shortest descending paths on polyhedral terrainalubiw/cs860/lecture10-class.pdfcs 860 fall 2014...
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CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
[HTML] Approximation algorithms for shortest descending paths in terrainsM Ahmed, S Das, S Lodha, A Lubiw… - … of Discrete Algorithms, 2010 - ElsevierFrom: http://scholar.google.ca/scholar?hl=en&q=Approximation+algorithms+for+shortest+descending+paths+in+terrains&btnG=&as_sdt=1%2C5&as_sdtp=
Shortest descending paths: Towards an exact algorithmM Ahmed, A Lubiw - International Journal of Computational …, 2011 - World ScientificFrom: http://scholar.google.ca/scholar?q=Shortest+descending+paths%3A+Towards+an+exact+algorithm&btnG=&hl=en&as_sdt=2005&sciodt=0%2C5&cites=15807659110083868733&scipsc=
Shortest descending paths on polyhedral terrain
in P or NP-hard? OPEN
CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
Shortest descending paths on polyhedral terrain
CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
Shortest descending paths on polyhedral terrain
unfold faces
shortest descending path is not straight in unfolding
CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
Shortest descending paths on polyhedral terrain
locally shortest path structure
s
shortest path (straight)contour line path
CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
Shortest descending paths on polyhedral terrain
locally shortest path structure
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jai
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mi sin ai + lij = mj sin bjgeneralized Snell’s Law
the first direction choice determines a locally shortest path
can we use Chen and Han?
CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
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Recall Chen and Han, Shortest paths on polyhedron surface.
CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
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Recall Chen and Han, Shortest paths on polyhedron surface.
expand all cones
trim using one-vertex one-cut
CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
Shortest descending paths on polyhedral terrain
Can we use Chan and Han?
• trace a locally shortest path
• trim the tree (one-vertex one-split)
• find path to v through unfolded face sequence
S
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CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
Shortest descending paths on polyhedral terrain
positive results for related problems (weighted region): approximation algorithms
On finding approximate optimal paths in weighted regionsZ Sun, JH Reif - Journal of Algorithms, 2006 - ElsevierFrom: http://scholar.google.ca/scholar?q=On+finding+approximate+optimal+paths+in+weighted+regions&btnG=&hl=en&as_sdt=0%2C5
Determining approximate shortest paths on weighted polyhedral surfacesL Aleksandrov, A Maheshwari, JR Sack - Journal of the ACM (JACM), 2005 - dl.acm.orgFrom: http://scholar.google.ca/scholar?q=Determining+approximate+shortest+paths+on+weighted+polyhedral+surfaces&btnG=&hl=en&as_sdt=0%2C5
add many Steiner points,use graph shortest path
CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
Shortest descending paths on polyhedral terrain
Approximation
Shortest Descending Paths on Terrains
Our Steiner point approach
place St. points uniformly or in geom. progression
Run time (as with other St. approx algs (not Papadimitriou’s)) depends on geometric parameters
when we add a Steiner point, add points on all edges at that height
place points uniformly on edges, or in geometric progression — run time will depend on geometric parameters
CS 860 Fall 2014 Anna Lubiw, U. WaterlooLecture 10 notes
Approximate shortest descending pathsSW Cheng, J Jin - SIAM Journal on Computing, 2014 - SIAMFrom: http://scholar.google.ca/scholar?q=Approximate+shortest+descending+paths&btnG=&hl=en&as_sdt=2005&sciodt=0%2C5&cites=15807659110083868733&scipsc=
possible paper to present (mentioned earlier)