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3D Shortest Paths

3D Shortest Paths. Lying on Polyhdral Surfaces. Reported by Shiqing Xin 2006-04-26. 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.

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3D Shortest Paths

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  1. 3D Shortest Paths Lying on Polyhdral Surfaces Reportedby 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

  4. 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 Dijkstra’s 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

  7. Constructing Approximate Shortest Pahs Maps

  8. Fast marching method • Theory Base • Process

  9. 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 & Han’s 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

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