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An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions

Yevgeny Schreiber, Micha Sharir
2007 Discrete & Computational Geometry  
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.  ...  The algorithm is based on the O(n log n) algorithm of Hershberger and Suri for shortest paths in the plane [11] , and similarly follows the continuous Dijkstra paradigm, which propagates a "wavefront"  ...  We are grateful to Haim Kaplan for his help in designing the data structures, to Joe O'Rourke for valuable comments and material on surface unfolding and overlapping, as well as for remarks on Kapoor's  ... 
doi:10.1007/s00454-007-9031-0 fatcat:d67mebu4m5d2xcwaeqdghd7mfq

An optimal-time algorithm for shortest paths on a convex polytope in three dimensions

Yevgeny Schreiber, Micha Sharir
2006 Proceedings of the twenty-second annual symposium on Computational geometry - SCG '06  
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.  ...  The algorithm is based on the O(n log n) algorithm of Hershberger and Suri for shortest paths in the plane [11] , and similarly follows the continuous Dijkstra paradigm, which propagates a "wavefront"  ...  We are grateful to Haim Kaplan for his help in designing the data structures, to Joe O'Rourke for valuable comments and material on surface unfolding and overlapping, as well as for remarks on Kapoor's  ... 
doi:10.1145/1137856.1137862 dblp:conf/compgeom/SchreiberS06 fatcat:4nbrn6574zatbjaf7c5jgfsedm

An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions [chapter]

Yevgeny Schreiber, Micha Sharir
Twentieth Anniversary Volume:  
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.  ...  The algorithm is based on the O(n log n) algorithm of Hershberger and Suri for shortest paths in the plane [11] , and similarly follows the continuous Dijkstra paradigm, which propagates a "wavefront"  ...  We are grateful to Haim Kaplan for his help in designing the data structures, to Joe O'Rourke for valuable comments and material on surface unfolding and overlapping, as well as for remarks on Kapoor's  ... 
doi:10.1007/978-0-387-87363-3_27 fatcat:vyfnrx3psfgytacj27cwj7shju

Practical methods for approximating shortest paths on a convex polytope in R3

John Hershberger, Subhash Suri
1998 Computational geometry  
We propose an extremely simple approximation scheme for computing shortest paths on the surface of a convex polytope in three dimensions.  ...  The best algorithms for computing an exact shortest path on a convex polytope take ~(n 2) time in the worst case; in addition, they are too complicated to be suitable in practice.  ...  Acknowledgements We thank Pankaj Agarwal and Boris Aronov for some discussions on the subject of this paper. They have independently derived results similar to Theorem 3.8 of our paper.  ... 
doi:10.1016/s0925-7721(97)00004-7 fatcat:clcqi4i7bvggtfl2wlravndpoq

Approximate Shortest Paths and Geodesic Diameter on a Convex Polytope in Three Dimensions

S. Har-Peled
1999 Discrete & Computational Geometry  
the shortest path between s and t on ∂ P.  ...  Our second related result is: Given a convex polytope P with n edges in R 3 , and a parameter 0 < ε ≤ 1, we present an O(n + 1/ε 5 )-time algorithm that computes two points s, t ∈ ∂ P such that d P (s,  ...  The author also wish to thank the referees for their comments and suggestions. Approximate Shortest Paths and Geodesic Diameter on a Convex Polytope  ... 
doi:10.1007/pl00009417 fatcat:2c4kh6qjrnbnroyqtwjaiv3woq

Page 3506 of Mathematical Reviews Vol. , Issue 94f [page]

1994 Mathematical Reviews  
In particular, for Minkowski addition in a fixed dimension d, they find a polynomial time algorithm for adding k convex d-polytopes with up to n vertices.  ...  The algorithm for the width problem uses O(an) space, supports updates in O(a log” n) time, and reports in O(alog’ n) time an approximation W to the width such that W/W < ,/1+tan?(z/4a).  ... 

Euclidean TSP on two polygons

Jeff Abrahamson, Ali Shokoufandeh
2010 Theoretical Computer Science  
We give an O(n 2 m + nm 2 + m 2 log m) time and O(n 2 + m 2 ) space algorithm for finding the shortest traveling salesman tour through the vertices of two simple polygonal obstacles in the Euclidean plane  ...  We also consider the problem's extension to higher dimensions, proving that, if P = NP, constructing a shortest TSP tour on the vertices of two non-intersecting polytopes is NP-hard if the polytopes are  ...  This work was funded in part by a grant from the Office of Naval Research (ONR-N000140410363).  ... 
doi:10.1016/j.tcs.2009.12.003 fatcat:k6zthvotfvgltdhvkmgouxpggm

Approximating shortest paths on a convex polytope in three dimensions

Pankaj K. Agarwal, Sariel Har-Peled, Micha Sharir, Kasturi R. Varadarajan
1997 Journal of the ACM  
Given a convex polytope P with n faces in ‫ޒ‬ 3 , points s, t ʦ ѨP, and a parameter 0 Ͻ ⑀ Յ 1, we present an algorithm that constructs a path on ѨP from s to t whose length is at most (1 ϩ ⑀)d P (s, t)  ...  We also present an extension of the algorithm that computes approximate shortest path distances from a given source point on ѨP to all vertices of P.  ...  In particular, the construction in the Appendix of shortest paths with arbitrarily large folding angles is a variant of a construction initially provided by János Pach.  ... 
doi:10.1145/263867.263869 fatcat:64svhgcz7jcq3el4rsdhglv6cm

Approximating shortest paths on a convex polytope in three dimensions

Sariel Har-Peled, Micha Sharir, Kasturi R. Varadarajan
1996 Proceedings of the twelfth annual symposium on Computational geometry - SCG '96  
Given a convex polytope P with n faces in ‫ޒ‬ 3 , points s, t ʦ ѨP, and a parameter 0 Ͻ ⑀ Յ 1, we present an algorithm that constructs a path on ѨP from s to t whose length is at most (1 ϩ ⑀)d P (s, t)  ...  We also present an extension of the algorithm that computes approximate shortest path distances from a given source point on ѨP to all vertices of P.  ...  In particular, the construction in the Appendix of shortest paths with arbitrarily large folding angles is a variant of a construction initially provided by János Pach.  ... 
doi:10.1145/237218.237402 dblp:conf/compgeom/Har-PeledSV96 fatcat:dhupemj53zg4vihvhyhaj7eljq

Computing approximate shortest paths on convex polytopes

Pankaj K. Agarwal, Sariel Har-Peled, Meetesh Karia
2000 Proceedings of the sixteenth annual symposium on Computational geometry - SCG '00  
Sharir and Schorr [SS] were the first to provide an efficient algorithm for computing a shortest path on convex polyhedral surfaces. 5 Their algorithm runs in O(n 3 log n) time and relies on the fact that  ...  We have developed and implemented a robust and efficient algorithm for computing approximate shortest paths on a convex polyhedral surface.  ...  Finally, the authors thank the anonymous referees for a number of useful comments.  ... 
doi:10.1145/336154.336213 dblp:conf/compgeom/AgarwalHK00 fatcat:p3c2dytaunhchiq23lpzs5jxru

Computing Approximate Shortest Paths on Convex Polytopes

P. K. Agarwal, S. Har-Peled, M. Karia
2002 Algorithmica  
Sharir and Schorr [SS] were the first to provide an efficient algorithm for computing a shortest path on convex polyhedral surfaces. 5 Their algorithm runs in O(n 3 log n) time and relies on the fact that  ...  We have developed and implemented a robust and efficient algorithm for computing approximate shortest paths on a convex polyhedral surface.  ...  Finally, the authors thank the anonymous referees for a number of useful comments.  ... 
doi:10.1007/s00453-001-0111-x fatcat:y3i6xfqaurebzeg5i35er5pwui

Algebraic and Topological Tools in Linear Optimization

Jesús A. De Loera
2019 Notices of the American Mathematical Society  
The Linear Optimization Problem This is a story about the significance of diverse viewpoints in mathematical research.  ...  I will discuss how the analysis of the linear optimization problem connects in elegant ways to algebra and topology. My presentation has two sections, grouped under the guiding light of these areas.  ...  The diameter is the length of the longest shortest path among all possible pairs of vertices; e.g., for a three-dimensional cube the diameter is three.  ... 
doi:10.1090/noti1907 fatcat:gguvxixppzanxb6vy2mjsh4mcu

Combinatorial Polytope Enumeration [article]

Sandeep Koranne, Anand Kulkarni
2009 arXiv   pre-print
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and  ...  Our generator has implications for several outstanding problems in polytope theory, including conjectures about the number of distinct polytopes, the edge expansion of polytopal graphs, and the d-step  ...  Figure 1 : 1 Generation of an n-facet polytope from an (n − 1)-facet polytope. Figure 2 : 2 Operation of the polytopal induction algorithm in three dimensions, up to n = 6 facets.  ... 
arXiv:0908.1619v1 fatcat:busjeyr2ajerlo2fjt55jrzloy

Page 8768 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
(IL-TLAV; Tel Aviv) Approximate shortest paths and geodesic diameter on a convex polytope in three dimensions. (English summary) Discrete Comput. Geom. 21 (1999), no. 2, 217-231.  ...  For the special case of a shortest path between two points along the surface of a single convex polytope there is an O(n) algorithm, but the existence of a subquadratic algorithm is open.  ... 

A PTAS for Euclidean TSP with Hyperplane Neighborhoods [article]

Antonios Antoniadis, Krzysztof Fleszar, Ruben Hoeksma, Kevin Schewior
2019 arXiv   pre-print
Our algorithm is based on approximating the convex hull of the optimal tour by a convex polytope of bounded complexity.  ...  To do so, we develop a novel and general sparsification technique to transform an arbitrary convex polytope into one with a constant number of vertices and, in turn, into one of bounded complexity in the  ...  For the case of lines in three dimensions, only a recent O(log 3 n)approximation algorithm by Dumitrescu and Tóth [16] is known.  ... 
arXiv:1804.03953v2 fatcat:wa2bub4xdnaublpxxl35rz5lp4
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