Computing optimal diameter-bounded polygon partitions

The minimum alpha-small partition problem is the problem of partitioning a given simple polygon into subpolygons, each with diameter at most alpha, for a given alpha > 0. This paper considers the version of this problem that disallows Steiner points. This problem is motivated by applications in mesh generation and collision detection. The main result in the paper is a polynomial-time algorithm that solves this problem, and either returns an optimal partition or reports the nonexistence of such a partition. This result contrasts with the recent NP-completeness result for the minimum a-small partition problem for polygons with holes (C. Worman, 15th Canadian Conference on Computational Geometry, 2003). Even though the running time of our algorithm is a polynomial in the input size, it is prohibitive for most real applications and we seek faster algorithms that approximate an optimal solution. We first present a faster 2-approximation algorithm for the problem for simple polygons and then a near linear-time algorithm for convex polygons that produces, for any epsilon > 0. an (alpha + epsilon)-small partition with no more polygons than in an optimal alpha-small partition. We also present an exact polynomial-time algorithm for the minimum alpha-small partition problem with the additional constraint that each piece in the partition be convex.