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On shortest paths in polyhedral spaces

Authors:

Micha Sharir(,

Amir SchorrAuthors Info & Claims

STOC '84: Proceedings of the sixteenth annual ACM symposium on Theory of computing

Pages 144 - 153

https://doi.org/10.1145/800057.808676

Published: 01 December 1984 Publication History

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Abstract

We consider the problem of computing the shortest path between two points in two- or three-dimensional space bounded by polyhedral surfaces. In the 2-D case the problem is easily solved in time O(n² log n).In the general 3-D case the problem is quite hard to solve, and is not even discrete; we present a doubly-exponential procedure for solving the discrete subproblem of determining the sequence of boundary edges through which the shortest path passes. The main result of this paper involves a favorable special case of the 3-D shortest path problem, namely that of finding the shortest path between two points along the surface of a convex polyhedron. We analyze this problem and solve it in time O(n³ log n).

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Meng WXin STu CChen SHe YWang W(2022)Geodesic Tracks: Computing Discrete Geodesics With Track-Based Steiner Point PropagationIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2021.310904228:12(4887-4901)Online publication date: 1-Dec-2022
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Huang STeo R(2019)Computationally Efficient Visibility Graph-Based Generation Of 3D Shortest Collision-Free Path Among Polyhedral Obstacles For Unmanned Aerial Vehicles2019 International Conference on Unmanned Aircraft Systems (ICUAS)10.1109/ICUAS.2019.8798322(1218-1223)Online publication date: Jun-2019
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Jiang KSeneviratne LEarles S(2016)Three-Dimensional Shortest Path Planning in the Presence of Polyhedral ObstaclesProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science10.1243/PIME_PROC_1996_210_209_02210:4(373-381)Online publication date: 11-Aug-2016
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Index Terms

On shortest paths in polyhedral spaces
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Paths and connectivity problems
2. Theory of computation
  1. Randomness, geometry and discrete structures

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STOC '84: Proceedings of the sixteenth annual ACM symposium on Theory of computing

December 1984

547 pages

ISBN:0897911334

DOI:10.1145/800057

Chairman:
Richard DeMillo

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Published: 01 December 1984

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https://doi.org/10.1109/TVCG.2021.3109042
Huang STeo R(2019)Computationally Efficient Visibility Graph-Based Generation Of 3D Shortest Collision-Free Path Among Polyhedral Obstacles For Unmanned Aerial Vehicles2019 International Conference on Unmanned Aircraft Systems (ICUAS)10.1109/ICUAS.2019.8798322(1218-1223)Online publication date: Jun-2019
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Recommendations

Approximating Shortest Paths on Weighted Polyhedral Surfaces

Improved algorithms for the k simple shortest paths and the replacement paths problems

Inapproximability results for the inverse shortest paths problem with integer lengths and unique shortest paths

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