TRAVERSAL OF A QUASI-PLANAR SUBDIVISION WITHOUT USING MARK BITS

Abstract

The problem of traversal of planar subdivisions or other graph-like structures without using mark bits is central to many real-world applications [7, 8, 11, 12, 13, 17, 18]. The first such algorithms developed were able to traverse triangulated subdivisions [10]. Later these algorithms were extended to traverse vertices of an arrangement or a convex polytope [3]. The research progress culminated to an algorithm that can traverse any planar subdivision [6, 9]. In this paper, we extend the notion of planar subdivision to quasi-planar subdivision in which we allow many edges to cross each other. We generalize the algorithm from [9] to traverse any quasi-planar subdivision that satisfies a simple geometric requirement. If we use techniques from [6] the worst case running time of our algorithm is O(|E| log |E|); matching the running time of the traversal algorithm for planar subdivisions [6].