Randomized External-Memory Algorithms for Some Geometric Problems

We show that the well-known random incremental construction of Clarkson and Shor18 can be adapted to provide efficient external-memory algorithms for some geometric problems. In particular, as the main result, we obtain an optimal randomized algorithm for the problem of computing the trapezoidal decomposition determined by a set of N line segments in the plane with K pairwise intersections, that requires [Formula: see text] expected disk accesses, where M is the size of the available internal memory and B is the size of the block transfer. The approach is sufficiently general to derive algorithms for other geometric problems: 3-d half-space intersections, 2-d and 3-d convex hulls, 2-d abstract Voronoi diagrams and batched planar point location; these algorithms require an optimal expected number of disk accesses and are simpler than the ones previously known. The results extend to an external-memory model with multiple disks.