Computing homotopic shortest paths in the plane

Bespamyatnikh, Sergei

doi:10.1016/s0196-6774(03)00090-7

Cited by 52 publications

(47 citation statements)

References 12 publications

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“…We consider general input paths, albeit with fixed homotopy classes, and study minimum-link rectilinear instead of shortest paths. There are several papers [2,6] that find shortest paths homotopic to a given collection of input paths. However, while a set of shortest paths homotopic to a set of non-crossing input paths is necessarily non-crossing, the same does not hold for minimumlink rectilinear paths.…”

Section: Introductionmentioning

confidence: 99%

Homotopic Rectilinear Routing with Few Links and Thick Edges

Speckmann

Verbeek

2010

LATIN 2010: Theoretical Informatics

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We study the problem of finding non-crossing thick minimum-link rectilinear paths homotopic to a set of input paths in an environment with rectangular obstacles. This problem occurs in the context of map schematization under geometric embedding restrictions, for example, when schematizing a highway network for use as a thematic layer. We present a 2-approximation algorithm that runs in O(n 3 + k in log n + k out ) time, where n is the total number of input paths and obstacles and k in and k out are the total complexities of the input and output paths, respectively. Our algorithm not only approximates the minimum number of links, but also minimizes the total length of the paths. An approximation factor of 2 is optimal when using smallest paths as lower bound.

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Section: Introductionmentioning

confidence: 99%

Homotopic Rectilinear Routing with Few Links and Thick Edges

Speckmann

Verbeek

2010

LATIN 2010: Theoretical Informatics

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“…with the following properties: (1) Any face of a simplex in X is another simplex in X ; (2) Any two simplices in X intersect in a common face. The k-skeleton of X is the subcomplex consisting of all simplices in X of dimension k or less.…”

Section: Preliminariesmentioning

confidence: 99%

Testing contractibility in planar rips complexes

Chambers

Erickson

Worah

2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

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The (Vietoris-)Rips complex of a discrete point-set P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Our algorithm requires O(m log n) time to preprocess a set of n points in the plane in which m pairs have distance at most 1; after preprocessing, deciding whether a cycle of k Rips edges is contractible requires O(k) time. We also describe an algorithm to compute the shortest non-contractible cycle in a planar Rips complex in O(n 2 log n + mn) time.

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“…8 The reconstructed epipolar geometry from the very wide baseline100 A. 9 The ratio of the height over the width of the window on the left is 1.5456 and the window on the right 1.8356, which gives us an indication of the reconstruction quality. .…”

Section: 1mentioning

confidence: 99%

Extending the Path-Planning Horizon

Nabbe¹,

Hebert

2007

The International Journal of Robotics Research

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SINCE typical mobile robotic vehicles have mobility sensors (such as LADAR or stereo) that can only acquire data up to a few tens of meters, a navigation system has no knowledge about the world beyond this sensing horizon. As a result, path planners that rely only on this knowledge to compute paths are unable to anticipate obstacles sufficiently early and has no choice than to plan inefficient paths that trace obstacle boundaries. To alleviate this problem, We present an opportunistic navigation and view planning strategy that incorporates look-ahead sensing of possible obstacle configurations. This planning strategy is based on a "what-if" analysis of hypothetical future configurations of the environment. Candidate vantage positions are evaluated based on their ability of observing anticipated obstacles. These vantage positions identified by this forward-simulation framework are used by the planner as intermediate waypoints.The validity of the strategy is supported by results from simulations as well as field experiments with a real robotic platform. These results also show that opportunistically significant reduction in path length can be achieved by using this framework. The reconstructed epipolar geometry from the very wide baseline100 A.9The ratio of the height over the width of the window on the left is 1.5456 and the window on the right 1.8356, which gives us an indication of the reconstruction quality. This effect is due to the fact that the planner is limited by the maximum range of the mobility sensors. Since typical mobility sensors, such as Laser Radar (LADAR) or passive stereo vision, will only acquire data up to a few tens of meters, the planner has no knowledge about what to encounter beyond the sensed perimeter. As a result, the planner is unable to anticipate obstacles sufficiently early and has no choice but to plan paths close to obstacle boundaries.For autonomous navigation over long distances, this issue degrades the performance of the system by greatly increasing the length of the path traveled by the vehicle. Consequently, the power consumed is increased and, more importantly, CHAPTER 1. INTRODUCTION FIGURE 1.1. Typical example of poor performance due to lack of sensor planning and mid-range sensing (Left: Overhead view of terrain; Right: Executed path with detected obstacles shown as shaded regions). The path from S1 to G1 intersects a large hill which is discovered only when the vehicle enters a large cul-de-sac, causing the executed path to be substantially more expensive than the path that would have been followed, had the obstruction been discovered earlier. (From [114]) the risk of exposure to threats is also increased. In addition, the relative short range of the mobility sensors forces the vehicle to drive closer to terrain obstructions than is safe or necessary.One solution is to use sensors with longer range to acquire information about the terrain further ahead. However, to use these sensors one needs to take into account certain constraints about the sensing geom...

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Computing homotopic shortest paths in the plane

Cited by 52 publications

References 12 publications

Homotopic Rectilinear Routing with Few Links and Thick Edges

Homotopic Rectilinear Routing with Few Links and Thick Edges

Testing contractibility in planar rips complexes

Extending the Path-Planning Horizon

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