Belangrijkste concepten
The authors present efficient data structures for maintaining the connectivity structure of dynamic disk graphs under insertions and deletions of sites.
Samenvatting
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.
Statistieken
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).
Citaten
"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."