data structures and Ambainis' variable time search