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26
Label Placement by Maximum Independent Set in Rectangles
 Computational Geometry: Theory and Applications
, 1997
"... Motivated by the problem of labeling maps, we investigate the problem of computing a large nonintersecting subset in a set of n rectangles in the plane. Our results are as follows. In O(n log n) time, we can find an O(log n)factor approximation of the maximum subset in a set of n arbitrary axispa ..."
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Cited by 72 (5 self)
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Motivated by the problem of labeling maps, we investigate the problem of computing a large nonintersecting subset in a set of n rectangles in the plane. Our results are as follows. In O(n log n) time, we can find an O(log n)factor approximation of the maximum subset in a set of n arbitrary axisparallel rectangles in the plane. If all rectangles have unit height, we can find a 2approximation in O(n log n) time. Extending this result, we obtain a (1 + 1 k )approximation in time O(n log n + n 2k\Gamma1 ) time, for any integer k 1. 1 Introduction Automated label placement is an important problem in geographic information systems (GIS), and has received considerable attention in recent years (for instance, see [6, 9]). The label placement problem includes positioning labels for area, line, and point features. The primary focus within the computational geometry community has been on labeling point features [5, 7, 17, 16]. A basic requirement in the label placement problem is that ...
PolynomialTime Approximation Schemes for Geometric Graphs
, 2001
"... A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomialtime approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weigh ..."
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Cited by 70 (4 self)
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A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomialtime approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weight) and for the minimum weight vertex cover problem in disk graphs. These are the first known PTASs for NPhard optimization problems on disk graphs. They are based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible. The PTASs for disk graphs represent a common generalization of previous results for planar graphs and unit disk graphs. They can be extended to intersections graphs of other "disklike" geometric objects (such as squares or regular polygons), also in higher dimensions.
Point Labeling with Sliding Labels
 Computational Geometry: Theory and Applications
, 1999
"... This paper discusses algorithms for labeling sets of points in the plane, where labels are not restricted to some nite number of positions. We show that continuously sliding labels allows more points to be labeled both in theory and in practice. We dene six dierent models of labeling, and analyze ho ..."
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Cited by 38 (11 self)
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This paper discusses algorithms for labeling sets of points in the plane, where labels are not restricted to some nite number of positions. We show that continuously sliding labels allows more points to be labeled both in theory and in practice. We dene six dierent models of labeling, and analyze how much better  more points get a label  one model can be than another. We show that maximizing the number of labeled points is NPhard in the most general of the new models. Nevertheless, we give a polynomialtime approximation scheme and a simple and ecient factor 1 2 approximation algorithm for each of the new models. Finally, we give experimental results based on the factor 1 2 approximation algorithm to compare the models in practice. We also compare this algorithm experimentally to other algorithms suggested in the literature. 1 Introduction Annotating sets of points is a common task to be performed in Geographic Information Systems. Cities on smallscale maps are shown as...
Property Testing with Geometric Queries
 In Proceedings of the 9th Annual European Symposium on Algorithms (ESA
, 2001
"... This paper investigates geometric problems in the context of property testing algorithms. Property testing is an emerging area in computer science in which one is aiming at verifying whether a given object has a predetermined property or is "far" from any object having the property. Although there ..."
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Cited by 17 (7 self)
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This paper investigates geometric problems in the context of property testing algorithms. Property testing is an emerging area in computer science in which one is aiming at verifying whether a given object has a predetermined property or is "far" from any object having the property. Although there has been some research previously done in testing geometric properties, prior works have been mostly dealing with the study of combinatorial notion of the distance defining whether an object is "far" or it is "close"; very little research has been done for geometric notion of distance measures, that is, distance measures that are based on the geometry underlying input objects. The main objective of this work is to develop sound models to study geometric problems in the context of property testing. Comparing to the previous work in property testing, there are two novel aspects developed in this paper: geometric measures of being close to an object having the predetermined property, and the use of geometric data structures as basic primitives to design the testers. We believe that the second aspect is of special importance in the context of property testing and that the use of specialized data structures as basic primitives in the testers can be applied to other important problems in this area. We shall discuss a number of models that in our opinion fit best geometric problems and apply them to study geometric properties for three very fundamental and representative problems in the area: testing convex position, testing map labeling, and testing clusterability. Research supported in part by an SBR grant and DFG grant Me872/71. 1
Practical Extensions of Point Labeling in the Slider Model
 In Proc. 7th ACM Symposium on Advances in Geographic Information Systems
, 1999
"... This paper extends on research by the authors together with Alexander Wol# on point label placement using a model where labels can be placed at any position that touches the point (the slider model). Such models have been shown to perform better than methods that allow only a fixed number of pos ..."
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Cited by 17 (2 self)
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This paper extends on research by the authors together with Alexander Wol# on point label placement using a model where labels can be placed at any position that touches the point (the slider model). Such models have been shown to perform better than methods that allow only a fixed number of positions per label. The novelties in this paper include respecting other map features that must be avoided by the labels, and incorporating labels with di#erent height. The result is an e#cient and simple O((n + m) log(n + m)) time algorithm with a performance guarantee for label placement in the slider model. Here n is the number of points to be labeled and m is the combinatorial complexity of the map features that must be avoided. Due to its e#ciency, the algorithm can be used in interactive and online mapping. 1 Introduction The automated map labeling problem is a wellknown problem for many years in cartographic and GIS research. Manual label placement is a timeconsuming task an...
A Combinatorial Framework for Map Labeling
, 1998
"... The general map labeling problem consists in labeling a set of sites (points, lines, regions) given a set of candidates (rectangles, circles, ellipses, irregularly shaped labels) for each site. A map can be a classical cartographical map, a diagram, a graph or any other figure that needs to be labe ..."
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Cited by 15 (3 self)
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The general map labeling problem consists in labeling a set of sites (points, lines, regions) given a set of candidates (rectangles, circles, ellipses, irregularly shaped labels) for each site. A map can be a classical cartographical map, a diagram, a graph or any other figure that needs to be labeled. A labeling is either a complete set of nonconflicting candidates, one per site, or a subset of maximum cardinality. Finding such a labeling is NPhard. We present a combinatorial framework to attack the problem in its full generality. The key idea is to separate the geometric from the combinatorial part of the problem. The latter is captured by the conflict graph of the candidates and by rules which successively simplify this graph towards a nearoptimal solution. We exemplify this framework at the problem of labeling point sets with axisparallel rectangles as candidates, four per point. We study competing algorithms and do a thorough empirical comparison. The new algorithm we sugges...
Point Set Labeling with Sliding Labels
 In Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... This paper discusses algorithms for labeling sets of points in the plane, where labels are not restricted to some finite number of positions. We show that continuously sliding labels allows more points to be labeled both in theory and in practice. We define six different models of labeling, and anal ..."
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Cited by 14 (3 self)
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This paper discusses algorithms for labeling sets of points in the plane, where labels are not restricted to some finite number of positions. We show that continuously sliding labels allows more points to be labeled both in theory and in practice. We define six different models of labeling, and analyze how much bettermore points get a labelone model can be than another. Maximizing the number of labeled points is NPhard, but we show that all models have a polynomialtime approximation scheme, and all models have a simple and efficient factor 1 2 approximation algorithm. Finally, we give experimental results based on the factor 1 2 approximation algorithm to compare the models in practice. 1 Introduction Annotating sets of points is a common task to be performed in Geographic Information Systems. Cities on smallscale maps are shown as points with the city's name attached (Figure 1 shows names as rectangles), points of altitude usually are small "+"signs with a value, and ...
Minimizing Movement
"... We give approximation algorithms and inapproximability results for a class of movement problems. In general, these problems involve planning the coordinated motion of a large collection of objects (representing anything from a robot swarm or firefighter team to map labels or network messages) to ach ..."
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Cited by 14 (2 self)
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We give approximation algorithms and inapproximability results for a class of movement problems. In general, these problems involve planning the coordinated motion of a large collection of objects (representing anything from a robot swarm or firefighter team to map labels or network messages) to achieve a global property of the network while minimizing the maximum or average movement. In particular, we consider the goals of achieving connectivity (undirected and directed), achieving connectivity between a given pair of vertices, achieving independence (a dispersion problem), and achieving a perfect matching (with applications to multicasting). This general family of movement problems encompass an intriguing range of graph and geometric algorithms, with several realworld applications and a surprising range of approximability. In some cases, we obtain tight approximation and inapproximability results using direct techniques (without use of PCP), assuming just that P != NP.
Polynomial Time Algorithms for Threelabel Point Labeling
, 2001
"... In this paper, we present an O(n log n) time solution for the following multilabel map labeling problem: Given a set S of n distinct sites in the plane, place at each site a triple of uniform squares of maximum possible size such that all the squares are axisparallel and a site is on the boun ..."
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Cited by 11 (0 self)
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In this paper, we present an O(n log n) time solution for the following multilabel map labeling problem: Given a set S of n distinct sites in the plane, place at each site a triple of uniform squares of maximum possible size such that all the squares are axisparallel and a site is on the boundaries of its three labeling squares. We also study the problem under the discrete model, i.e., a site must be at the corners of its three label squares. We obtain an optimal \Theta(n log n) time algorithm for the latter problem.
Three Rules Suffice for Good Label Placement
 Algorithmica Special Issue on GIS
, 2000
"... The general labelplacement problem consists in labeling a set of features (points, lines, regions) given a set of candidates (rectangles, circles, ellipses, irregularly shaped labels) for each feature. The problem arises when annotating classical cartographical maps, diagrams, or graph drawings. Th ..."
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Cited by 10 (1 self)
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The general labelplacement problem consists in labeling a set of features (points, lines, regions) given a set of candidates (rectangles, circles, ellipses, irregularly shaped labels) for each feature. The problem arises when annotating classical cartographical maps, diagrams, or graph drawings. The size of a labeling is the number of features that receive pairwise nonintersecting candidates. Finding an optimal solution, i.e. a labeling of maximum size, is NPhard. We present an approach to attack the problem in its full generality. The key idea is to separate the geometric part from the combinatorial part of the problem. The latter is captured by the conflict graph of the candidates. We present a set of rules that simplify the conflict graph without reducing the size of an optimal solution. Combining the application of these rules with a simple heuristic yields nearoptimal solutions. We study competing algorithms and do a thorough empirical comparison on pointlabeling data. The new algorithm we suggest is fast, simple, and effective.