## An optimal-time algorithm for shortest paths on a convex polytope in three dimensions (2006)

### Cached

### Download Links

Venue: | IN PROC. 22ND ACM SYMPOS. COMPUT. GEOM |

Citations: | 8 - 0 self |

### BibTeX

@INPROCEEDINGS{Schreiber06anoptimal-time,

author = {Yevgeny Schreiber and Micha Sharir},

title = {An optimal-time algorithm for shortest paths on a convex polytope in three dimensions },

booktitle = {IN PROC. 22ND ACM SYMPOS. COMPUT. GEOM},

year = {2006},

publisher = {}

}

### OpenURL

### Abstract

We present an optimal-time algorithm for computing (an implicit representation of) the shortest-path map from a fixed source s on the surface of a convex polytope P in three dimensions. Our algorithm runs in O(n log n) time and requires O(n log n) space, where n is the number of edges of P. The algorithm is based on the O(n log n) algorithm of Hershberger and Suri for shortest paths in the plane [22], and similarly follows the continuous Dijkstra paradigm, which propagates a “wavefront” from s along ∂P. This is effected by generalizing the concept of conforming subdivision of the free space used in [22], and by adapting it for the case of a convex polytope in R³, allowing the algorithm to accomplish the propagation in discrete steps, between the “transparent” edges of the subdivision. The algorithm constructs a dynamic version of Mount’s data structure [32] that implicitly encodes the shortest paths from s to all other points of the surface. This structure allows us to answer single-source shortest-path queries, where the length of the path, as well as its combinatorial type, can be reported in O(log n) time; the actual path can be reported in additional O(k) time, where k is the number

### Citations

620 | Data Structures and Network Algorithms - Tarjan - 1983 |

264 | Optimal search in planar subdivisions - Kirkpatrick - 1983 |

250 | S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields - Callahan, Kosaraju - 1995 |

237 | A dichromatic framework for balanced trees - Guibas, Sedgewick - 1978 |

159 | Optimal point location in a monotone subdivision - Edelsbrunner, Guibas, et al. - 1986 |

159 | The discrete geodesic problem
- Mitchell, Mount, et al.
- 1987
(Show Context)
Citation Context ...test-path queries. However, the question whether this data structure can be constructed in subquadratic time, has been left open. For a general, possibly nonconvex polyhedral surface, Mitchell et al. =-=[14]-=- presented an O(n 2 log n) algorithm for the single source shortest path problem (improving an earlier solution in [19]), extending the technique of Mount [15]. All algorithms in [14, 15, 23] use the ... |

120 | Symmetric binary b-trees: Data structure and maintenance algorithms - Bayer - 1972 |

102 |
On shortest paths in polyhedral spaces
- Sharir, Schorr
- 1986
(Show Context)
Citation Context ...y of Mitchell [13] for many variants and extensions; here we mention only the results that are most relevant to our specific problem. Its first study in computational geometry is by Sharir and Schorr =-=[23]-=-. Their algorithm runs in O(n 3 log n) time, where n is the number of vertices of P. The algorithm constructs a planar layout of the shortest path map, and then the shortest path from the fixed source... |

93 | An optimal algorithm for Euclidean shortest paths in the plane
- Hershberger, Suri
- 1999
(Show Context)
Citation Context ...n be judged at all. The algorithm of Hershberger and Suri for polygonal domains. A dramatic breakthrough on a loosely related problem has taken place in 1995, when Hershberger and Suri [10] (see also =-=[9, 11]-=-) have obtained an O(n log n)time algorithm for computing the shortest path map in the plane in the presence of polygonal obstacles (where n is the number of obstacle vertices). Shortest path queries ... |

82 | Linear time algorithms for visibility and shortest path problems inside simple polygons - Guibas, Hershberger, et al. - 1986 |

73 |
Shortest paths and networks
- Mitchell
- 2004
(Show Context)
Citation Context ...ean shortest obstacle-avoiding path between two given points, or, more generally, compute a compact representation of all such paths that emanate from a fixed source point. See the survey of Mitchell =-=[13]-=- for many variants and extensions; here we mention only the results that are most relevant to our specific problem. Its first study in computational geometry is by Sharir and Schorr [23]. Their algori... |

56 | Folding and unfolding in computational geometry
- O’Rourke
- 1998
(Show Context)
Citation Context ...al difficulties. On top of the main problem that a surface cell may intersect up to Ω(n) facets of P, it can in general be unfolded in more than one way, and such an unfolding may overlap itself (see =-=[18, 26]-=- for description of this problem). To overcome this difficulty, we introduce a Riemann structure, constructed by subdividing each surface cell into O(1) simple building blocks, whose planar unfolding ... |

46 | Data structures and algorithms for disjoint set union problems - Galil, Italiano - 1991 |

45 | Efficient computation of geodesic shortest paths - Kapoor - 1999 |

40 | Schevon, Star unfolding of a polytope with applications - Agarwal, Aronov, et al. - 1997 |

38 | Approximating shortest paths on a convex polytope in three dimensions
- Agarwal, Har-Peled, et al.
- 1997
(Show Context)
Citation Context ...ime, has been left open. The problem has been more or less “stuck” after Chen and Han’s paper, and the quadratic-time barrier seemed very difficult to break. For this and other reasons, several works =-=[1, 2, 7, 8, 24]-=- have presented approximate algorithms for the 3-dimensional shortest path problem. Nevertheless, the major problem of obtaining a subquadratic, or even near-linear, exact algorithm has remained open.... |

35 | A new algorithm for computing shortest paths in weighted planar subdivisions - Mata, Mitchell - 1997 |

30 | Constructing approximate shortest path maps in three dimensions
- Har-Peled
- 1999
(Show Context)
Citation Context ...ime, has been left open. The problem has been more or less “stuck” after Chen and Han’s paper, and the quadratic-time barrier seemed very difficult to break. For this and other reasons, several works =-=[1, 2, 7, 8, 24]-=- have presented approximate algorithms for the 3-dimensional shortest path problem. Nevertheless, the major problem of obtaining a subquadratic, or even near-linear, exact algorithm has remained open.... |

29 | Determining approximate shortest paths on weighted polyhedral surfaces - Aleksandrov, Maheshwari, et al. |

27 | Approximate shortest-path and geodesic diameter on convex polytopes
- Har-Peled
- 1999
(Show Context)
Citation Context ...ime, has been left open. The problem has been more or less “stuck” after Chen and Han’s paper, and the quadratic-time barrier seemed very difficult to break. For this and other reasons, several works =-=[1, 2, 7, 8, 24]-=- have presented approximate algorithms for the 3-dimensional shortest path problem. Nevertheless, the major problem of obtaining a subquadratic, or even near-linear, exact algorithm has remained open.... |

25 |
MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com
- Weisstein
- 1999
(Show Context)
Citation Context ...urface structure of e; it will be used in Sections 4 and 5 for wavefront propagation block-by-block from e in all directions. This structure is indeed similar to standard Riemann surfaces (see, e.g., =-=[25]-=-); its main purpose is to handle effectively (i) the possibility of overlap between distinct portions of ∂P when unfolded onto some plane, and (ii) the possibility that shortest paths may traverse a c... |

24 | An Epsilon-Approximation Algorithm for Weighted - Aleksandrov, Lanthier, et al. - 1998 |

22 | Shortest paths on a polyhedron; part i: computing shortest paths
- Chen, Han
- 1996
(Show Context)
Citation Context ... on the surface whoseshortest path distance to the source s has the same value t, and maintains this “wavefront” as t increases. The same general approach is also used in our algorithm. Chen and Han =-=[4]-=- use a rather different approach (for a not necessarily convex polyhedral surface). Their algorithm builds a shortest path sequence tree, using an observation that they call “one angle one split” to b... |

22 | Fully persistent lists with catenation
- Driscoll, Sleator, et al.
- 1994
(Show Context)
Citation Context ... the transformations stored in the nodes v1 = root, v2, . . . , vk = leaf storing si, of the path from the leaf vk storing si to the root. 2 We require the data structure to be confluently persistent =-=[6]-=-, since we need the ability to operate on and modify past versions of any list (wavefront), and we need the ability to merge existing distinct versions into a new version. To perform the search operat... |

20 | Two-dimensional and three-dimensional point location in rectangular subdivisions - Berg, Kreveld, et al. - 1995 |

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

20 | Approximating shortest paths on a nonconvex polyhedron - Varadarajan, Agarwal - 1997 |

16 |
On finding shortest paths on convex polyhedra
- Mount
- 1985
(Show Context)
Citation Context ... source point s to any given query point q can be computed in O(k+log n) time, where k is the number of edges of the polytope that are crossed by the shortest path from s to q. Soon afterwards, Mount =-=[15]-=- gave an improved algorithm for convex polytopes with running time O(n 2 log n). Moreover, in [16], Mount has shown that the problem of storing shortest path information can be treated separately from... |

13 | Minimum spanning trees in d dimensions - Krznaric, Levcopoulos - 1997 |

13 | On the development of the intersection of a plane with a polytope - O’Rourke |

10 |
An Improved Approximation Algorithm for Computing Geometric Shortest Paths
- Aleksandrov, Maheshwari, et al.
(Show Context)
Citation Context |

7 | An ε-approximation algorithm for weighted shortest path queries on polyhedral surfaces - Aleksandrov, Lanthier, et al. - 1998 |

7 | Unfolding polyhedral bands - Aloupis, Demaine, et al. - 2004 |

7 |
Non overlap of the star unfolding, Discrete Comput
- Aronov, O’Rourke
- 1992
(Show Context)
Citation Context ...similarly for answering shortest path queries in O(k + log n) time. (Their algorithm is somewhat simpler for the case of a convex polytope, relying on the property, established by Aronov and O‘Rourke =-=[3]-=-, that this layout does not overlap itself.) In [5], Chen and Han follow the general idea of Mount [16] to solve the problem of storing shortest path information separately, for a general, possibly no... |

7 |
Storing the subdivision of a polyhedral surface
- Mount
- 1987
(Show Context)
Citation Context ... R 3 , allowing the algorithm to accomplish the propagation in discrete steps, between the “transparent” edges of the subdivision. The algorithm constructs a dynamic version of Mount’s data structure =-=[16]-=- that implicitly encodes the shortest paths from s to all other points of the surface. This structure allows us to answer single-source shortest-path queries, where the length of the path, as well as ... |

5 |
Shortest paths on polyhedral surfaces
- O’Rourke, Suri, et al.
- 1985
(Show Context)
Citation Context ...left open. For a general, possibly nonconvex polyhedral surface, Mitchell et al. [14] presented an O(n 2 log n) algorithm for the single source shortest path problem (improving an earlier solution in =-=[19]-=-), extending the technique of Mount [15]. All algorithms in [14, 15, 23] use the same general approach, called “continuous Dijkstra”, first formalized in [14]. The technique keeps track of all the poi... |

2 | Topics in Data Structures - Italiano, Raman - 1998 |

1 |
Shortest paths on a polyhedron, Part II: Storing shortest paths
- Chen, Han
- 1990
(Show Context)
Citation Context ...k + log n) time. (Their algorithm is somewhat simpler for the case of a convex polytope, relying on the property, established by Aronov and O‘Rourke [3], that this layout does not overlap itself.) In =-=[5]-=-, Chen and Han follow the general idea of Mount [16] to solve the problem of storing shortest path information separately, for a general, possibly nonconvex polyhedral surface. They obtain a tradeoff ... |

1 | A note on the problems in connection with graphs - Dijkstra - 1959 |

1 | Computational geometry column 35, Internat - O’Rourke - 1999 |

1 | Storing the subdivision of a polyhedral surface, Discrete Comput. Geom - Mount - 1987 |