An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling

Abstract

Map labeling is a classical problem in cartography and geographic information systems that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent (i.e., overlap-free) labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each timestep, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sublinear update time. Motivated by this, we study the maximal independent set ( MIS ) and maximum independent set ( Max-IS ) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types: (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs. We present the first deterministic algorithm for maintaining an MIS (and thus a 4-approximate Max-IS ) of a dynamic set of uniform rectangles with polylogarithmic update time. This breaks the natural barrier of \( \Omega (\Delta) \) update time (where \( \Delta \) 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 \( 2(1+\frac{1}{k}) \) and update time \( O(k^2\log n) \) , for \( k\in \mathbb {N} \) . We conclude with an algorithm that maintains a 2-approximate Max-IS for dynamic sets of unit-height and arbitrary-width rectangles with \( O(\log ^2 n + \omega \log n) \) update time, where \( \omega \) is the maximum size of an independent set of rectangles stabbed by any horizontal line. We implement our algorithms and report the results of an experimental comparison exploring the trade-off between solution quality and update time for synthetic and real-world map labeling instances. We made several major observations in our empirical study. First, the original approximations are well above their respective worst-case ratios. Second, in comparison with the static approaches, the dynamic approaches show a significant speedup in practice. Third, the approximation algorithms show their predicted relative behavior. The better the solution quality, the worse the update times. Fourth, a simple greedy augmentation to the approximate solutions of the algorithms boost the solution sizes significantly in practice.