3D Shortest Paths

1
3D Shortest Paths

• Lying on Polyhdral
• Surfaces

Reported by Shiqing Xin 2006-04-26
2
Problem Description

• Source and destination
• Lying on the surface
• Classification
• One source, one destination
• One source, any destination
• Any source, any destination
• One source, many destinations

3
Examples
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Methods Classification

• Exact algorithms
• Approximate algorithms
• Controlled by an error bound
• Relying on practical experience
• Theorital algorithms
• Conceptial extension of computational geometry.
  For example, Polthier and Schmies, 1998.

5
Exact algorithms

• Continuous Dijkstras algorithm
• Mitchell et al., 1987, SIAM J. Comput.
• Building a sequence tree
• Chen and Han, 1990, SCG '90
• Wavefront propagation
• Sanjiv Kapoor, 1999, STOC '99

6
Approximate algorithms

• Based on theory in geometry
• Hershberger, Suri, 1998, Com. Geo.
• S. Har-Peled, 1999, Dis. Com. Geo.
• Converting into 2D problem
• Varadarajan, Agarwal, 1997
• Aleksandrov et al., 2003, FCT
• S. Har-Peled, 1999, SIAM, J. Com

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Constructing Approximate Shortest Pahs Maps
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Fast marching method

• Theory Base
• Process

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Algorithm Implementation

• Lanthier et al., 1997, ACM
• Kaneva, O'Rourke, 2000, Proc. of the 12th
  Canadian Conference on Computational Geometry
• Surazhsky et al., 2005, ACM
• Common conclusion
• Exact algorithms are space-consuming

10
My current work

• Improve Chen Hans algorithm
• Implement CH algorithm fully
• Make comparison between them
• Apply ICH into LESP
• Make comparison between MMP and the improved CH
  algorithm
• Segment polyhedral surface and find shortest
  paths step by step
• Heuristically compute leve by level
• Find LESP with mountain climbing