Maintaining the Union of Unit Discs under Insertions with Near-Optimal Overhead

03/20/2019
by   Pankaj K. Agarwal, et al.
0

We present efficient data structures for problems on unit discs and arcs of their boundary in the plane. (i) We give an output-sensitive algorithm for the dynamic maintenance of the union of n unit discs under insertions in O(k ^2 n) update time and O(n) space, where k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. (ii) As part of the solution of (i) we devise a fully dynamic data structure for the maintenance of lower envelopes of pseudo-lines, which we believe is of independent interest. The structure has O(^2 n) update time and O( n) vertical ray shooting query time. To achieve this performance, we devise a new algorithm for finding the intersection between two lower envelopes of pseudo-lines in O( n) time, using tentative binary search; the lower envelopes are special in that at x=-∞ any pseudo-line contributing to the first envelope lies below every pseudo-line contributing to the second envelope. (iii) We also present a dynamic range searching structure for a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), where the ranges are unit discs, with O(n n) preprocessing time, O(n^1/2+ε + ℓ) query time and O(^2 n) amortized update time, where ℓ is the size of the output and for any ε>0. The structure requires O(n) storage space.

READ FULL TEXT
research
02/25/2019

Dynamic Maintenance of the Lower Envelope of Pseudo-Lines

We present a fully dynamic data structure for the maintenance of lower e...
research
04/19/2022

Unit-Disk Range Searching and Applications

Given a set P of n points in the plane, we consider the problem of compu...
research
03/01/2023

Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain

We devise a data structure that can answer shortest path queries for two...
research
02/15/2021

On Ray Shooting for Triangles in 3-Space and Related Problems

We consider several problems that involve lines in three dimensions, and...
research
02/25/2020

Algorithms for Subpath Convex Hull Queries and Ray-Shooting Among Segments

In this paper, we first consider the subpath convex hull query problem: ...
research
11/05/2021

Hopcroft's Problem, Log-Star Shaving, 2D Fractional Cascading, and Decision Trees

We revisit Hopcroft's problem and related fundamental problems about geo...
research
12/20/2014

SVEN: Informative Visual Representation of Complex Dynamic Structure

Graphs change over time, and typically variations on the small multiples...

Please sign up or login with your details

Forgot password? Click here to reset