APPROXIMATION OF A TESSELLATION OF THE PLANE BY A VORONOI DIAGRAM

Abstract

In this paper the problem of obtaining the Voronei diagram which approximates a given tessellation of the plane is forrnulated as the optimization problem, where the objective function is the discrepancy of the Voronoi diagrani and the given tessellation. The ebjective funetion is generally non-convex and nondifferentiable, so we adopt the primitive descent algorithrn and its variants as a solution algorithnL, Of course, we have to be content with the locally minimum solutions. However the resuits of the computational examples suggest that satisfactory good solutions can be obtained by eur algorithn, This problem includes the problem to restore the generators from a given Voronoi diagram (i.e., the inverse problem of constructing a Voronoi diagrarn from the given points) when the given diagram 2s itself a Voronoi diagram, We can get the ELpproximate position of the generators from a given VoTonoi diagrain in practical time; it takes dbout 10s to re/store the generators from a Voronei diagram generated from thirty-two points on a computer of speecl about 17 MPS. Two other practical examples are presented where our algorithrn is efficient, one being a pToblem in ecology and the other being one in urban planning. We can get the Voronoi diagrams which approximate the giyen tessellations Cwhich have 32 regions and are defined by 172 points in the former example, 1l regions and 192 points in the latter example) within 10s in these two examples on the same computer.