Proximity and Motion Planning on l_1-Embeddable Tilings

A motion planning problem on plane crystallographic graphs turned out to be important in nanotechnology due to recent innovation in techniques of handling real atoms on a physical lattice. The motion planning problem on graphs are well studied and it was shown that fast algorithms for proximity problems on graphs lead to fast motion planning algorithms. On the other hand, tilings are well studied as a model for plane crystallographic graphs and l₁-embeddable tilings are enumerated by Deza, Grishukhin and Shtogrin. In this paper, we focus on the geometry of tilings and propose two proximity algorithms on l₁-embeddable tilings for the application to nanotechnology. We show that Voronoi diagrams on tilings embeddable in the 3-dimensional l₁ lattice can be implicitly describes by Voronoi diagrams on the plane under appropriate convex piecewise linear functions with extra elaborations. We also propose a fast algorithm for another proximity problem, which we call nearest pair problem, on l₁-embeddable tilings. Using these algorithms, we propose algorithms for the motion planning problem on l₁-embeddable tilings.