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## Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces (2008)

Citations: | 5 - 0 self |

### Citations

456 | A separator theorem for planar graphs
- Lipton, Tarjan
- 1977
(Show Context)
Citation Context ...oblem formulation is rather general and many of the known graph separator results can be viewed as solutions of its particular cases. For example, the well known separator result by Lipton and Tarjan =-=[34]-=- can be viewed as a solution of the problem formulated here, where P ∗ is planar, t = 2/3, and all vertices have unit cost. An other important example is the separator result by Frederickson [26] obta... |

215 | New lower-bound techniques for robot motion-planning problems - Canny, Reif - 1987 |

199 | S.: Faster shortest-path algorithms for planar graphs
- Henzinger, Klein, et al.
- 1997
(Show Context)
Citation Context ...hniques have been successfully applied to shortest path problems in both graphs and geometric environments. Examples include solutions of shortest path problems in planar graphs and in planar domains =-=[11, 16, 26, 27, 33]-=-. In this paper, we develop and apply a partitioning technique to solve the approximate APQ problem on weighted polyhedral surfaces, as sketched next. First, we compute a special set of faces S, calle... |

179 | The discrete geodesic problem
- Mitchell, Mount, et al.
- 1987
(Show Context)
Citation Context ... finding of exact or approximate distances is required. Work on exact problems: A solution to the exact SSQ problem on general polyhedral surfaces was presented initially in [41] and then improved in =-=[35]-=-. For a surface of size n and a source point s, the algorithm in [35] takes O(n 2 log n) preprocessing time to construct a diagram that can be used to answer distance queries to s in O(log n) time. Ap... |

148 | Fast algorithms for shortest paths in planar graphs, with applications
- Frederickson
- 1987
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Citation Context ...hniques have been successfully applied to shortest path problems in both graphs and geometric environments. Examples include solutions of shortest path problems in planar graphs and in planar domains =-=[11, 16, 26, 27, 33]-=-. In this paper, we develop and apply a partitioning technique to solve the approximate APQ problem on weighted polyhedral surfaces, as sketched next. First, we compute a special set of faces S, calle... |

120 |
On shortest paths in polyhedral spaces
- Sharir, Schorr
- 1984
(Show Context)
Citation Context ...ferring to the cases where finding of exact or approximate distances is required. Work on exact problems: A solution to the exact SSQ problem on general polyhedral surfaces was presented initially in =-=[41]-=- and then improved in [35]. For a surface of size n and a source point s, the algorithm in [35] takes O(n 2 log n) preprocessing time to construct a diagram that can be used to answer distance queries... |

112 | An optimal algorithm for Euclidean shortest paths in the plane
- Hershberger, Suri
- 1999
(Show Context)
Citation Context ...special attention to finding efficient solutions under certain restrictions imposed on the 3Problem Geom. Model Preprocessing Space Query Remarks Reference SSQ Planar domain O(n log n) O(n) O(log n) =-=[32]-=- SSQ Convex Surf. O(n log n) O(n) O(log n) [40] SSQ General Surf. O(n2 ) O(n log n/ logd) O(d log n/ log d) 1 ≤ d ≤ n [17] APQ Planar domain O(n5+10δ1+δ ) O(n5+10δ1+δ ) O(n1−δ log n) 0 < δ1 ≤ 1, [21] ... |

104 | Approximation algorithms for shortest path motion planning
- Clarkson
- 1987
(Show Context)
Citation Context ...aining the query point). Moreover, a solution of the approximate SSQ problem on the surface of a convex polytope with improved preprocessing time and data structure size has been proposed in [31]. In =-=[22]-=-, Clarkson presented an algorithm for solving the approximate APQ problem in planar polygonal domains. For an approximation parameter 0 < ε ≤ π and a domain of size n, the algorithm builds, in O( n lo... |

92 |
A separator theorem for graphs of bounded genus
- Gilbert, Hutchinson, et al.
- 1984
(Show Context)
Citation Context ...e separator result by Frederickson [26] obtained for planar graphs with unit costs, where an upper bound on the size of B is provided. Other separator results related to the above formulation include =-=[4, 5, 23, 28]-=-. In this paper, we show the existence of t-separators for the class of graphs dual to triangulated polyhedral surfaces with weights and costs assigned to their vertices. We establish bounds on the co... |

91 |
Optimal shortest path queries in a simple polygon
- Guibas, Hershberger
- 1989
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Citation Context ... 1, [21] δ > 0 APQ Convex Surf. O(n6m1+δ ) O(n6m1+δ ) O( √ n m1/4 log n) 1 ≤ m ≤ n2 , δ > 0 [1] Table 1: Selected exact solutions to the Euclidean SSQ and APQ problems. domain. Guibas and Hershberger =-=[29]-=- presented an asymptotically optimal solution of the exact APQ problem inside a simple polygon in the plane. In O(n) time their algorithm constructs a data structure that allows answering distance que... |

87 |
Shortest paths on a polyhedron
- Chen, Han
- 1990
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Citation Context ...t s, the algorithm in [35] takes O(n 2 log n) preprocessing time to construct a diagram that can be used to answer distance queries to s in O(log n) time. Applying an alternate approach, Chen and Han =-=[17, 18]-=- obtained an algorithm with improved preprocessing time and controllable query time. For a surface of size n and an input parameter 1 ≤ d ≤ n, their algorithm takes O(n 2 ) preprocessing time to const... |

80 |
Metric entropy of some classes of sets with differentiable boundaries
- Dudley
- 1974
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Citation Context ...pproximation, except for the case p = 1. Approximation problems on the surface of a convex polytope P have been extensively studied. A series of results have been obtained based on a result of Dudley =-=[25]-=- about Hausdorff approximation of convex sets. Dudley’s result shows that P can be approximated by a convex polytope Q of size O(1/ε 3 2) so that distances on Q provide ε-approximation to the distance... |

41 | Approximate shortest paths on a convex polytope in three dimensions
- Agarwal, Har-Peled, et al.
- 1997
(Show Context)
Citation Context ...ets. Dudley’s result shows that P can be approximated by a convex polytope Q of size O(1/ε 3 2) so that distances on Q provide ε-approximation to the distances on P. This was used in the algorithm of =-=[3]-=- to compute an ε-approximate shortest path in O( 1 ε3 +n log 1) time. Later, in [30], this algorithm was extended to solve the ε approximate APQ problem in O(n) preprocessing time and size of the data... |

39 | N.: Planar graph decomposition and all pairs shortest paths - Frederickson - 1991 |

38 | Star unfolding of a polytope with applications
- Agarwal, Aronov, et al.
- 1997
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Citation Context ...edral surfaces is believed to be complex, both theoretically and computationally. To the best of our knowledge no complete solution to this problem has been published so far. It had been announced in =-=[1, 21]-=- that methods applied to special cases, such as planar polygonal domain [21] and surface of convex polytopes [1], extend to the general case. These potential solutions represent a theoretical interest... |

37 | Determining approximate shortest paths on weighted polyhedral surfaces
- Aleksandrov, Maheshwari, et al.
(Show Context)
Citation Context ...A. The remaining authors are from Carleton University, Ottawa, Canada. 1in various applications such as motion/route planning, navigation, graphics, seismology, injection molding (for references see =-=[10]-=-). Aside from the importance of shortest path problems in their own right, they often appear in the solutions to other problems. Unlike the well-studied graph version of the shortest path problem, com... |

32 | Constructing Approximate Shortest Path Maps in Three Dimensions
- Har-Peled
- 1999
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Citation Context ...te solution for the SSQ problem as well. To the best of our knowledge these are the first solutions to the APQ and SSQ problems on weighted surfaces. These problems were mentioned as open problems in =-=[31]-=-; it is easy to see that they are natural ones to consider once we know how to answer queries on surfaces under Euclidean metric. To place our work in the context of the literature we next state some ... |

29 | Linear algorithms for partitioning embedded graphs of bounded genus
- Aleksandrov, Djidjev
- 1996
(Show Context)
Citation Context ...e separator result by Frederickson [26] obtained for planar graphs with unit costs, where an upper bound on the size of B is provided. Other separator results related to the above formulation include =-=[4, 5, 23, 28]-=-. In this paper, we show the existence of t-separators for the class of graphs dual to triangulated polyhedral surfaces with weights and costs assigned to their vertices. We establish bounds on the co... |

29 | Approximate shortest paths and geodesic diameters on convex polytopes in three dimensions
- Har-Peled
- 1999
(Show Context)
Citation Context ...e O(1/ε 3 2) so that distances on Q provide ε-approximation to the distances on P. This was used in the algorithm of [3] to compute an ε-approximate shortest path in O( 1 ε3 +n log 1) time. Later, in =-=[30]-=-, this algorithm was extended to solve the ε approximate APQ problem in O(n) preprocessing time and size of the data structure and in log n O( ε3/2 + 1 ε3) query time. Practical issues about this appr... |

27 | Sublinear geometric algorithms
- Chazelle, Liu, et al.
(Show Context)
Citation Context ...ell as an implementation and experiments have been discussed in [2]. That algorithm finds an ε-approximate distance between any pair of points on P in O( n √ + ε 1 ε4) time. Recently, Chazelle et al. =-=[13]-=- proposed a randomized solution to the approximate APQ problem on convex surfaces. See Table 2 for more details. The weighted case: In the weighted surface model it is assumed that positive weights ar... |

23 | Two-point Euclidean shortest path queries in the plane
- Chiang, Mitchell
- 1999
(Show Context)
Citation Context ...edral surfaces is believed to be complex, both theoretically and computationally. To the best of our knowledge no complete solution to this problem has been published so far. It had been announced in =-=[1, 21]-=- that methods applied to special cases, such as planar polygonal domain [21] and surface of convex polytopes [1], extend to the general case. These potential solutions represent a theoretical interest... |

20 | On finding approximate optimal paths in weighted regions
- Sun, Reif
(Show Context)
Citation Context ...SQ and APQ problems. Problem Geom. Model Preprocessing Space Query Reference SSSP tree Weighted Surf. O( n ε2 log n log 1 ε ) O(n) O(1) [8] SSSP tree Weighted Surf. O( n n 1 ε log ε log ε ) O(n) O(1) =-=[37, 38]-=- ) O(n) O(1) [9, 10] SSSP tree Weighted Surf. O( n √ log ε n ε SSQ Weighted pl. reg. O(n8 log n SSQ Weighted pl. reg. O( n4 ε2 log SSQ Weighted Surf. O( n √ ε log n ε APQ APQ Weighted Surf. of genus g... |

19 | An Efficient Approximation Algorithm Of Weighted Region Shortest Path Problem
- Reif, Sun
(Show Context)
Citation Context ...SQ and APQ problems. Problem Geom. Model Preprocessing Space Query Reference SSSP tree Weighted Surf. O( n ε2 log n log 1 ε ) O(n) O(1) [8] SSSP tree Weighted Surf. O( n n 1 ε log ε log ε ) O(n) O(1) =-=[37, 38]-=- ) O(n) O(1) [9, 10] SSSP tree Weighted Surf. O( n √ log ε n ε SSQ Weighted pl. reg. O(n8 log n SSQ Weighted pl. reg. O( n4 ε2 log SSQ Weighted Surf. O( n √ ε log n ε APQ APQ Weighted Surf. of genus g... |

18 | Computing approximate shortest paths on convex polytopes
- Agarwal, Har-Peled, et al.
(Show Context)
Citation Context ...(n) preprocessing time and size of the data structure and in log n O( ε3/2 + 1 ε3) query time. Practical issues about this approach as well as an implementation and experiments have been discussed in =-=[2]-=-. That algorithm finds an ε-approximate distance between any pair of points on P in O( n √ + ε 1 ε4) time. Recently, Chazelle et al. [13] proposed a randomized solution to the approximate APQ problem ... |

17 | Approximate Shortest Paths in Anisotropic Regions
- Cheng, Na, et al.
- 2008
(Show Context)
Citation Context ...ct ratios, which is unrealistic in practice. For example, if none of the angles of P is smaller than 6 ◦ , then Γ is less than 10. During the write-up of the journal version of [6, 7], the results in =-=[19, 20]-=- with respect to the weighted region problem in planar domains have appeared. This work is inspired by our work and that of Reif and Sun [38], and the authors were able to remove the dependence on the... |

14 |
On the a&pairs Euclidean short path problem
- Chen
- 1995
(Show Context)
Citation Context ... π and a domain of size n, the algorithm builds, in O( n log n) time, a data structure of size O(n) that supports ε ε approximate shortest path queries in O(n( 1 + log n)) time. As noticed by Chen in =-=[14]-=-, ε Clarkson’s approach can be used to obtain an algorithm with query time reduced to O( 1 ε (1 ε + log n)) at the expense of a preprocessing time increased to O(n2 ( 1 + log n)) and a size of ε the d... |

13 | An optimal-time algorithm for shortest paths on a convex polytope in three dimensions
- Schreiber, Sharir
- 2008
(Show Context)
Citation Context ...s under certain restrictions imposed on the 3Problem Geom. Model Preprocessing Space Query Remarks Reference SSQ Planar domain O(n log n) O(n) O(log n) [32] SSQ Convex Surf. O(n log n) O(n) O(log n) =-=[40]-=- SSQ General Surf. O(n2 ) O(n log n/ logd) O(d log n/ log d) 1 ≤ d ≤ n [17] APQ Planar domain O(n5+10δ1+δ ) O(n5+10δ1+δ ) O(n1−δ log n) 0 < δ1 ≤ 1, [21] δ > 0 APQ Convex Surf. O(n6m1+δ ) O(n6m1+δ ) O(... |

12 | Shortest path queries in planar graphs
- Chen, Xu
- 2000
(Show Context)
Citation Context ...hniques have been successfully applied to shortest path problems in both graphs and geometric environments. Examples include solutions of shortest path problems in planar graphs and in planar domains =-=[11, 16, 26, 27, 33]-=-. In this paper, we develop and apply a partitioning technique to solve the approximate APQ problem on weighted polyhedral surfaces, as sketched next. First, we compute a special set of faces S, calle... |

10 |
An improved approximation algorithm for computing geometric shortest paths
- Aleksandrow, Maheshwari, et al.
(Show Context)
Citation Context ...Problem Geom. Model Preprocessing Space Query Reference SSSP tree Weighted Surf. O( n ε2 log n log 1 ε ) O(n) O(1) [8] SSSP tree Weighted Surf. O( n n 1 ε log ε log ε ) O(n) O(1) [37, 38] ) O(n) O(1) =-=[9, 10]-=- SSSP tree Weighted Surf. O( n √ log ε n ε SSQ Weighted pl. reg. O(n8 log n SSQ Weighted pl. reg. O( n4 ε2 log SSQ Weighted Surf. O( n √ ε log n ε APQ APQ Weighted Surf. of genus g Weighted Surf. of g... |

9 |
Partitioning planar graphs with costs and weights
- Aleksandrov, Djidjev, et al.
- 2006
(Show Context)
Citation Context ...e separator result by Frederickson [26] obtained for planar graphs with unit costs, where an upper bound on the size of B is provided. Other separator results related to the above formulation include =-=[4, 5, 23, 28]-=-. In this paper, we show the existence of t-separators for the class of graphs dual to triangulated polyhedral surfaces with weights and costs assigned to their vertices. We establish bounds on the co... |

7 | On geometric path query problems
- Daescu, KIenk
- 1997
(Show Context)
Citation Context ...e space for the corresponding data structures vary from O(n 5 ) to O(n 15 ). An algorithm for solving the exact APQ problem with O(n 2 ) bound on the preprocessing time and space has been proposed in =-=[15]-=-. The bound on the query time of this algorithm depends on the position of the query points and is O(n log n) in the worst case. An optimal solution of the exact SSQ problem on the surface of a convex... |

6 | Approximate shortest path queries on weighted polyhedral surfaces - Aleksandrov, Djidjev, et al. - 2006 |

6 | An ε-approximation algorithm for weighted shortest path queries on polyhedral surfaces
- Aleksandrov, Lanthier, et al.
- 1998
(Show Context)
Citation Context ...∗ Research supported by Natural Sciences and Engineering Research Council of Canada and US Department of Energy under contract W-705-ENG-36. Preliminary research leading to this paper has appeared in =-=[6, 7]-=-. Research was carried out in part while the first and second authors were visiting Carleton University. The first author is from the Bulgarian Academy of Sciences, Sofia, Bulgaria. The second is from... |

3 | Querying approximate shortest paths in anisotropic regions
- Cheng, Na, et al.
- 2005
(Show Context)
Citation Context ...4 ) O(n7 log n ε 2 n ε ) n4 O( ε2 log n ε ) n O(log ε 1 log ε ) O( n √ log ε 1 ε ) 1 O(log ε ) n 1 log ε log4 ε ) O( n2 ε2 n 1 q log q log ε log4 ε ) O((g+1)n2 ε 3/2 q n2 O( ε3q2 log 2 q log ) [36] ) =-=[20]-=- here 1 log4 ε ) O(q) here 6 1 ε ) O(q) here Table 3: Selected solutions to SSSP tree, SSQ, and APQ problems on weighted surfaces. It is assumed that query consists of query points as well as pointers... |

3 |
Linear algorithms for graph separation problems
- Djidjev
- 1988
(Show Context)
Citation Context |

2 |
Spatial point location and its applications
- Tan, Hirata, et al.
- 1990
(Show Context)
Citation Context ... then answered in O(log ε ε ε ) time. This assumes that the face containing the query point is specified with the query - otherwise, a point location data structure needs to be used (e.g., the one in =-=[42]-=- can be used which requires O(n2 log n) time and O(n2 ) space for preprocessing and O(log n) time for locating the face containing the query point). Moreover, a solution of the approximate SSQ problem... |

1 |
The weighted region problem: Fiding shortest paths through a weighted planar subdivision
- Mitchell, Papadimitriou
- 1991
(Show Context)
Citation Context ...surface faces. Cost of a path on the weighted surface is computed as the weighted sums of the Euclidean lengths of their portions inside the faces of the surface. The weighted model was introduced in =-=[36]-=- for planar subdivisions. Solutions of the exact problems in the weighted case seem to be infeasible, since computing of exact distances even on very simple weighted surfaces with just a few faces req... |