Labeling Points with Weights |
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Authors: | Sheung-Hung Poon Chan-Su Shin Tycho Strijk Takeaki Uno and Alexander Wolff |
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Affiliation: | (1) Department of Computer Science, HKUST, Hong Kong;(2) School of Electronics and Information Engineering, Hankuk University of Foreign Studies, Korea;(3) ASM Lithography, Veldhoven, The Netherlands;(4) National Institute of Informatics, 2-1-2 Chiyoda-ku, Tokyo 101-8430, Japan;(5) Faculty of Computer Science, Karlsruhe University, P.O. Box 6980, 76128 Karlsruhe, Germany |
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Abstract: | Annotating maps, graphs, and diagrams with pieces of text is an
important step in information visualization that is usually referred
to as label placement. We define nine label-placement models for
labeling points with axis-parallel rectangles given a weight for
each point. There are two groups: fixed-position models and slider
models. We aim to maximize the weight sum of those points that
receive a label.
We first compare our models by giving bounds for the ratios between
the weights of maximum-weight labelings in different models. Then
we present algorithms for labeling n points with unit-height
rectangles. We show how an O(n\log n)-time factor-2 approximation
algorithm and a PTAS for fixed-position models can be extended to
handle the weighted case. Our main contribution is the first
algorithm for weighted sliding labels. Its approximation factor is
(2+\varepsilon), it runs in O(n
2/\varepsilon) time and uses
O(n/\varepsilon) space.
We show that other than for fixed-position models even the
projection to one dimension remains NP-hard.
For slider models we also investigate some special cases, namely
(a) the number of different point weights is bounded, (b) all labels
are unit squares, and (c) the ratio between maximum and minimum
label height is bounded. |
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Keywords: | Computational geometry GIS Label placement Sliding labels Combinatorial optimization Job scheduling Throughput maximization Maximum weight independent set |
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