Parallel searching on m rays☆☆This research is supported by the DFG-Project “Diskrete Probleme”, No. Ot 64/8-3.

AbstractWe investigate parallel searching on m concurrent rays. We assume that a target t is located somewhere on one of the rays; we are given a group of m point robots each of which has to reach t. Furthermore, we assume that the robots have no way of communicating over distance. Given a strategy S we are interested in the competitive ratio defined as the ratio of the time needed by the robots to reach t using S and the time needed to reach t if the location of t is known in advance.If a lower bound on the distance to the target is known, then there is a simple strategy which achieves a competitive ratio of 9—independent of m. We show that 9 is a lower bound on the competitive ratio for two large classes of strategies if m⩾2.If the minimum distance to the target is not known in advance, we show a lower bound on the competitive ratio of 1+2(k+1)k+1/kk where k=⌈logm⌉ where log is used to denote the base-2 logarithm. We also give a strategy that obtains this ratio

Metadata of the chapter that will be visualized in SpringerLink Book Title Combinatorial Optimization and Applications Series Title Chapter Title Optimal Strategy for Walking in Streets with Minimum Number of Turns for a Simple Robot Optimal Strategy for

Abstract We consider the problem of walking a simple robot in an unknown street. The robot that cannot infer any geometric properties of the street traverses the environment to reach a target , starting from a point . The robot has a minimal sensing capability that can only report the discontinuities in the depth information (gaps), and location of the target point once it enters in its visibility region. Also, the robot can only move towards the gaps while moving along straight lines is cheap, but rotation is expensive for the robot. We maintain the location of some gaps in a tree data structure of constant size. The tree is dynamically updated during the movement. Using the data structure, we present an online strategy that generates a search path for the robot with optimal number of turns. Keywords (separated by '-') Computational geometry -Minimum link path -Simple robot -Street polygon -Unknown environment Abstract. We consider the problem of walking a simple robot in an unknown street. The robot that cannot infer any geometric properties of the street traverses the environment to reach a target t, starting from a point s. The robot has a minimal sensing capability that can only report the discontinuities in the depth information (gaps), and location of the target point once it enters in its visibility region. Also, the robot can only move towards the gaps while moving along straight lines is cheap, but rotation is expensive for the robot. We maintain the location of some gaps in a tree data structure of constant size. The tree is dynamically updated during the movement. Using the data structure, we present an online strategy that generates a search path for the robot with optimal number of turns

Competitive Searching over Terrains

We study a variant of the searching problem where the environment consists of a known terrain and the goal is to obtain visibility of an unknown target point on the surface of the terrain. The searcher starts on the surface of the terrain and is allowed to fly above the terrain. The goal is to devise a searching strategy that minimizes the competitive ratio, that is, the worst-case ratio between the distance traveled by the searching strategy and the minimum travel distance needed to detect the target. For $1.5$D terrains we show that any searching strategy has a competitive ratio of at least $\sqrt{82}$ and we present a nearly-optimal searching strategy that achieves a competitive ratio of $3\sqrt{19/2} \approx \sqrt{82} + 0.19$. This strategy extends directly to the case where the searcher has no knowledge of the terrain beforehand. For $2.5$D terrains we show that the optimal competitive ratio depends on the maximum slope $\lambda$ of the terrain, and is hence unbounded in general. Specifically, we provide a lower bound on the competitive ratio of $\Omega(\sqrt{\lambda})$. Finally, we complement the lower bound with a searching strategy based on the maximum slope of the known terrain, which achieves a competitive ratio of $O(\sqrt{\lambda})$

Geometric On-line Ray Searching Under Probability of Placement Scenarios

Online computation is a model for formulating decision making under uncertainty. In an online problem, the algorithm does not know the entire input from the beginning; the input is revealed in a sequence of steps. At each step, the algorithm should make its decisions based on the past and without any knowledge about the future. Many important real-life problems such as robot navigation are intrinsically online and thus the design and analysis of online algorithms is one of the main research areas in theoretical computer science. Competitive analysis is the standard measure for analysis of online algorithms. It has been applied to many online problems in diverse areas ranging from robot navigation, to network routing, to scheduling, to online graph coloring. In this thesis, we first survey three classic online problems, namely the cow-path problem, the Processor-Allocation problem and the Robots-Search-Rays problem and highlight connections between them. Second, the main result is for the One-Robot-Searches-Two-Rays problem for which we consider the weighted scenario, in which the robot is located on a ray with a preferential probability p. We term the One-Robot-Searches-Two-Rays-And-Weighted problem as 1-STRAW (and in general k-STRAW for k searchers). In the 1-STRAW problem, we propose a search strategy which is optimal among weighted geometric states. In addition, we prove a tight lower bound of the worst case competitive ratio and conjecture a lower bound of the average case competitive ratio for the 1-STRAW problem. Additionally, we compare our search strategy and its performance with the doubling strategy and the SmartCow algorithm

Polygon reconstruction from visibility information

vii, 78 leaves ; 28 cm.Reconstruction results attempt to rebuild polygons from visibility information. Reconstruction of a general polygon from its visibility graph is still open and only known to be in PSPACE; thus additional information, such as the ordering of the edges around nodes that corresponds to the order of the visibilities around vertices is frequently added. The first section of this thesis extracts, in o(E) time, the Hamiltonian cycle that corresponds to the boundary of the polygon from the polygon's ordered visibility graph. Also, it converts an unordered visibility graph and Hamiltonian cycle to the ordered visibility graph for that polygon in O(E) time. The secod, and major result is an algorithm to reconstruct an arthogonal polygon that is consistent with the Hamiltonian cylce and visibility stabs of the sides of an unknown polygon. The algorithm uses O(nlogn) time, assuming there are no collinear sides, and )(n2) time otherwise

On-line algorithms for robot navigation and server problems

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1994.Includes bibliographical references (p. 83-88).by Jon Michael Kleinberg.M.S