Independent Sets of Dynamic Rectangles: Algorithms and Experiments
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We study the maximal independent set (MIS) and maximum independent set (MAX-IS) problems on dynamic sets of O(n) axis-parallel rectangles, which can be modeled as dynamic rectangle intersection graphs. We consider the fully dynamic vertex update (insertion/deletion) model for two types of rectangles: (i) uniform height and width and (ii) uniform height and arbitrary width. These types of dynamic vertex update problems arise, e.g., in interactive map labeling. We present the first deterministic algorithm for maintaining a MIS (and thus a 4-approximate MAX-IS) of a dynamic set of uniform rectangles with amortized sub-logarithmic update time. This breaks the natural barrier of O(Δ) update time (where Δ is the maximum degree in the graph) for vertex updates presented by Assadi et al. (STOC 2018). We continue by investigating MAX-IS and provide a series of deterministic dynamic approximation schemes. For uniform rectangles, we first give an algorithm that maintains a 4-approximate MAX-IS with O(1) update time. In a subsequent algorithm, we establish the trade-off between approximation quality 2(1+1/k) and update time O(k^2log n) for k∈ℕ. We conclude with an algorithm that maintains a 2-approximate MAX-IS for dynamic sets of uniform height and arbitrary width rectangles with O(ωlog n) update time, where ω is the largest number of maximal cliques stabbed by any axis-parallel line. We have implemented our algorithms and report the results of an experimental comparison exploring the trade-off between solution size and update time for synthetic and real-world map labeling data sets.
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