A packing problem with applications to lettering of maps
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Computers and Intractability; A Guide to the Theory of NP-Completeness
Computers and Intractability; A Guide to the Theory of NP-Completeness
Boundary labeling: Models and efficient algorithms for rectangular maps
Computational Geometry: Theory and Applications
ISAAC '01 Proceedings of the 12th International Symposium on Algorithms and Computation
Efficient Algorithms
Boundary Labeling with Octilinear Leaders
Algorithmica - Special Issue: Scandinavian Workshop on Algorithm Theory; Guest Editor: Joachim Gudmundsson
Area-Feature Boundary Labeling1
The Computer Journal
Dynamic one-sided boundary labeling
Proceedings of the 18th SIGSPATIAL International Conference on Advances in Geographic Information Systems
Multi-stack boundary labeling problems
FSTTCS'06 Proceedings of the 26th international conference on Foundations of Software Technology and Theoretical Computer Science
Two-Sided boundary labeling with adjacent sides
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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Boundary labeling deals with placing annotations for objects in an image on the boundary of that image. This problem occurs frequently in situations where placing labels directly in the image is impossible or produces too much visual clutter. Previous algorithmic results for boundary labeling consider a single layer of labels along some or all sides of a rectangular image. If, however, the number of labels is large or labels are too long, multiple layers of labels are needed. In this paper we study boundary labeling for panorama images, where n points in a rectangle R are to be annotated by disjoint unit-height rectangular labels placed above R in k different rows (or layers). Each point is connected to its label by a vertical leader that does not intersect any other label. We present polynomial-time algorithms based on dynamic programming that either minimize the number of rows to place all n labels, or maximize the number (or total weight) of labels that can be placed in k rows for a given integer k. For weighted labels, the problem is shown to be (weakly) NP-hard, and we give a pseudo-polynomial algorithm to maximize the weight of the selected labels. We have implemented our algorithms; the experimental results show that solutions for realistically-sized instances are computed instantaneously.