Fast algorithms for finding nearest common ancestors
SIAM Journal on Computing
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
IEEE Transactions on Visualization and Computer Graphics
Boundary labeling: Models and efficient algorithms for rectangular maps
Computational Geometry: Theory and Applications
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Efficient Algorithms
Optimizing active ranges for consistent dynamic map labeling
Computational Geometry: Theory and Applications
Boundary Labeling with Octilinear Leaders
Algorithmica - Special Issue: Scandinavian Workshop on Algorithm Theory; Guest Editor: Joachim Gudmundsson
Multi-stack boundary labeling problems
FSTTCS'06 Proceedings of the 26th international conference on Foundations of Software Technology and Theoretical Computer Science
Consistent labeling of rotating maps
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Boundary-labeling algorithms for panorama images
Proceedings of the 19th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
Two-Sided boundary labeling with adjacent sides
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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In boundary labeling, features on a map are connected to a stack of labels on the map boundary, using simple polylines called leaders. We consider the setting that the labels are axis-aligned non-overlapping rectangles placed on one side of the map, and leaders are rectilinear polylines with at most one bend. The goal is to find a labeling that minimizes the total length of the leaders. We introduce three extensions of the one-sided boundary labeling problem: (i) a dynamic setting for continuous scale changes, (ii) a clustered setting for multiple label stacks, and (iii) a combined dynamic clustered setting. We obtain the following results: • Optimal label placement as a function of map scale can be computed in O(n log n + σ log n) time, where σ is the number of "combinatorially different" labelings that occur during zooming. • In a map with fixed scale, an optimal clustered label placement can be found in O(n log n) time. • In O(n log2 n + γ log n) time one can build a structure of size O(γ) representing the optimal clustered label placement for all possible map scales; here γ is, again, the number of combinatorially different labelings. We further extend our basic model to the case where labeled features enter or leave the viewport due to map panning and zooming. Our algorithms are based on combining standard computational-geometry tools and have been implemented in a Java applet (available online), which indicates that the algorithms are fast enough for interactive use without delays.