Efficient Maintenance of Connectivity in Dynamic Disk Graphs

Core Concepts

The authors present efficient data structures for maintaining the connectivity structure of dynamic disk graphs under insertions and deletions of sites.

Abstract

The authors consider the problem of designing data structures that maintain the connectivity structure of disk graphs while allowing the insertion and deletion of sites. They consider three variants of disk graphs: unit disk graphs, disk graphs with bounded radius ratio, and disk graphs with unbounded radius ratio.
For unit disk graphs, the authors describe a data structure that has O(log^2 n) amortized update time and O(log n / log log n) amortized query time.
For disk graphs with bounded radius ratio Ψ, the authors consider the decremental, incremental, and fully dynamic cases separately. In the fully dynamic case, they achieve amortized O(Ψλ^6(log n) log^9 n) update time and O(log n) query time, improving the update time of the currently best known data structure by a factor of Ψ at the cost of an additional O(log log n) factor in the query time.
In the incremental case for bounded radius ratio, the authors manage to achieve a logarithmic dependency on Ψ with a data structure with O(α(n)) query and O(log Ψλ^6(log n) log^9 n) update time.
For the decremental setting with bounded radius ratio, the authors first develop a new dynamic data structure that allows them to efficiently report all disks in one set that no longer intersect any disk in another set when a disk is deleted from the first set. Using this data structure, they obtain decremental data structures with an amortized query time of O(log n / log log n) supporting m deletions in O((n log^5 n + m log^9 n)λ^6(log n) + n log Ψ log^4 n) overall time.
For the general case of unbounded radius ratio, the authors obtain decremental data structures with an amortized query time of O(log n / log log n) and overall time of O((n log^6 n + m log^10 n)λ^6(log n)) for m deletions.

Stats

The authors use the following key metrics and figures:

The size of the site set S is denoted by n.
The ratio between the largest and smallest radius is denoted by Ψ.
The maximum length of a Davenport-Schinzel sequence of order s on n symbols is denoted by λ_s(n).

Quotes

"Let S be a set of sites, each associated with a point in R^2 and a radius r_s and let D(S) be the intersection graph of the disks defined by the sites and radii. We consider the problem of designing data structures that maintain the connectivity structure of D(S) while allowing the insertion and deletion of sites."
"For unit disk graphs we describe a data structure that has O(log^2 n) amortized update time and O(log n / log log n) amortized query time."
"For disk graphs where the ratio Ψ between the largest and smallest radius is bounded, we consider the decremental and the incremental case separately, in addition to the fully dynamic case."

Key Insights Distilled From

Dynamic Connectivity in Disk Graphs

by Alexander Ba... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2106.14935.pdf

Deeper Inquiries

How can the data structures be extended to handle other types of geometric objects beyond disks, such as ellipses or polygons

To extend the data structures to handle other types of geometric objects beyond disks, such as ellipses or polygons, we can modify the data structure to accommodate the specific geometric properties of these shapes.
For ellipses, we can represent them using their center, major and minor axes, and orientation. The connectivity between ellipses can be determined based on their proximity and intersection. We would need to adapt the data structure to handle the unique characteristics of ellipses, such as their eccentricity and orientation.
Similarly, for polygons, we can represent them using their vertices or edges. The connectivity between polygons can be established based on their overlapping regions or shared vertices. The data structure would need to incorporate algorithms for polygon intersection and containment to determine connectivity.
In both cases, the key challenge would be to efficiently update the connectivity structure when objects are inserted or deleted. This may involve maintaining additional information about the geometric objects, such as bounding boxes or convex hulls, to optimize the update process while ensuring accurate connectivity information.

What are the implications of the authors' results on the design of efficient algorithms for other problems on dynamic disk graphs, such as finding the maximum independent set or the minimum dominating set

The results of the authors have significant implications for the design of efficient algorithms for other problems on dynamic disk graphs.
For example, in the case of finding the maximum independent set on dynamic disk graphs, the data structures developed by the authors can be leveraged to efficiently update the independent set as disks are inserted or deleted. By maintaining the connectivity structure of the graph, one can identify independent sets based on the connected components and adjust them dynamically.
Similarly, for the minimum dominating set problem, the connectivity information provided by the data structures can be used to identify dominating sets that cover all vertices in the graph. By efficiently updating the dominating set based on changes in the graph, the algorithms can adapt to dynamic scenarios and maintain the minimum dominating set optimally.
Overall, the efficient data structures for dynamic disk graphs enable the development of algorithms for various graph problems that require real-time updates and adjustments based on changes in the graph structure.

Can the techniques used in the authors' data structures be applied to maintain other properties of dynamic disk graphs, such as the diameter or the number of connected components, in an efficient manner

The techniques used in the authors' data structures can be applied to maintain other properties of dynamic disk graphs in an efficient manner, such as the diameter or the number of connected components.
For maintaining the diameter of a dynamic disk graph, one can utilize the connectivity information to determine the shortest path between the farthest pair of vertices. By updating the connectivity structure efficiently, one can track changes in the diameter of the graph as disks are inserted or deleted.
Similarly, for tracking the number of connected components in a dynamic disk graph, the data structures can be adapted to identify and update the components based on changes in the connectivity. By efficiently maintaining the information about connected components, one can determine the number of disjoint subsets in the graph and adjust them dynamically as the graph evolves.
Overall, the techniques employed in the data structures can be extended to handle various properties of dynamic disk graphs, providing efficient solutions for maintaining and updating key graph characteristics.