On the Number of Regular Vertices of the Union of Jordan Regions

Let C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. Let U denote the union of C. If the boundaries of two sets in C cross exactly twice, then their intersection points are called regular vertices of the arrangement A(C). Let R(C) denote the set of regular vertices on the boundary of the union of C. We present several bounds on jR(C)j, determined by the type of the sets of C. (i) If each set of C is convex, then jR(C)j = O(n1:5+" ) for any " > 0.1 (ii) If C consists of two collections C1 and C2 where C1 be a collection of n convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of each pair of them intersect at most twice), and C2 is a collection of convex polygons with a total of n sides, then jR(C)j = O(n4=3), and this bound is tight in the worst case. (iii) If no further assumptions are made on the sets of C, then w...

• Type: Conference
• Year: 1998
• Where: SWAT
• Authors: Boris Aronov, Alon Efrat, Dan Halperin, Micha Sharir
• Keywords: Algorithms, Binational Science Foundation, Jordan Regions, Regular Vertices, SWAT 1998