Competitive on-line coverage of grid environments by a mobile robot

We describe in this paper two on-line algorithms for covering planar areas by a square-shaped tool attached to a mobile robot. Let D be the tool size. The algorithms, called Spanning Tree Covering (STC) algorithms, incrementally subdivide the planar area into a grid of D-size cells, while following a spanning tree of a grid graph whose nodes are 2D-size cells. The two STC algorithms cover general planar grids. The first, Spiral-STC, employs uniform weights on the grid-graph edges and generates spiral-like covering patterns. The second, Scan-STC, assigns lower weights to edges aligned with a particular direction and generates scan-like covering patterns along this direction. Both algorithms cover any planar grid using a path whose length is at most (n+m)D, where n is the total number of D-size cells and mn is the number of boundary cells, defined as cells that share at least one point with the grid boundary. We also demonstrate that any on-line coverage algorithm generates a covering path whose length is at least (2−)l_opt in worst case, where l_opt is the length of the optimal off-line covering path. Since (n+m)D2l_opt, the bound is tight and the STC algorithms are worst-case optimal. Moreover, in practical environments mn, and the STC algorithms generate close-to-optimal covering paths in such environments.