## Approximation algorithms for shortest path motion planning (1987)

Venue: | In 19th ACM Symposium on Theory of Computing (STOC'87 |

Citations: | 78 - 0 self |

### BibTeX

@INPROCEEDINGS{Clarkson87approximationalgorithms,

author = {Kenneth L. Clarkson},

title = {Approximation algorithms for shortest path motion planning},

booktitle = {In 19th ACM Symposium on Theory of Computing (STOC'87},

year = {1987},

pages = {56--65}

}

### OpenURL

### Abstract

This paper gives approximation algorithms for solving the following motion planning problem: Given a set of polyhedral obstacles and points s and t, find a shortest path from s to t that avoids the obstacles. The paths found by the algorithms are piecewise linear, and the length of a path is the sum of the lengths of the line segments making up the path. Approximation algorithms will be given for versions of this problem in the plane and in three-dimensional space. The algorithms return an ɛ-short path, that is, a path with length within (1 + ɛ) of shortest. Let n be the total number of faces of the polyhedral obstacles, and ɛ a given value satisfying 0 < ɛ ≤ π. The algorithm for the planar case requires O(n log n)/ɛ time to build a data structure of size O(n/ɛ). Given points s and t, an ɛ-short path from s to t can be found with the use of the data structure in time O(n/ɛ + n log n). The data structure is associated with a new variety of Voronoi diagram. Given obstacles S ⊂ E 3 and points s, t ∈ E 3, an ɛ-short path between s and t can be found in O(n 2 λ(n) log(n/ɛ)/ɛ 4 + n 2 log nρ log(n log ρ)) time, where ρ is the ratio of the length of the longest obstacle edge to the distance between s and t. The function λ(n) = α(n) O(α(n)O(1)), where the α(n) is a form of inverse of Ackermann’s function. For log(1/ɛ) and log ρ that are O(log n), this bound is O(n 2 (log 2 n)λ(n)/ɛ 4). 1

### Citations

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- 1987
(Show Context)
Citation Context ...nd every C 2 F/ . Then Theorem 2.5 holds for the graph V ffl augmented in this way, and a shortest path in V ffl from s to t is an ffl-short path from s to t. Using the algorithm of Fredman and Tarjan=-=[FT84]-=-, such a path can be found in O(n log n + n=ffl) time. How can such an augmented subgraph V ffl be found quickly? It is shown in this section that this can be done using conical Voronoi diagrams, or C... |

333 |
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(Show Context)
Citation Context ...n edges, a C-VoD of S can be computed by a sweepline algorithm in O(n log n) time. Proof. Omitted. Sweepline algorithms are well known. For this problem, a sweepline algorithm somewhat like Fortune's =-=[For86]-=- can be used. Fortune's algorithm uses a geometric transformation, so that the transform of the Voronoi region for a site is not encountered in the sweep until the site is. This transformation is not ... |

187 |
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- 1982
(Show Context)
Citation Context ...s. a b c Figure 1: Cones for edges ofsG incident to a Chew has applied this approach to motion planning [Che86], and it has been previously been applied to the geometric minimum spanning tree problem =-=[Yao82]-=-. In the geometric minimum spanning tree problem, a set S of n points is given in the d-dimensional Euclidean space E d , and a minimum spanning tree (MST) is desired for the weighted undirected graph... |

119 |
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- 1989
(Show Context)
Citation Context ...his work has shown that O(n 2 log n) time suffices for this problem[SS84], and with sophisticated improvements, an O(n 2 ) bound on time and space can be attained [AAG + 85]. Recently, Chew has shown =-=[Che86]-=- that O(n 2 ) time and O(n) space are enough to find a ( p 10 \Gamma 1)-short path. In this paper, an algorithm is given that requires O(n log n=ffl) time to build a data structure of size O(n=ffl), s... |

95 |
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(Show Context)
Citation Context ...ed to the plane, previous work has generally focused on finding exact algorithms: those finding a shortest path between two points. This work has shown that O(n 2 log n) time suffices for this problem=-=[SS84]-=-, and with sophisticated improvements, an O(n 2 ) bound on time and space can be attained [AAG + 85]. Recently, Chew has shown [Che86] that O(n 2 ) time and O(n) space are enough to find a ( p 10 \Gam... |

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Citation Context ...wn exact algorithms for this version require exponential time, although some polynomial-time algorithms are known for special cases [Mou84, SB86, RS85]. As for approximation algorithms, Papadimitriou =-=[Pap85]-=- has shown that O(n 3 (L+log(n=ffl)) 2 =ffl) time suffices to find an ffl-short path between two points. Here L is a bound on the number of bits in an integer specifying the coordinates of a point in ... |

27 |
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- 1986
(Show Context)
Citation Context ...the problem of determining a path by which an object can be moved from place to place while avoiding obstacles. A survey of the substantial literature on various cases of this problem can be found in =-=[Yap85]-=-. This paper describes algorithms for the problem of moving in a short path from one point to another in a way that avoids given polyhedral obstacles. In the case where the obstacles, points, and path... |

18 | On shortest paths amidst convex polyhedra - Sharir - 1987 |

16 | On finding shortest paths on convex polyhedra - Mount - 1985 |

12 | Shortest Paths in Euclidean Spaces with Polyhedral Obstacles - Reif, Storer - 1985 |

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1 | Visibility-polygon search and Euclidean shortest paths - Chew - 1985 |

1 | A sweepline algorithm for Voronoi diagrams - Convexity - 1958 |