Automatic Classification of Restricted Lattice Walks

We propose an experimental mathematics approach leading to the computer-driven discovery of various structural properties of general counting functions coming from enumeration of walks

10481 Abstracts Collection -- Computational Counting

From November 28 to December 3 2010, the Dagstuhl Seminar 10481 ``Computational Counting\u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

New steps in walks with small steps in the quarter plane

In this article we obtain new expressions for the generating functions counting (non-singular) walks with small steps in the quarter plane. Those are given in terms of infinite series, while in the literature, the standard expressions use solutions to boundary value problems. We illustrate our results with three examples (an algebraic case, a transcendental D-finite case, and an infinite group model).Comment: 47 pages, 8 figures, to appear in Annals of Combinatoric

Nonholonomic Motion Planning as Efficient as Piano Mover's

We present an algorithm for non-holonomic motion planning (or 'parking a car') that is as computationally efficient as a simple approach to solving the famous Piano-mover's problem, where the non-holonomic constraints are ignored. The core of the approach is a graph-discretization of the problem. The graph-discretization is provably accurate in modeling the non-holonomic constraints, and yet is nearly as small as the straightforward regular grid discretization of the Piano-mover's problem into a 3D volume of 2D position plus angular orientation. Where the Piano mover's graph has one vertex and edges to six neighbors each, we have three vertices with a total of ten edges, increasing the graph size by less than a factor of two, and this factor does not depend on spatial or angular resolution. The local edge connections are organized so that they represent globally consistent turn and straight segments. The graph can be used with Dijkstra's algorithm, A*, value iteration or any other graph algorithm. Furthermore, the graph has a structure that lends itself to processing with deterministic massive parallelism. The turn and straight curves divide the configuration space into many parallel groups. We use this to develop a customized 'kernel-style' graph processing method. It results in an N-turn planner that requires no heuristics or load balancing and is as efficient as a simple solution to the Piano mover's problem even in sequential form. In parallel form it is many times faster than the sequential processing of the graph, and can run many times a second on a consumer grade GPU while exploring a configuration space pose grid with very high spatial and angular resolution. We prove approximation quality and computational complexity and demonstrate that it is a flexible, practical, reliable, and efficient component for a production solution.Comment: 34 pages, 37 figures, 9 tables, 4 graphs, 8 insert

Analytic combinatorics : functional equations, rational and algebraic functions

This report is part of a series whose aim is to present in a synthetic way the major methods and models in analytic combinatorics. Here, we detail the case of rational and algebraic functions and discuss systematically closure properties, the location of singularities, and consequences regarding combinatorial enumeration. The theory is applied to regular and context-free languages, finite state models, paths in graphs, locally constrained permutati- ons, lattice paths and walks, trees, and planar maps