Computing a Face in an Arrangement of Line Segments and Related Problems

This paper presents a randomized incremental algorithm for computing a single face in an arrangement of n line segments in the plane that is fairly simple to implement. The expected running time of the algorithm is O (not (n) log n). The analysis of the algorithm uses a novel approach that generalizes and extends the Clarkson-Shor analysis technique [in Discrete Comput. Geom., 4 (1989), pp. 387-421 ]. A few extensions of the technique, obtaining efficient randomized incremental algorithms for constructing the entire arrangement of a collection of line segments and for computing a single face in an arrangement of Jordan arcs are also presented. Key words, computational geometry, arrangements, randomized incremental algorithms, probabilistic backwards analysis, Davenport-Schinzel sequences AMS subject classifications. 68P05, 68Q20, 68R99, 51M99