Exploring Simple Grid Polygons

We investigate the online exploration problem of a shortsighted 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 S ≤ C + 1 2 E − 3, 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 4 3 , and give a lower bound of 7 6 for our problem. This leaves a gap of only 1 6 between the lower and the upper bound.