Christiane Schmidt

Exact solutions and bounds for general art gallery problems

The classical Art Gallery Problem asks for the minimum number of guards that achieve visibility coverage of a given polygon. This problem is known to be NP-hard, even for very restricted and discrete special cases. For the case of vertex guards and simple orthogonal polygons, Cuoto et al. have recently developed an exact method that is based on a set-cover approach. For the general problem (in which both the set of possible guard positions and the point set to be guarded are uncountable), neither constant-factor approximation algorithms nor exact solution methods are known. We present a primal-dual algorithm based on linear programming that provides lower bounds on the necessary number of guards in every step and—in case of convergence and integrality—ends with an optimal solution. We describe our implementation and give experimental results for an assortment of polygons, including nonorthogonal polygons with holes.

Point guards and point clouds: solving general art gallery problems

In this video, we illustrate how one of the classical areas of computational geometry has gained in practical relevance, which in turn gives rise to new, fascinating geometric problems. In particular, we demonstrate how the robot platform IRMA3D can produce high-resolution, virtual 3D environments, based on a limited number of laser scans. Computing an optimal set of scans amounts to solving an instance of the Art Gallery Problem (AGP): Place a minimum number of stationary guards in a polygonal region P, such that all points in P are guarded.

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 Nuchter 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 optimization 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(logA)-competitive strategy for simple rectilinear polygons, using the assumption that each edge of the polygon has to be fully visible from some scan point.

Empowered by wireless communication: Distributed methods for self-organizing traffic collectives

In recent years, tremendous progress has been made in understanding the dynamics of vehicle traffic flow and traffic congestion by interpreting traffic as a multiparticle system. This helps to explain the onset and persistence of many undesired phenomena, for example, traffic jams. It also reflects the apparent helplessness of drivers in traffic, who feel like passive particles that are pushed around by exterior forces; one of the crucial aspects is the inability to communicate and coordinate with other traffic participants. We present distributed methods for solving these fundamental problems, employing modern wireless, ad-hoc, multi-hop networks. The underlying idea is to use these capabilities as the basis for self-organizing methods for coordinating data collection and processing, recognizing traffic phenomena, and changing their structure by coordinated behavior. The overall objective is a multi-level approach that reaches from protocols for local wireless communication, data dissemination, pattern recognition, over hierarchical structuring and coordinated behavior, all the way to large-scale traffic regulation. In this article, we describe three types of results: (i) self-organizing and distributed methods for maintaining and collecting data (using our concept of Hovering Data Clouds); (ii) adaptive data dissemination for traffic information systems; (iii) methods for self-recognition of traffic jams. We conclude by describing higher-level aspects of our work.

Emergent algorithms for centroid and orientation detection in high-performance embedded cameras

Due to increasing speed and capabilities of production machines, the need for extremely fast and robust observation, classification, and error handling is vital to industrial image processing. We present an emergent algorithmic computing scheme and a corresponding embedded massively-parallel hardware architecture for these problems. They offer the potential to turn CMOS-camera-chips into intelligent vision devices which carry out tasks without help of a central processor, only based on local interaction of agents crawling on a large field of processing elements. It also constitutes a breakthrough for understanding sensor devices as a decentralized concept, resulting in much faster computation evading communication bottlenecks of classic approaches that become an ever-growing impediment to scalability. Here, in contrast, the number of agents and the field size and thus the computable image resolution is extremely scalable and therefore promises even more benefit with future hardware development. The results are based on novel algorithmic solutions allowing processor elements to compute center points, moments, and orientation of multiple image objects in parallel, which is of central importance to e.g. robotics. We finally present the algorithms capabilities if realized in state-of-the-art FPGAs and ASICs.

Exploring and triangulating a region by a swarm of robots

We consider online and offline problems related to exploring and surveying a region by a swarm of robots with limited communication range. The minimum relay triangulation problem (MRTP) asks for placing a minimum number of robots, such that their communication graph is a triangulated cover of the region. The maximum area triangulation problem (MATP) aims at finding a placement of n robots such that their communication graph contains a root and forms a triangulated cover of a maximum possible amount of area. Both problems are geometric versions of natural graph optimization problems. The offline version of both problems share a decision problem, which we prove to be NP-hard. For the online version of the MRTP, we give a lower bound of 6/5 for the competitive ratio, and a strategy that achieves a ratio of 3; for different offline versions, we describe polynomial-time approximation schemes. For the MATP we show that no competitive ratio exists for the online problem, and give polynomial-time approximation schemes for offline versions.

The continuous 1.5D terrain guarding problem: Discretization, optimal solutions, and PTAS

In the NP-hard continuous 1.5D Terrain Guarding Problem (TGP) we are given an

A Novel Efficient Approach for Solving the Art Gallery Problem

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Triangulating unknown environments using robot swarms

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Designing a Decentralized Traffic Information System — AutoNomos

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