On Chamfer Distances on the Square and Body-Centered Cubic Grids: An Operational Research Approach

Figure 3

The feasible bases for BCC grid with their relation. Bases that can be optimal are marked with double frame. Black arrows show how one can shift from one basis to other by replacing a vector. Condition on the arrow shows when the change provides a better solution. Opposite condition implies that the change going opposite direction gives a better solution. For instance, from the basis (1, 2, 3), in case 3b > 2a, one gets a better solution by changing vector 3 to 7 obtaining basis (1, 2, 7). However, in case 3b < 2a, reverse direction changes, i.e., from basis (1, 2, 7) to (1, 2, 3) provides a better solution. Green arrows mean trivial conditions, e.g., a > 0; thus, these arrows can be used only in one direction and they always lead to a possible optimal basis. The orange lines show indifferent changes, i.e., the solutions of the related two basis are equivalent.