Sampling Hyperplanes and Revealing Disks

We present a data structure which allows detecting when disks of a set B are no longer intersected by disks of set of disk A when deleting its disks. Preprocessing A, B and deleting n disks of A with detecting all newly revealed disks of B requires O(|B| log4 |A| + |A| log(|A|)λ6(log |A|) + n log(|A|)λ6(log |A|)) expected time, where λ6(·) is the Davenport-Schinzel bound of order 6. To construct this data structure, we extend known dynamic lower envelope data structures for hyperplanes and bivariate functions of constant description complexity with a linear size lower envelope in R3, such that they allow sampling of a random element not above a given point in O(log3 n) expected time. 1 From Graph Connectivity to Disk Sampling Graph connectivity plays a fundamental role in algorithms and data structures. The dynamic variant where edges can be inserted or deleted is reasonably well understood [6–8,11,14], with data structures supporting updates and queries determining the connectivity of two vertices in polylogarithmic time. Updating vertices seems significantly harder, as a single update can have a large impact on the overall structure. Chan et al. [4, Theorem 1] presented a data structure allowing vertex updates in Õ(m2/3) amortized time and queries in Õ(m1/3) time, where m is the number of possible edges of the graph that need to be known in advance. The vertices of geometric intersection graphs correspond to geometric objects and its edges to intersections. As they restrict possible graphs, faster solutions may be within reach. However, work is required to find edges affected by an update. Chan et al. [4, Theorem 5] gave a general method for various objects with sub-linear update and query times. In this work we present a data structure, which allows detecting when a disk is no longer intersected by any disk of a set after deletions. We call such a disk revealed. This data structure can be used as a component in maintaining connectivity information in deletion-only disk intersection graphs [9]. To construct it, we first describe data structures for randomly sampling a hyperplane or a continuous function of constant description complexity not above a given point in R3, which might be of interest of its own. ∗ Supported in part by grant 1367/2016 from the German-Israeli Science Foundation (GIF). † Supported in part by grant 1367/2016 from the German-Israeli Science Foundation (GIF) and by the German Research Foundation within the collaborative DACH project Arrangements and Drawings as DFG Project MU 3501/3-1. ‡ Supported in part by ERC StG 757609. 37th European Workshop on Computational Geometry, St. Petersburg, Russia, April 7–9, 2021. This is an extended abstract of a presentation given at EuroCG’21. It has been made public for the benefit of the community and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expected to appear eventually in more final form at a conference with formal proceedings and/or in a journal. 63:2 Sampling Hyperplanes and Revealing Disks Figure 1 When removing a red disk, we want to obtain all blue disks intersecting this red disk but no other red disk. After removing the dashed red disk, the dashed blue disks need to be obtained.