Minimizing Turns in Watchman Robot Navigation: Strategies and Solutions

The Orthogonal Watchman Route Problem (OWRP) entails the search for the shortest path, known as the watchman route, that a robot must follow within a polygonal environment. The primary objective is to ensure that every point in the environment remains visible from at least one point on the route, allowing the robot to survey the entire area in a single, continuous sweep. This research places particular emphasis on reducing the number of turns in the route, as it is crucial for optimizing navigation in watchman routes within the field of robotics. The cost associated with changing direction is of significant importance, especially for specific types of robots. This paper introduces an efficient linear-time algorithm for solving the OWRP under the assumption that the environment is monotone. The findings of this study contribute to the progress of robotic systems by enabling the design of more streamlined patrol robots. These robots are capable of efficiently navigating complex environments while minimizing the number of turns. This advancement enhances their coverage and surveillance capabilities, making them highly effective in various real-world applications.Comment: 6 pages, 3 figure

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

Securing Pathways with Orthogonal Robots

The protection of pathways holds immense significance across various domains, including urban planning, transportation, surveillance, and security. This article introduces a groundbreaking approach to safeguarding pathways by employing orthogonal robots. The study specifically addresses the challenge of efficiently guarding orthogonal areas with the minimum number of orthogonal robots. The primary focus is on orthogonal pathways, characterized by a path-like dual graph of vertical decomposition. It is demonstrated that determining the minimum number of orthogonal robots for pathways can be achieved in linear time. However, it is essential to note that the general problem of finding the minimum number of robots for simple polygons with general visibility, even in the orthogonal case, is known to be NP-hard. Emphasis is placed on the flexibility of placing robots anywhere within the polygon, whether on the boundary or in the interior.Comment: 8 pages, 5 figure

Guarding and Searching Polyhedra

Guarding and searching problems have been of fundamental interest since the early years of Computational Geometry. Both are well-developed areas of research and have been thoroughly studied in planar polygonal settings. In this thesis we tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra. We solve the Art Gallery Problem with reflex edge guards in orthogonal polyhedra having reflex edges in just two directions: generalizing a classic theorem by O'Rourke, we prove that r/2 + 1 reflex edge guards are sufficient and occasionally necessary, where r is the number of reflex edges. We also show how to compute guard locations in O(n log n) time. Then we investigate the Art Gallery Problem with mutually parallel edge guards in orthogonal polyhedra with e edges, showing that 11e/72 edge guards are always sufficient and can be found in linear time, improving upon the previous state of the art, which was e/6. We also give tight inequalities relating e with the number of reflex edges r, obtaining an upper bound on the guard number of 7r/12 + 1. We further study the Art Gallery Problem with edge guards in polyhedra having faces oriented in just four directions, obtaining a lower bound of e/6 - 1 edge guards and an upper bound of (e+r)/6 edge guards. All the previously mentioned results hold for polyhedra of any genus. Additionally, several guard types and guarding modes are discussed, namely open and closed edge guards, and orthogonal and non-orthogonal guarding. Next, we model the Searchlight Scheduling Problem, the problem of searching a given polyhedron by suitably turning some half-planes around their axes, in order to catch an evasive intruder. After discussing several generalizations of classic theorems, we study the problem of efficiently placing guards in a given polyhedron, in order to make it searchable. For general polyhedra, we give an upper bound of r^2 on the number of guards, which reduces to r for orthogonal polyhedra. Then we prove that it is strongly NP-hard to decide if a given polyhedron is entirely searchable by a given set of guards. We further prove that, even under the assumption that an orthogonal polyhedron is searchable, approximating the minimum search time within a small-enough constant factor to the optimum is still strongly NP-hard. Finally, we show that deciding if a specific region of an orthogonal polyhedron is searchable is strongly PSPACE-hard. By further improving our construction, we show that the same problem is strongly PSPACE-complete even for planar orthogonal polygons. Our last results are especially meaningful because no similar hardness theorems for 2-dimensional scenarios were previously known

Visibility Domains and Complexity

Two problems in discrete and computational geometry are considered that are related to questions about the combinatorial complexity of arrangements of visibility domains and about the hardness of path planning under cost measures defined using visibility domains. The first problem is to estimate the VC-dimension of visibility domains. The VC-dimension is a fundamental parameter of every range space that is typically used to derive upper bounds on the size of hitting sets. Better bounds on the VC-dimension directly translate into better bounds on the size of hitting sets. Estimating the VC-dimension of visibility domains has proven to be a hard problem. In this thesis, new tools to tackle this problem are developed. Encircling arguments are combined with decomposition techniques of a new kind. The main ingredient of the novel approach is the idea of relativization that makes it possible to replace in the analysis of intersections the complicated visibility domains by simpler geometric ranges. The main result here is the new upper bound of 14 on the VC-dimension of visibility polygons in simple polygons that improves significantly upon the previously known best upper bound of 23. For the VC-dimension of perimeter visibility domains, the new techniques yield an upper bound of 7 that leaves only a very small gap to the best known lower bound of 5. The second problem considered is to compute the barrier resilience of visibility domains. In barrier resilience problems, one is given a set of barriers and two points s and t in R^d. The task is to find the minimum number of barriers one has to remove such that there is a way between s and t that does not cross a barrier. In the field of sensor networks, the barriers are interpreted as sensor ranges and the barrier resilience of a network is a measure for its vulnerability. In this thesis the very natural special case where the barriers are visibility domains is investigated. It can also be formulated in terms of finding a so-called minimum witness path. For visibility domains in simple polygons it is shown that one can find an optimal path efficiently. For polygons with holes an approximation hardness result is shown that is stronger than previous hardness results in geometric settings. Two different three-dimensional settings are considered and their respective relations to the Minimum Neighborhood Path problem and the Minimum Color Path problem in graphs are demonstrated. For one of the three-dimensional problems a 2-approximation algorithm is designed. For the general problem of finding minimum witness paths among polyhedral obstacles it turns out that it is not approximable in a strong sense

Minimizing Turns in Single and Multi Robot Coverage Path Planning

Spurred by declining costs of robotics, automation is becoming a prevalent area of interest for many industries. In some cases, it even makes economic sense to use a team of robots to achieve a goal faster. In this thesis we study sweep coverage path planning, in which a robot or a team of robots must cover all points in a workspace with its footprint. In many coverage applications, including cleaning and monitoring, it is beneficial to use coverage paths with minimal robot turns. In the first part of the thesis, we address this for a single robot by providing an efficient method to compute the minimum altitude of a non-convex polygonal region, which captures the number of parallel line segments, and thus turns, needed to cover the region. Then, given a non-convex polygon, we provide a method to cut the polygon into two pieces that minimizes the sum of their altitudes. Given an initial convex decomposition of a workspace, we apply this method to iteratively re-optimize and delete cuts of the decomposition. Finally, we compute a coverage path of the workspace by placing parallel line segments in each region, and then computing a tour of the segments of minimum cost. We present simulation results on several workspaces with obstacles, which demonstrate improvements in both the number of turns in the final coverage path and runtime. In the second part of the thesis, we extend the concepts developed for a single robot coverage to a multi robot case. We provide a metric χ that approximates the cost of a coverage path, which accounts for the cost of turns. Given a polygon, we provide a method for cutting a polygon into two that would minimize the maximum cost χ between the two polygons. Provided with an initial n-cell decomposition, we apply this method in the iterative manner to re-optimize cuts in order to minimize the maximum cost χ over all cells in the decomposition. For each cell in the re-optimized n-cell decomposition, a single robot coverage path is computed using the minimum altitude decomposition. We present the simulation results that demonstrate improvements in the maximum cost as well as the range of costs over all robots in the team

Photophysics and light induced photobiology of antiviral and antitumor agents, hypericin and hypocrellin

The dissertation mainly focuses on the excited-state photophysics and the light-induced biological activity of hypericin and its analogs.;Femtosecond laser technology has provided the opportunity to investigate the rapid dynamics of these molecules. Fluorescence upconversion measurements, which monitor only emission from the fluorescent singlet state, demonstrate that hexamethoxy hypericin, which possesses no labile protons, has an instantaneous rise time for its transient response. On the other hand, hypericin shows a clear 10-ps rise time. This confirms excited-state H-atom transfer as the primary photophysical process in hypericin.;Femtosecond transient absorption spectroscopy technique is used to determine if excited state H-atom transfer is concerted. Previous studies using human serum albumin (HSA) and hypericin suggested that excited state H-atom transfer is concerted, but the results from the hypericin in reverse micelles show no evidence for a concerted hydrogen atom transfer mechanism. We are, however, unable to conclude if only one hydrogen atom is transferred or if two are transferred in a stepwise fashion.;By means of time-resolved infrared spectroscopy, ab initio quantum mechanical calculations, and synthetic organic chemistry, a region in the infrared spectrum, between 1400 and 1500 cm-1, of triplet hypericin has been found corresponding to translocation of the hydrogen atom between the enol and the keto oxygens, O &cdots; H &cdots; O. This result is discussed in the context of the photophysics of hypericin and of eventual measurements to observe directly the excited-state H-atom transfer.;Light-induced antiviral activity of hypericin and hypocrellin is compared in normoxic and hypoxic conditions. Although both molecules require oxygen to show full virucidal effects, hypericin is still effective at low oxygen level where hypocrellin is not. Since the singlet oxygen yield of hypericin is about half of that of hypocrellin, this result cannot be explained by a traditional Type II mechanism. We propose that the ejected proton upon illumination might enhance the activity of activated oxygen species.;A series of hypericin analogs were found to differ in their cytotoxic activity induced by ambient light levels. These analogs vary in their ability to partition into cells, to generate singlet oxygen as well as in other photophysical properties. The percent distribution of hypericin and its analogs in cells are measured using a steady state absorption technique. We attempt to find a relationship between those results and the exact localization of the drug at subcellular level

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version