Meshes Preserving Minimum Feature Size

Abstract

Abstract. The minimum feature size of a planar straight-line graph is the minimum distance between a vertex and a nonincident edge. When such a graph is partitioned into a mesh, the degradation is the ratio of original to final minimum feature size. For an n-vertex input, we give a triangulation (meshing) algorithm that limits degradation to only a constant factor, as long as Steiner points are allowed on the sides of triangles. If such Steiner points are not allowed, our algorithm realizes O(lg n) degradation. This addresses a 14-year-old open problem by Bern, Dobkin, and Eppstein.

Dissection with the Fewest Pieces is Hard, Even to Approximate

“…-Can we prove stronger hardness of approximation, or find an approximation algorithm, for Min Piece Dissection? The current best known algorithm for finding a dissection is a worst-case bound of a pseudopolynomial number of pieces [1]. -Is k-Piece Dissection solvable in polynomial time for constant k?…”