shortest descending paths on polyhedral terrainalubiw/cs860/lecture10-class.pdfcs 860 fall 2014...

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



unfold faces

shortest descending path is not straight in unfolding



locally shortest path structure

s

shortest path (straight)contour line path



locally shortest path structure

i

jai

bj

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?


I

e

ProjIe

I

Shadow of

I

e1

e2

e3

e1

e2

e3

Proj Ie1

Proj Ie1

Proj Ie1

Proj Ie2

Proj Ie3

Proj Ie2

I

e

ProjIe

I

Shadow of

I

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e2

e3

e1

e2

e3

Proj Ie1

Proj Ie1

Proj Ie1

Proj Ie2

Proj Ie3

Proj Ie2

Recall Chen and Han, Shortest paths on polyhedron surface.


1 0

18

1

1811

1

11

15

1

15

5

1

5

8

1

8

6

1

6

16

1

1600

16

17

16

6

14

6

8

13

8

5

4

5

15

10

15

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3

1118

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12

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10

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4

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613

14

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16

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Recall Chen and Han, Shortest paths on polyhedron surface.

expand all cones

trim using one-vertex one-cut



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

T

T

T



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



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


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)

shortest descending paths on polyhedral terrainalubiw/cs860/lecture10-class.pdfcs 860 fall 2014...

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