Partitioning Meshes with Lines and Planes

We investigate several geometric methods for dividing an irregular mesh into pieces of roughly equal size with few interconnecting edges. All these methods are based on cutting a mesh with a line (in two dimensions) or a hyperplane (in any dimension). Line cuts have often been used in practice, but their quality varies widely. Until now, no theory has existed to predict the eeectiveness of any line-cut algorithm. We make two main contributions: First, we give rigorous (and tight) bounds on the quality of line cuts for meshes of bounded aspect ratio in terms of a parameter we call grading that measures the nonuniformity of the mesh. Our bound on line cut quality diiers from the known bound on circle cut quality by a factor that grows only as the 1=d power of the logarithm of the mesh grading. Second, we give an upper bound on the quality of an exact 50:50 line cut. This is the rst proof of a cut size guarantee for any exact bisection induced by a single geometric cut. We use these bounds to design simple and eecient algorithms that guarantee to nd good line cuts.