Local polyhedra and geometric graphs

To appear in Computational Geometry: Theory and Applications,
(Special issue of invited papers from the 19th Annual ACM Symposium on Computational Geometry)

Proceedings of the 19th Annual ACM Symposium on Computational Geometry, 171-180, 2003.

We introduce a new realistic input model for geometric graphs and nonconvex polyhedra. A geometric graph is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor and (2) the lengths of the longest and shortest edges differ by at most a polynomial factor. A polyhedron is local if all its faces are simplices and its edges form a local geometric graph. We show that any boolean combination of any two local polyhedra in Rd, each with n vertices, can be computed in O(n log n) time, using a standard hierarchy of axis-aligned bounding boxes. Using results of de Berg, we also show that any local polyhedron in Rd has a binary space partition tree of size O(n logd-2 n) and depth O(log n); these bounds are tight in the worst case when d≤3. Finally, we describe efficient algorithms for computing Minkowski sums of local polyhedra in two and three dimensions.

Publications - Jeff Erickson (jeffe@cs.uiuc.edu) 14 Sep 2004