L. Aleksandrov, M. Lanthier, A. Maheshwari, and J.-R. Sack. An ffl-approximation algorithm for weighted shortest paths on polyhedral surfaces. Lecture Notes in Computer Science, 1432:11--22, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and the metric is not the same for all regions, the straight line segment st may no longer be an optimal path from s to t, even if st lies in the free space. Therefore, many techniques developed in previous motion planning works are no longer valid. In the weighted region optimal path problem ([5, 4, 3, 1, 6, 2]) the entire free space is divided into polygonal regions each of which is associated with a unit weight. The cost of a path p is defined to be the weighted sum of the lengths of the segments of p inside each region. Another example is the flow problem ( 7] Supported by NSF ITR EIA 0086015, ....

....Office of Naval Research Contract N00014 99 1 0406. where inside each region there is a flow defined by a vector, and the cost of path p is the total travel time on p by robot with a fixed maximum velocity. To solve the weighted region optimal path problem, a number of previous works ([3, 1, 2]) used a discretization of the problem based on edge subdivision, and Dijkstra s algorithm to find an optimal path in the graph induced by discretization. Aleksandrov et al. [1] proposed a logarithmic discretization that guarantees an ffl short approximation, where m = O( 1 ffl log 1 ffl ) ....

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L. Aleksandrov, M. Lanthier, A. Maheshwari, and J.-R. Sack. An ffl-approximation algorithm for weighted shortest paths on polyhedral surfaces. Lecture Notes in Computer Science, 1432:11--22, 1998.

.... cost path problem, and approximation algorithms for the weighted region minimal cost path problem include the continuous Dijkstra method of Mitchell and Papadimitriou [9] as well as a variety of other discretization algorithms given by Mata and Mitchell [7] Lanthier et al. [6] Aleksandrov et al. [1]. More recent works include Reif and Sun [16, 15] and Aleksandrov et al. [1, 2] In Section 1, we have defined and motivated the flow path problem, and stated our results. In Section 2, we provide some preliminary results on the geometry of optimum paths for flow path problems, including a simple ....

.... cost path problem include the continuous Dijkstra method of Mitchell and Papadimitriou [9] as well as a variety of other discretization algorithms given by Mata and Mitchell [7] Lanthier et al. [6] Aleksandrov et al. [1] More recent works include Reif and Sun [16, 15] and Aleksandrov et al. [1, 2]. In Section 1, we have defined and motivated the flow path problem, and stated our results. In Section 2, we provide some preliminary results on the geometry of optimum paths for flow path problems, including a simple example where the 2D version of the for flow path problem consists of an ....

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L. Aleksandrov, M. Lanthier, A. Maheshwari, and J.-R. Sack. An ffl-approximation algorithm for weighted shortest paths on polyhedral surfaces. Lecture Notes in Computer Science, 1432:11--22, 1998.

....is associated with a positive unit weight, and two points s and t, find a path from s to t with the minimum weight, where the weight of a path is defined to be the weighted sum of the lengths of the sub paths within each region. Some previous algorithms (Lanthier et al. [9] and Aleksandrov et al. [1]) took a discretization approach by introducing m Steiner points on each edge. A discrete graph is constructed by adding edges connecting Steiner points in the same triangular region and an optimal path is computed in the resulting discrete graph using Dijkstra s algorithm. To avoid high time ....

....approach by introducing m Steiner points on each edge. A discrete graph is constructed by adding edges connecting Steiner points in the same triangular region and an optimal path is computed in the resulting discrete graph using Dijkstra s algorithm. To avoid high time complexity, both [9] and [1] use a subgraph of the complete graph in each triangular region. As a result, in the discrete graph only an approximate optimal path can be achieved, whose error is proportional to the weight of the optimal path. This approximate optimal path then is used to approximate the optimal path in the ....

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L. Aleksandrov, M. Lanthier, A. Maheshwari, and J.- R. Sack. An ffl-approximation algorithm for weighted shortest paths on polyhedral surfaces. Lecture Notes in Computer Science, 1432:11--22, 1998.

.... graph of size O(nk) is constructed and then an approximate optimal path with relative error of ffl is computed, where ffl = O( Wmax=Wmin k min ) The time complexity, in terms of ffl, n and other geometric parameters, is O( n 3 Wmax=Wmin ffl min ) Lanthier et al. [7] and Aleksandrov et al. [1] both adopted a natural approach to solving this problem which discretizes the polygonal subdivision by placing Steiner points along the edges of polygonal regions. Both algorithms construct a dense discrete graph by connecting (in a specified way) the Steiner points that share the same region and ....

....in the discrete graph. The relative error ffl is treated as a constant in their complexity analysis. Including ffl also as an input parameter, the complexity is O( n 3 ffl n 3 log n) as in the discrete graph there will be O(n 3 ) vertices and O( n 3 ffl ) edges. Aleksandrov et al. [1] proposed a logarithmic discretization that can guarantee a relative error of ffl. The time complexity of this algorithm is O(mn log(mn) where m = O(log ffi (L=r) is the number of points added on each edge. Here r is the minimum distance from any point to the boundary of the faces adjacent to ....

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L. Aleksandrov, M. Lanthier, A. Maheshwari, and J.-R. Sack. An ffl-approximation algorithm for weighted shortest paths on polyhedral surfaces. Lecture Notes in Computer Science, 1432:11--22, 1998.

....and Agarwal [VA97] described a subquadratic algorithm for a constant factor approximate shortest path on a polyhedral terrain. Recently, there has been more practical work on developing and implementing simple approximation algorithms for shortest paths on polyhedral surfaces. Aleksandrov et al. [ALMS98], Mata and Mitchell [MM97] Lanthier et al. LMS97] proposed a number of approximation algorithms for computing a shortest path on a polyhedral surface, especially for unweighted and weighted terrains. Using an idea of Papadimitriou [Pap85] they place Steiner points along the edges of P , ....

L. Aleksandrov, M. Lanthier, A. Maheshwari, and J.-R. Sack. An ffl-approximation algorithm for weighted shortest paths on polyhedral surfaces. In Proc. 6th Scand. Workshop Algorithm Theory, volume 1432 of Lecture Notes Comput. Sci., pages 11--22. Springer-Verlag, 1998.

....of Mitchell and Papadimitriou [292] the worst case dependence on n is much better. If, as in [292] the coordinates have integral values at most N , then sin = O(1=N 2 ) and h = O(1=N ) making the time bound roughly O( N 4 W 2 n ffl 2 ) An improved variant of their result ([12]) searches a reduced subgraph, allowing them to remove the additive term nM 2 in the complexity, resulting in time bound O(Mn log Mn) roughly O( N 2 Wn ffl ) Several other papers have also addressed practical and effective (possibly heuristic) methods for the WRP; see the work by ....

....O(Mn log Mn nM 2 ) and space O(nM 2 ) where M = O( 1 ffl sin log hffl ) is the length of a longest edge, h is the minimum altitude of a triangular facet, is the smallest angle of any triangular facet, and 0 ffl 2 3 . By searching a sparser subgraph, they have recently ([12]) improved the time bound to O(Mn log Mn) 6.4 Other Metrics Link distance in a polyhedral domain in d can be approximated (within factor 2) in polynomial time, by searching a weak visibility graph whose nodes correspond to simplices in a simplicial decomposition of the domain. The ....

L. Aleksandrov, M. Lanthier, A. Maheshwari, and J.-R. Sack. An ffl-approximation algorithm for weighted shortest paths on polyhedral surfaces. In Proc. 6th Scand. Workshop Algorithm Theory, volume ?? of Lecture Notes Comput. Sci., page to appear. Springer-Verlag, 1998.

....allowing reasonably efficient solutions to two point queries. The reported path can also be postprocessed with a local optimality procedure that results in a solution even closer to optimal. Using a slightly different discrete graph than the edge subdivision graph of [246, 273] Aleksandrov et al. [11] give alternative time bounds that depend on other parameters related to the fatness of the triangular facets of a weighted polyhedral surface. They place Steiner points along edges in a geometric progression, as Papadimitriou [317] has done for approximating shortest paths in three dimensions ....

....path obtained can be postprocessed with a local optimality procedure that pulls the path taut within the sleeve of facets that it crosses, resulting in a solution even closer to optimal. Using a slightly different discrete graph than the edge subdivision graph of [246, 273] Aleksandrov et al. [11] give alternative time bounds that depend on other parameters related to the fatness of the triangular facets of a polyhedral surface. They place Steiner points along edges in a geometric progression, as in Papadimitriou [317] This allows one to compute a (1 ffl) approximate shortest path ....

L. Aleksandrov, M. Lanthier, A. Maheshwari, and J.-R. Sack. An ffl-approximation algorithm for weighted shortest path queries on polyhedral surfaces. In Abstracts 14th European Workshop Comput. Geom., pages 19--21, 1998.

....paths on weighted polyhedra. More specifically, the algorithms compute paths from the source vertex s to all vertices, Steiner points which are introduced on edges of the polyhedron. The techniques described in this paper can be used to derive algorithms for shortest path queries, as discussed in [1]. An alternative approach, which we are investigating, is to compute the relevant portion of the subgraphs G i on the fly. It is clear that in Dijkstra s algorithm when the current vertex v (with least cost) explores the edges incident to it, we don t have to explore all of them because of the ....

L. Aleksandrov, M. Lanthier, A. Maheshwari and J.-R. Sack, "An ffl-Approximation Algorithm for Weighted Shortest Path Queries on Polyhedral Surfaces", to appear 14th European Workshop on Computational Geometry, Barcelona, Spain, 1998.

....complexities of 3 d shortest paths algorithms even for special problem instances have motivated the search for approximate solutions to the shortest path problem. For weighted shortest path approximations on planar subdivisions or polyhedra, more recently, several algorithms have been proposed [1, 2, 6 8]. In the model introduced by [9] the direction of travel along a face is not captured. The direction of travel plays an important role in determining the physical effects incurred on a vehicle (e.g. car, truck, robot, or even person) traveling along a terrain surface. Through anisotropism, we ....

....vu 0 w 0 of vuw. The spherical cap C v around v consists of all such sub faces incident at v. 1. 3 Overview of Our Approach Our approach is to discretize the polyhedral terrain in a natural way, by placing Steiner points along the edges of the polyhedron (as in our earlier subdivision approach [2, 6] but with substantial differences as illustrated below) We construct a graph G containing the Steiner points as vertices and edges as those interconnections between Steiner points that correspond to segments which lie completely in the triangular faces of the polyhedron. The geometric shortest ....

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L. Aleksandrov, M. Lanthier, A. Maheshwari and J.-R. Sack, "An ffl-Approximation Algorithm for Weighted Shortest Paths on Polyhedral Surfaces", SWAT '98, Stockholm, Sweden, 1998.

.... A large body of work has centered around the computation of Euclidean shortest paths (we refer the reader to the survey in [10] For terrains, Sharir and Schorr presented an algorithm for computing Euclidean shortest paths [12] and now we know of a number of different algorithms (see cf. [1, 6, 10]) Weighted shortest paths (introduced by [9] provide more realism in that they can incorporate terrain attributes such as variable costs for different regions. This allows Research supported in part by NSERC. 5 We encountered several shortest path related problems in our R D on GIS (see ....

....complexities of 3 d shortest paths algorithms even for special problem instances have motivated the search for approximate solutions to the shortest path problem. For weighted shortest path approximations on planar subdivisions or polyhedra, more recently, several algorithms have been proposed [1, 2, 6 8]. In the model introduced by [9] the direction of travel along a face is not captured. The direction of travel plays an important role in determining the physical effects incurred on a vehicle (e.g. car, truck, robot, or even person) traveling along a terrain surface. Through anisotropism, we ....

[Article contains additional citation context not shown here]

L. Aleksandrov, M. Lanthier, A. Maheshwari and J.-R. Sack, "An ffl-Approximation Algorithm for Weighted Shortest Path Queries on Polyhedral Surfaces", 14th European Workshop on Computational Geometry, Barcelona, Spain, 1998.

....cost plus fiWL, where fi 1 is an adjustable constant. Mata and Mitchell [12] presented an algorithm that constructs a graph (pathnet) which can be searched to obtain an approximate path, but they do not state any bounds on the accuracy of the path obtained. Very recently, Aleksandrov et al. [2] presented an ffl approximation algorithm for weighted shortest paths between two vertices on a polyhedral surface. Their algorithm runs in O(mn log mn nm 2 ) time where m = O(log ffi (L=r) and L is the length of the longest edge and r = f(ffl) times the minimum distance from any vertex to ....

....sin 4 , where is the minimum angle between any two adjacent edges of P . In the weighted case, f(ffl) min( ffl 2 3(W=w) 1=6) and ffi 1 ffl sin 2 3(W=w) where W and w are the largest and smallest face weights of P , respectively. The work presented here is a generalizetion of [2], which is extended to allow arbitrary query points, both source and destination) and is easily parallelizable. Our techniques also allow for the computation of a weighted shortest path map. This work represents the first ffl approximation algorithm for weighted shortest path queries on a ....

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L. Aleksandrov, M. Lanthier, A. Maheshwari and J.-R. Sack, "An ffl-Approximation Algorithm for Weighted Shortest Paths on Polyhedral Surfaces", Technical Report, Carleton University, December 1997.

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