Path Refinement in Weighted Regions

Gheibi, Amin; Maheshwari, Anil; Sack, Jörg-Rüdiger; Scheffer, Christian

doi:10.1007/s00453-018-0414-9

Cited by 3 publications

(4 citation statements)

References 15 publications

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“…grassland, blacktop, water, et cetera). Other practical applications of the WRP have been proposed for instance in geographical information systems (GIS) and in Seismology (see [19]).…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, the WRP with constraints on the number of links studied in [11] is a variant that requires a specific structure for the generated path. Another variant in this group is the one in [19], where some additional qualitative criteria are required on a path for the WRP. These qualitative criteria, expressed as geometric rules that the path must obey, are based on the discussions of the authors with Seismologists and Geophysicists, and can be related with the criteria of smoothness and sharp turns avoidance previously considered in the modeling of paths in other real applications [9,28,31].…”

Section: Introductionmentioning

confidence: 99%

“…Snell's law is the physical law that governs the phenomenon of refraction for light and other waves (see Section 2.2 for a formal statement). As pointed out in [19], among all the qualitative criteria for a weighted Euclidean shortest path, Snell's law is the most prominent one. The relation between some classes of geodesic paths and Snell's law has also been pointed out in [5,22,25,27,41] among others.…”

Section: Introductionmentioning

confidence: 99%

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Shortest paths and location problems in a continuous framework with different $\ell_p$-norms on different regions

Labbé,

Puerto,

Rodríguez-Madrena

2021

Preprint

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In this paper we address two different related problems. We first study the problem of finding a simple shortest path in a d-dimensional real space subdivided in several polyhedra endowed with different p-norms. This problem is a variant of the weighted region problem, a classical path problem in computational geometry introduced in Mitchell and Papadimitriou (JACM 38(1):18-73, 1991). As done in the literature for other geodesic path problems, we relate its local optimality condition with Snell's law and provide an extension of this law in our framework space. We propose a solution scheme based on the representation of the problem as a mixed-integer second order cone problem (MISOCP) using the p-norm modeling procedure given in Blanco et al. (Comput Optim Appl 58(3):563-595, 2014). We derive two different MISOCPs formulations, theoretically compare the lower bounds provided by their continuous relaxations, and propose a preprocessing scheme to improve their performance. The usefulness of this approach is validated through computational experiments. The formulations provided are flexible since some extensions of the problem can be handled by transforming the input data in a simple way. The second problem that we consider is the Weber problem that results in this subdivision of p-normed polyhedra. To solve it, we adapt the solution scheme that we developed for the shortest path problem and validate our methodology with extensive computational experiments.

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“…grassland, blacktop, water, et cetera). Other practical applications of the WRP have been proposed for instance in geographical information systems (GIS) and in Seismology (see [19]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Shortest paths and location problems in a continuous framework with different $\ell_p$-norms on different regions

Labbé,

Puerto,

Rodríguez-Madrena

2021

Preprint

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show abstract

“…An interesting point made by Rowe and Kindl (2012) is the following: There is a misperception that the WRP has been "solved" since it has been discussed for a long time, but this is incorrect because current practice heavily uses approximation algorithms. The works of (Gheibi et al 2018;Mitchell 2017) also claim that there is no known exact solution for WRP. To our knowledge, we still have no practical solution that can compute a very-close optimum path in reasonable time.…”

Section: Introductionmentioning

confidence: 99%

Computing Close to Optimal Weighted Shortest Paths in Practice

Tran

Dinneen

Linz

2020

ICAPS

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This paper proposes a new practical method for the weighted region problem (WRP). The objective of WRP is to find a minimum cost path between two vertices among different regions where each region incurs a traversal cost per unit distance. Currently, there is no practical algorithm that solves this problem exactly. Among the approximation methods that solve instances of WRP, there is a limited number of algorithms that compute paths whose lengths are close to optimal, which we call very-close optimum paths. However, they are considered as theoretical methods. On the other hand, algorithms for solving WRP that can be applied to practical data sets (using decomposition ideas or heuristics) are not guaranteed to find a very-close optimum path within an acceptable amount of time. In this paper, we consider an alternative method for solving WRP that exploits Snell's law of physical refraction. We compare the performance of our new algorithm with that of two existing algorithms, using at least 500 test cases for each such comparison. The experimental results show that our algorithm returns a very-close optimum weighted shortest path in reasonable time.

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Close Weighted Shortest Paths on 3D Terrain Surfaces

Tran

Dinneen

Linz

2020

Proceedings of the 28th International Conference on Advances in Geographic Information Systems

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This paper proposes an efficient method for the weighted region problem (WRP) on the surface of three-dimensional terrains. WRP is a classical path planning problem, asking for the minimum cost path between two given points crossing different regions in which each region is assigned a traversal cost per unit distance. Although WRP has been studied for decades, the exact solution for WRP, even in a two-dimensional environment, is unknown. Thus, the existing solutions for WRP are all approximations with decomposition-based and heuristic methods being the most widely-used in practice. However, when a very-close to optimal path is required, especially on real terrains with many regions, these approaches are not guaranteed or cannot return a satisfactory result in reasonable time. In this paper, we first present a new algorithm of finding a very-close optimal path, based on a user-defined parameter , between two points, crossing the surface of a sequence of regions in 3D, using Snell's law of physical refraction. We then show how to combine this algorithm with one existing decomposition-based method to compute a close optimal path over the whole terrain. In addition to a theoretical analysis, with an extensive set of test cases, the practicality and feasibility of our method are confirmed by that, our method always runs faster and returns closer to optimal paths in comparison with the existing ones.

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Path Refinement in Weighted Regions

Cited by 3 publications

References 15 publications

Shortest paths and location problems in a continuous framework with different $\ell_p$-norms on different regions

Shortest paths and location problems in a continuous framework with different $\ell_p$-norms on different regions

Computing Close to Optimal Weighted Shortest Paths in Practice

Close Weighted Shortest Paths on 3D Terrain Surfaces

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