An Elementary Algorithm for Reporting Intersections of Red/Blue Curve Segments

Let and be two sets of -monotone and non-intersecting curve segments, and . We give a new sweep-line algorithm that reports the intersecting pairs of segments of . Our algorithm uses only three simple predicates that allow to decide if two segments intersect, if a point is left or right to another point, and if a point is above, below or on a segment. These three predicates seem to be the simplest predicates that lead to subquadratic algorithms. Our algorithm is almost optimal in this restricted model of computation. Its time complexity is ! " # $ % % '& and it requires '& space.