Exploring Simple Grid Polygons

• Published in: Computing and Combinatorics pp. 524 - 533
• Main Authors: Icking, Christian, Kamphans, Tom, Klein, Rolf, Langetepe, Elmar
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2005; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Applied sciences; competitive analysis; Computer science; control theory; systems; covering; Exact sciences and technology; exploration; Flows in networks. Combinatorial problems; grid polygons; Information retrieval. Graph; lower bounds; online algorithms; Operational research and scientific management; Operational research. Management science; Robot navigation; Theoretical computing; Lower bound; Perimeter; Upper bound; Moving robot; Combinatorial problem; Measurement sensor; Grid; Polygon; Robotics
• ISBN: 9783540280613; 3540280618
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11533719_53

We investigate the online exploration problem of a short-sighted mobile robot moving in an unknown cellular room without obstacles. The robot has a very limited sensor; it can determine only which of the four cells adjacent to its current position are free and which are blocked, i.e., unaccessible for the robot. Therefore, the robot must enter a cell in order to explore it. The robot has to visit each cell and to return to the start. Our interest is in a short exploration tour, i.e., in keeping the number of multiple cell visits small. For abitrary environments without holes we provide a strategy producing tours of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $S \leq C + \frac{1}{2} E -- 3$ \end{document}, where C denotes the number of cells – the area – , and E denotes the number of boundary edges – the perimeter – of the given environment. Further, we show that our strategy is competitive with a factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\frac43$ \end{document}, and give a lower bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\frac76$ \end{document} for our problem. This leaves a gap of only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\frac16$ \end{document} between the lower and the upper bound.