An elementary algorithm for reporting intersections of red/blue curve segments

Abstract

Let E-r and E-b be two sets of x -monotone and non-intersecting curve segments, E = E-r boolean OR E-b and \E\ = n. We give a new sweep-line algorithm that reports the k intersecting pairs of segments of E. 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 O(n log n + k log log n) and it requires O(n) space.