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

2007
*
Discrete & Computational Geometry
*

We present

doi:10.1007/s00454-007-9031-0
fatcat:d67mebu4m5d2xcwaeqdghd7mfq
*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 ...##
###
An optimal-time algorithm for shortest paths on a convex polytope in three dimensions

2006
*
Proceedings of the twenty-second annual symposium on Computational geometry - SCG '06
*

We present

doi:10.1145/1137856.1137862
dblp:conf/compgeom/SchreiberS06
fatcat:4nbrn6574zatbjaf7c5jgfsedm
*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 ...##
###
An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions
[chapter]

*
Twentieth Anniversary Volume:
*

We present

doi:10.1007/978-0-387-87363-3_27
fatcat:vyfnrx3psfgytacj27cwj7shju
*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 ...##
###
Practical methods for approximating shortest paths on a convex polytope in R3

1998
*
Computational geometry
*

We propose

doi:10.1016/s0925-7721(97)00004-7
fatcat:clcqi4i7bvggtfl2wlravndpoq
*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. ...##
###
Approximate Shortest Paths and Geodesic Diameter on a Convex Polytope in Three Dimensions

1999
*
Discrete & Computational Geometry
*

the

doi:10.1007/pl00009417
fatcat:2c4kh6qjrnbnroyqtwjaiv3woq
*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*...##
###
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

2010
*
Theoretical Computer Science
*

We give

doi:10.1016/j.tcs.2009.12.003
fatcat:k6zthvotfvgltdhvkmgouxpggm
*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). ...##
###
Approximating shortest paths on a convex polytope in three dimensions

1997
*
Journal of the ACM
*

Given

doi:10.1145/263867.263869
fatcat:64svhgcz7jcq3el4rsdhglv6cm
*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. ...##
###
Approximating shortest paths on a convex polytope in three dimensions

1996
*
Proceedings of the twelfth annual symposium on Computational geometry - SCG '96
*

Given

doi:10.1145/237218.237402
dblp:conf/compgeom/Har-PeledSV96
fatcat:dhupemj53zg4vihvhyhaj7eljq
*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. ...##
###
Computing approximate shortest paths on convex polytopes

2000
*
Proceedings of the sixteenth annual symposium on Computational geometry - SCG '00
*

Sharir and Schorr [SS] were the first to provide

doi:10.1145/336154.336213
dblp:conf/compgeom/AgarwalHK00
fatcat:p3c2dytaunhchiq23lpzs5jxru
*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. ...##
###
Computing Approximate Shortest Paths on Convex Polytopes

2002
*
Algorithmica
*

Sharir and Schorr [SS] were the first to provide

doi:10.1007/s00453-001-0111-x
fatcat:y3i6xfqaurebzeg5i35er5pwui
*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. ...##
###
Algebraic and Topological Tools in Linear Optimization

2019
*
Notices of the American Mathematical Society
*

The Linear

doi:10.1090/noti1907
fatcat:gguvxixppzanxb6vy2mjsh4mcu
*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*. ...##
###
Combinatorial Polytope Enumeration
[article]

2009
*
arXiv
*
pre-print

We describe

arXiv:0908.1619v1
fatcat:busjeyr2ajerlo2fjt55jrzloy
*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. ...##
###
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]

2019
*
arXiv
*
pre-print

Our

arXiv:1804.03953v2
fatcat:wa2bub4xdnaublpxxl35rz5lp4
*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. ...
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