Homotopic rectilinear routing with few links and thick edges

We study the problem of finding non-crossing thick minimum-link rectilinear paths homotopic to a set of input paths in an environment with rectangular obstacles. This problem occurs in the context of map schematization under geometric embedding restrictions, for example, when schematizing a highway network for use as a thematic layer. We present a 2-approximation algorithm that runs in O(n3 +kin log n + kout) time, where n is the total number of input paths and obstacles and kin and kout are the total complexities of the input and output paths, respectively. Our algorithm not only approximates the minimum number of links, but also minimizes the total length of the paths. An approximation factor of 2 is optimal when using smallest paths as lower bound

Polygon Exploration with Time-Discrete Vision

With the advent of autonomous robots with two- and three-dimensional scanning capabilities, classical visibility-based exploration methods from computational geometry have gained in practical importance. However, real-life laser scanning of useful accuracy does not allow the robot to scan continuously while in motion; instead, it has to stop each time it surveys its environment. This requirement was studied by Fekete, Klein and Nuechter for the subproblem of looking around a corner, but until now has not been considered in an online setting for whole polygonal regions. We give the first algorithmic results for this important algorithmic problem that combines stationary art gallery-type aspects with watchman-type issues in an online scenario: We demonstrate that even for orthoconvex polygons, a competitive strategy can be achieved only for limited aspect ratio A (the ratio of the maximum and minimum edge length of the polygon), i.e., for a given lower bound on the size of an edge; we give a matching upper bound by providing an O(log A)-competitive strategy for simple rectilinear polygons, using the assumption that each edge of the polygon has to be fully visible from some scan point.Comment: 28 pages, 17 figures, 2 photographs, 3 tables, Latex. Updated some details (title, figures and text) for final journal revision, including explicit assumption of full edge visibilit

Bidirected minimum Manhattan network problem

In the bidirected minimum Manhattan network problem, given a set T of n terminals in the plane, we need to construct a network N(T) of minimum total length with the property that the edges of N(T) are axis-parallel and oriented in a such a way that every ordered pair of terminals is connected in N(T) by a directed Manhattan path. In this paper, we present a polynomial factor 2 approximation algorithm for the bidirected minimum Manhattan network problem.Comment: 14 pages, 16 figure