On Computing Connected Components of Line Segments

It is shown that given a set of n line segments their connected components can be computed in time O n log n A bound of o n for this problem would imply a similar bound for detecting for a given set of n points and n lines whether some point lies on some of the lines This problem known as Hopcroft s problem is believed to have a lower bound of n For the special case when the endpoints of each segment fall inside the same face of the arrangement induced by the set of segments we give a faster algorithm that runs in O n log n time