# The Discrete Geodesic Problem

```@article{Mitchell1987TheDG,
title={The Discrete Geodesic Problem},
author={Joseph S. B. Mitchell and David M. Mount and Christos H. Papadimitriou},
journal={SIAM J. Comput.},
year={1987},
volume={16},
pages={647-668}
}```
• Published 1 August 1987
• Mathematics, Computer Science
• SIAM J. Comput.
We present an algorithm for determining the shortest path between a source and a destination on an arbitrary (possibly nonconvex) polyhedral surface. The path is constrained to lie on the surface, and distances are measured according to the Euclidean metric. Our algorithm runs in time O(n log n) and requires O(n2) space, where n is the number ofedges ofthe surface. Afterwe run our algorithm, the distance from the source to any other destination may be determined using standard techniques in… Expand
666 Citations

#### Paper Mentions

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#### References

SHOWING 1-10 OF 17 REFERENCES
Voronoi Diagrams on the Surface of a Polyhedron.
Abstract : This document presents an algorithm that computes the Voronol diagram of a set of point lying on the surface of a possibly nonconvex polyhedron. Distances are measured in the EuclideanExpand
Euclidean shortest paths in the presence of rectilinear barriers
• Mathematics, Computer Science
• Networks
• 1984
The goal is to find interesting cases for which the solution can be obtained without the explicit construction of the entire visibility graph, which solve the problems by constructing the shortest-path tree from the source to all the vertices of the obstacles and to the destination. Expand
On Finding Shortest Paths on Convex Polyhedra.
Abstract : Applications in robotics and autonomous navigation have motivated the study of motion planning and obstacle avoidance algorithms. The special case considered here is that of moving a pointExpand
On Shortest Paths in Polyhedral Spaces
• Mathematics, Computer Science
• SIAM J. Comput.
• 1986
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, is considered, which can be solved in time \$O(n^3 \log n)\$. Expand
Storing the subdivision of a polyhedral surface
• D. Mount
• Mathematics, Computer Science
• Discret. Comput. Geom.
• 1987
This work considers a natural generalization of a subdivision of a plane defined by the faces of a straight-line planar graph on a polyhedral surface and provides an efficient solution to the nearest-neighbor query problem on polyhedral surfaces. Expand
Shortest Paths on Polyhedral Surfaces
• Mathematics, Computer Science
• STACS
• 1985
An algorithm is presented that finds the shortest path between two points on a polyhedral surface in O(n5) time, where n is the number of vertices on the surface, thereby establishing that theExpand
Shortest Paths in Euclidean Space with Polyhedral Obstacles.
• Mathematics
• 1985
Abstract : This document considers the problem of finding a minimum length path between two points in Euclidean space which avoids a set (not necessarily convex) polyhedral obstacles; we let n denoteExpand
A New Approach to Planar Point Location
• F. Preparata
• Mathematics, Computer Science
• SIAM J. Comput.
• 1981
This paper presents a practical algorithm which runs in less than \$6\lceil {\log _2 n} \rceil \$ comparisons on a data structure which uses O(n\log n) storage, in the worst case. Expand
A note on two problems in connexion with graphs
• E. Dijkstra
• Mathematics, Computer Science
• Numerische Mathematik
• 1959
A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given. Expand
Optimal Search in Planar Subdivisions
This work presents a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely \$O(\log n)\$ search time with \$O(n)\$ storage. Expand