## A New Algorithm for Computing Shortest Paths in Weighted Planar Subdivisions (Extended Abstract) (1997)

Venue: | In Proc. 13th Annu. ACM Sympos. Comput. Geom |

Citations: | 37 - 3 self |

### BibTeX

@INPROCEEDINGS{Mata97anew,

author = {Cristian S. Mata and J.S.B. Mitchell},

title = {A New Algorithm for Computing Shortest Paths in Weighted Planar Subdivisions (Extended Abstract)},

booktitle = {In Proc. 13th Annu. ACM Sympos. Comput. Geom},

year = {1997},

pages = {264--273},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

) Cristian S. Mata Joseph S. B. Mitchell y Abstract We present a practical new algorithm for the problem of computing low-cost paths in a weighted planar subdivision or on a weighted polyhedral surface. The algorithm is based on constructing a relatively sparse graph, a "pathnet", that links selected pairs of subdivision vertices with locally optimal paths. The pathnet can be searched for paths that are provably close to optimal and approach optimal, as one varies the parameter that controls the sparsity of the pathnet. We analyze our algorithm both analytically and experimentally. We report on the results of a set of experiments comparing the new algorithm with other standard methods. 1 Introduction For a given weight function, F : ! 2 ! !, the weighted length of an s-t path ß in the plane is the path integral, R ß F (x; y)doe, of the weight function along the path ß, linking the start s to the goal t. The weighted region metric associated with F defines the distance dF (...

### Citations

507 | Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams
- Guibas, Stol
- 1985
(Show Context)
Citation Context ...y (e.g., by triangulation).) We will assume that S is given to us in a convenient data structure that allows the usual basic operations to be done in constant time; e.g., the quad-edge data structure =-=[7]-=- is one possibility (a variant of which is used by our code). Let n denote the total number of vertices of S. Then S also has O(n) faces and edges. Each face f has an associated integer weight ff f 2 ... |

201 | On constructing minimum spanning trees in k-dimensional space and related problems
- YAO
- 1982
(Show Context)
Citation Context ...gly motivated by results on approximating Euclidean shortest paths among obstacles (as in Clarkson [3] and Mitchell [16]), t-spanners [9], and approximation methods devised for minimum spanning trees =-=[24]-=-. 5 Other Algorithms In order to determine the practicality and relative effectiveness of our new algorithm, we implemented several other methods, for experimental comparison. 5.1 Grid-Based Algorithm... |

132 |
Classes of graphs which approximate the complete euclidean graph
- Keil, Gutwin
- 1992
(Show Context)
Citation Context ...fl) of optimal, where ffl = O( W=w k` min ). Our approach is strongly motivated by results on approximating Euclidean shortest paths among obstacles (as in Clarkson [3] and Mitchell [16]), t-spanners =-=[9]-=-, and approximation methods devised for minimum spanning trees [24]. 5 Other Algorithms In order to determine the practicality and relative effectiveness of our new algorithm, we implemented several o... |

99 |
The weighted region problem: finding shortest paths through a weighted planar subdivision
- Mitchell, Papadimitriou
- 1991
(Show Context)
Citation Context ...ted terrain maps. We refer the reader to the survey [15] for pointers to the literature. The main theoretical results known for the weighted region problem are presented in Mitchell and Papadimitriou =-=[17]. They giv-=-e an algorithm, based on the "continuous Dijkstra method", to find a path whose weighted length is guaranteed to be within a factor (1 + ffl) of optimal, where ffl ? 0 is any user-specified ... |

82 | Approximation algorithms for shortest path motion planning
- Clarkson
- 1987
(Show Context)
Citation Context ...paths that are within factor (1 + ffl) of optimal, where ffl = O( W=w k` min ). Our approach is strongly motivated by results on approximating Euclidean shortest paths among obstacles (as in Clarkson =-=[3]-=- and Mitchell [16]), t-spanners [9], and approximation methods devised for minimum spanning trees [24]. 5 Other Algorithms In order to determine the practicality and relative effectiveness of our new ... |

58 |
An algorithm for shortest path motion in three dimensions
- Papadimitriou
- 1985
(Show Context)
Citation Context ... parameter ffi (or one can specify a maximum number of Steiner points on any one edge); and (2) how to space the points on an edge, e.g., evenly (as we do), in geometric progression (as Papadimitriou =-=[19]-=- does for shortest paths in ! 3 ), or in some other way. We have not yet seen how to use uneven spacing to achieve provably better approximation bounds; this may be a subject of future experimental an... |

51 | Approximating weighted shortest paths on polyhedral surfaces
- Lanthier, Maheshwari, et al.
- 1997
(Show Context)
Citation Context ...s part of a system built at Hughes, for the DARPA ALV project in 1985), Johansson [8] (who implemented it for use in fluid flow computations for injection molding), and Lanthier, Maheshwari, and Sack =-=[12]-=- (who, in indpendent work closely related work our own, have conducted experiments on this, and related, practical methods for the WRP). The basic edge subdivision algorithm is as follows. For each ed... |

36 |
L1 shortest paths among polygonal obstacles in the plane
- Mitchell
- 1992
(Show Context)
Citation Context ...thin factor (1 + ffl) of optimal, where ffl = O( W=w k` min ). Our approach is strongly motivated by results on approximating Euclidean shortest paths among obstacles (as in Clarkson [3] and Mitchell =-=[16]-=-), t-spanners [9], and approximation methods devised for minimum spanning trees [24]. 5 Other Algorithms In order to determine the practicality and relative effectiveness of our new algorithm, we impl... |

32 |
An Algorithm Approach to Some Problems in Terrain Navigation
- Mitchell
- 1988
(Show Context)
Citation Context ...e of terrain has an assigned weight equal to the reciprocal of the maximum speed of traversal for the robot. It also arises in numerous other applications involving route planning in geographic data (=-=[5, 15, 23]-=-), military mission planning and decision support ([1, 10, 11, 13, 14, 15, 21, 22]), and fluid flow in injection molding [8]. We assume that f is specified by a triangulation having n vertices, with e... |

22 |
Shortest path queries among weighted obstacles in the rectilinear plane
- Chen, Klenk, et al.
(Show Context)
Citation Context ...sion is rectilinear, and path length is measured according to weighted L1 length, then efficient algorithms for single-source and two-point queries can be based upon searching a path-preserving graph =-=[2]-=-. Similarly, if the region weights are restricted to f0; 1; 1g (while edges may have arbitrary (nonnegative) weights), then an O(n 2 ) algorithm can be based on constructing a path-preserving graph si... |

18 |
Planning and reasoning for autonomous vehicle control
- Mitchell, Payton, et al.
- 1987
(Show Context)
Citation Context ...stance instead of Euclidean distance. For 4-connectivity this "digitization bias" (aliasing) results in an error factor of p 2; for 8-connectivity, it results in a factor of ( p 2+1)= p 5 = =-=1:08. See [18, 20]-=- for discussions on the digitization bias problem, and on different distance transforms that can be used to address it. One approach to reducing grid bias is to increase the connectivity of the grid. ... |

13 |
Path planning by optimal-path-map construction for homogeneous-cost two-dimensional regions
- Alexander, Rowe
- 1990
(Show Context)
Citation Context ... the maximum speed of traversal for the robot. It also arises in numerous other applications involving route planning in geographic data ([5, 15, 23]), military mission planning and decision support (=-=[1, 10, 11, 13, 14, 15, 21, 22]-=-), and fluid flow in injection molding [8]. We assume that f is specified by a triangulation having n vertices, with each face assigned an integer weight ff 2 f0; 1; : : : ; W;+1g. We can allow edges ... |

8 | A Stochastic Approach to the Weighted-Region Problem: Design and Testing of a Path Annealing Algorithm,” Naval Center for Applied Research
- Kindl, Shing, et al.
- 1991
(Show Context)
Citation Context ... the maximum speed of traversal for the robot. It also arises in numerous other applications involving route planning in geographic data ([5, 15, 23]), military mission planning and decision support (=-=[1, 10, 11, 13, 14, 15, 21, 22]-=-), and fluid flow in injection molding [8]. We assume that f is specified by a triangulation having n vertices, with each face assigned an integer weight ff 2 f0; 1; : : : ; W;+1g. We can allow edges ... |

8 |
Concealed routes in ModSAF
- Longtin, Megherbi
- 1995
(Show Context)
Citation Context ... the maximum speed of traversal for the robot. It also arises in numerous other applications involving route planning in geographic data ([5, 15, 23]), military mission planning and decision support (=-=[1, 10, 11, 13, 14, 15, 21, 22]-=-), and fluid flow in injection molding [8]. We assume that f is specified by a triangulation having n vertices, with each face assigned an integer weight ff 2 f0; 1; : : : ; W;+1g. We can allow edges ... |

7 |
Robust and efficient implementation of the Delaunay tree
- Devillers
- 1992
(Show Context)
Citation Context ...es, while constraining the perturbation so that all faces of the subdivision remain convex. 2. A second method uses Delaunay triangulations of point sets (based on the algorithm and code of Devillers =-=[4]-=-). In one set of datasets the vertices of the subdivision are generated uniformly at random within a square. Another set of triangulations is designed to yield nonuniform point distributions, by conce... |

7 |
Path planning in 0=1=1 weighted regions with applications.ORSA
- Gewali, Meng, et al.
- 1990
(Show Context)
Citation Context ...eights are restricted to f0; 1; 1g (while edges may have arbitrary (nonnegative) weights), then an O(n 2 ) algorithm can be based on constructing a path-preserving graph similar to a visibility graph =-=[6]-=-. This also leads to an efficient method for performing lexicographic optimization, in which one prioritizes various types of regions according to which is most important for path length minimization.... |

7 |
An Efficient Snell’s Law Method for Optimal-Path Planning across Multiple Two-Dimensional
- 9Rowe, Richbourg
- 1990
(Show Context)
Citation Context |

6 |
Transportation, social physics, and the law of refraction
- Warntz
- 1957
(Show Context)
Citation Context ...e of terrain has an assigned weight equal to the reciprocal of the maximum speed of traversal for the robot. It also arises in numerous other applications involving route planning in geographic data (=-=[5, 15, 23]-=-), military mission planning and decision support ([1, 10, 11, 13, 14, 15, 21, 22]), and fluid flow in injection molding [8]. We assume that f is specified by a triangulation having n vertices, with e... |

5 |
Cover and concealment in ModSAF
- Longtin
- 1994
(Show Context)
Citation Context |

5 |
Solving Global, Two-Dimensional Routing Problems using Snell’s Law and A
- 8Richbourg, Rowe, et al.
- 1987
(Show Context)
Citation Context |

3 |
On a weighted distance model for injection moulding. Linkoping
- Johansson
- 1997
(Show Context)
Citation Context ...ous other applications involving route planning in geographic data ([5, 15, 23]), military mission planning and decision support ([1, 10, 11, 13, 14, 15, 21, 22]), and fluid flow in injection molding =-=[8]-=-. We assume that f is specified by a triangulation having n vertices, with each face assigned an integer weight ff 2 f0; 1; : : : ; W;+1g. We can allow edges of the triangulation to have a weight that... |

2 |
Least cost path in geographic information systems. Research note no
- Douglas
- 1993
(Show Context)
Citation Context ...e of terrain has an assigned weight equal to the reciprocal of the maximum speed of traversal for the robot. It also arises in numerous other applications involving route planning in geographic data (=-=[5, 15, 23]-=-), military mission planning and decision support ([1, 10, 11, 13, 14, 15, 21, 22]), and fluid flow in injection molding [8]. We assume that f is specified by a triangulation having n vertices, with e... |

1 |
The Euclidean distance transform. Linkoping
- Ragnemalm
- 1993
(Show Context)
Citation Context ...stance instead of Euclidean distance. For 4-connectivity this "digitization bias" (aliasing) results in an error factor of p 2; for 8-connectivity, it results in a factor of ( p 2+1)= p 5 = =-=1:08. See [18, 20]-=- for discussions on the digitization bias problem, and on different distance transforms that can be used to address it. One approach to reducing grid bias is to increase the connectivity of the grid. ... |