Stabbing Circles for Sets of Segments in the Plane

Abstract

Abstract. Stabbing a set S of n segments in the plane by a line is a well-known 8 problem. In this paper we consider the variation where the stabbing object is a cir-9 cle instead of a line. We show that the problem is tightly connected to two cluster

Searching Edges in the Overlap of Two Plane Graphs

“…For arbitrary point clusters, the diagram may have complexity Θ(n 2 ) and can be computed in O(n 2 ) time and space [1,11], where n is the total number of points in all clusters. When clusters are pairs of endpoints of n parallel line segments, the diagram has O(n) complexity and can be constructed in O(n log n) time and O(n) space [8]. In this paper, we broaden the class of inputs, for which the diagram can be constructed in subquadratic time.…”

“…A blue bundle at time i is a maximal contiguous sequence of blue segments that intersect the pseudoline i . 8 See Figure 2, right. A red bundle is defined analogously.…”

“…Since the diagram is not an instance of farthest abstract Voronoi diagrams [17], studying particular families of clusters (proving better bounds on the complexity of the diagram, and finding faster construction algorithms) is a non-trivial task. Recently it was shown that if the clusters in S are pairs of endpoints of disjoint segments parallel to each other, FCVD(S) has O(n) complexity and can be computed in optimal O(n log n) time [8].…”

“…Often it is guaranteed that each red edge contains at most one sought red-blue intersection, and an oracle is provided, that, given a red-blue intersection, is able to quickly determine to which side of that intersection the sought intersection lies along the same red edge (see Section 3.1 for more details on this setting). A particular case, when the red graph consists of a unique edge, appeared under the name of segment query in the randomized incremental construction algorithm for the Hausdorff Voronoi diagram [5], and under the name of find-change query in an algorithm to solve the stabbing circle problem for a set of line segments [8]. If the blue graph is a tree, it can be preprocessed in O(n log n) time using the centroid decomposition [2,5].…”

“…Centroid decomposition supports segment (or find-change) queries for arbitrary line segments, requiring only O(log n) queries to the oracle [5,7]. If the blue graph is not a tree, then in O(n log n) time it can be preprocessed for point location, and a nested point location along the red edge is performed, which requires O(log 2 n) queries to the oracle [6,8]. For two general plane straight-line graphs (where the red graph is not necessarily one edge) the problem is called batched red-blue intersection problem (see Problem 2).…”