Guarding Scenes against Invasive Hypercubes - Semantic Scholar
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Guarding Scenes against Invasive Hypercubes Mark de Berg1 Department of Computer Science, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, the Netherlands e-mail: [email protected] Haggai David2 Department of Mathematics and Computer Science, Ben-Gurion University of the Negev Beer-Sheva 84105, Israel e-mail: [email protected] Matthew J. Katz2 Department of Mathematics and Computer Science, Ben-Gurion University of the Negev Beer-Sheva 84105, Israel e-mail: [email protected] Mark Overmars1 Department of Computer Science, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, the Netherlands e-mail: [email protected] A. Frank van der Stappen1 Department of Computer Science, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, the Netherlands e-mail: [email protected] and Jules Vleugels3 Department of Computer Science, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, the Netherlands e-mail: [email protected] ABSTRACT A set of points G is a -guarding set for a set of objects O, if any hypercube not containing a point from G in its interior intersects at most objects of O. This de nition underlies a new input model, that is both more general than de Berg's unclutteredness, and retains its main property: a d-dimensional scene satisfying the new model's requirements is known to have a linear-size binary space partition. We propose several algorithms for computing -guarding sets, and evaluate them experimentally. One of them appears to be quite practical.
1. Introduction Recently de Berg et al. [4] brought together several of the realistic input models that have been proposed in the literature, namely fatness, low density, unclutteredness, and small simple-cover 1 2 3
Supported by the ESPRIT IV LTR Project No. 21957 (CGAL). Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Supported by the Netherlands' Organization for Scienti c Research (NWO).
Guarding Scenes against Invasive Hypercubes
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complexity (see de nitions below). They showed that these models form a strict hierarchy in the sense that fatness implies low density, which in turn implies unclutteredness, which implies small simple-cover complexity, and that the reverse implications are false. For each of the models above, data structures and algorithms were proposed [1, 7, 8, 9, 10, 11], that perform better than their more general counterparts. Assume an ecient algorithm exists for some problem under one of the models, say, M1 . A natural question that arises is: Does there also exist an algorithm for the same problem under the more general model M2 , that is more or less comparable to the former algorithm in terms of eciency. Clearly, it would be useful to have such an algorithm. In [1] de Berg presents an algorithm for computing a linear-size binary space partition (BSP) for uncluttered d-dimensional scenes, which enables him (see [2]) to construct a linear-size data structure supporting logarithmic-time point location for such scenes. We have de ned a new input model which is both more general than unclutteredness, and retains the property that a d-dimensional scene satisfying the new model's requirements is known to have a linear-size BSP, and consequently a linear-size data structure supporting logarithmic-time point location. However, in order to compute a linear-size BSP for such a scene, using de Berg's algorithm, we must rst compute a linear-size guarding set for the scene (see de nition below). This paper deals primarily with the problem of computing a small guarding set for such scenes. We describe three algorithms for this task and evaluate them both theoretically, but mainly experimentally. (We suspect that the problem of computing a smallest guarding set for such scenes is NP-complete.) The notion of guarding sets was introduced very recently by de Berg et al. [3]. They showed that having a linear-size guarding set is essentially equivalent to having small simple-cover complexity, and used this result to prove that the complexity of the free space of a bounded-reach robot with f degrees of freedom moving in a planar uncluttered scene or in a planar scene with small simple-cover complexity is (nf=2 ).
2. Preliminaries We rst de ne the two more general input models in the hierarchy of [4]. Unclutteredness was introduced by de Berg [1] under the name bounding-box- tness condition. The model is de ned as follows. (Throughout the paper, whenever we mention a square, cube, rectangle, etc., we assume it is axis-parallel.)
De nition 2.1. Let O be a set of objects in
Guarding Scenes against Invasive Hypercubes Mark de Berg1 Department of Computer Science, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, the Netherlands e-mail: [email protected] Haggai David2 Department of Mathematics and Computer Science, Ben-Gurion University of the Negev Beer-Sheva 84105, Israel e-mail: [email protected] Matthew J. Katz2 Department of Mathematics and Computer Science, Ben-Gurion University of the Negev Beer-Sheva 84105, Israel e-mail: [email protected] Mark Overmars1 Department of Computer Science, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, the Netherlands e-mail: [email protected] A. Frank van der Stappen1 Department of Computer Science, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, the Netherlands e-mail: [email protected] and Jules Vleugels3 Department of Computer Science, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, the Netherlands e-mail: [email protected] ABSTRACT A set of points G is a -guarding set for a set of objects O, if any hypercube not containing a point from G in its interior intersects at most objects of O. This de nition underlies a new input model, that is both more general than de Berg's unclutteredness, and retains its main property: a d-dimensional scene satisfying the new model's requirements is known to have a linear-size binary space partition. We propose several algorithms for computing -guarding sets, and evaluate them experimentally. One of them appears to be quite practical.
1. Introduction Recently de Berg et al. [4] brought together several of the realistic input models that have been proposed in the literature, namely fatness, low density, unclutteredness, and small simple-cover 1 2 3
Supported by the ESPRIT IV LTR Project No. 21957 (CGAL). Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Supported by the Netherlands' Organization for Scienti c Research (NWO).
Guarding Scenes against Invasive Hypercubes
2
complexity (see de nitions below). They showed that these models form a strict hierarchy in the sense that fatness implies low density, which in turn implies unclutteredness, which implies small simple-cover complexity, and that the reverse implications are false. For each of the models above, data structures and algorithms were proposed [1, 7, 8, 9, 10, 11], that perform better than their more general counterparts. Assume an ecient algorithm exists for some problem under one of the models, say, M1 . A natural question that arises is: Does there also exist an algorithm for the same problem under the more general model M2 , that is more or less comparable to the former algorithm in terms of eciency. Clearly, it would be useful to have such an algorithm. In [1] de Berg presents an algorithm for computing a linear-size binary space partition (BSP) for uncluttered d-dimensional scenes, which enables him (see [2]) to construct a linear-size data structure supporting logarithmic-time point location for such scenes. We have de ned a new input model which is both more general than unclutteredness, and retains the property that a d-dimensional scene satisfying the new model's requirements is known to have a linear-size BSP, and consequently a linear-size data structure supporting logarithmic-time point location. However, in order to compute a linear-size BSP for such a scene, using de Berg's algorithm, we must rst compute a linear-size guarding set for the scene (see de nition below). This paper deals primarily with the problem of computing a small guarding set for such scenes. We describe three algorithms for this task and evaluate them both theoretically, but mainly experimentally. (We suspect that the problem of computing a smallest guarding set for such scenes is NP-complete.) The notion of guarding sets was introduced very recently by de Berg et al. [3]. They showed that having a linear-size guarding set is essentially equivalent to having small simple-cover complexity, and used this result to prove that the complexity of the free space of a bounded-reach robot with f degrees of freedom moving in a planar uncluttered scene or in a planar scene with small simple-cover complexity is (nf=2 ).
2. Preliminaries We rst de ne the two more general input models in the hierarchy of [4]. Unclutteredness was introduced by de Berg [1] under the name bounding-box- tness condition. The model is de ned as follows. (Throughout the paper, whenever we mention a square, cube, rectangle, etc., we assume it is axis-parallel.)
De nition 2.1. Let O be a set of objects in
