thick non crossing paths in polygonal domain joint work with n. mirehii shahid beheshti uni. 1 thick...

1

Thick non crossing pathsIn polygonal domain

Joint work with N. Mirehii

Shahid Beheshti Uni. Thick path planning - M.Tahmasbi

2Shahid Beheshti Uni. Thick path planning - M.Tahmasbi

3

Motivatin

• Our particular motivation for the problem comes from ATM applications in routing safe lanes (“flows”) of air tra c through sectors or “flow fficonstrained areas”(FCAs) while avoiding certain constraints –hazardous weather systems, no-fly zones, regions of congestion, etc. Each lane is a thick path, determined by the protected air space zone (PAZ) that specifies the horizontal separation standard for flights.


Airplane routes

• The path for each airplane is a thick path

Shahid Beheshti Uni. Thick path planning - M.Tahmasbi 4

5

Other applications

• VLSI rouing and design• Fat edge graph drawing• Sensor networks


6

Modeling

• We model the airspace as a simple polygon,

hazardous weather systems or no-fly zones as

polygonal holes inside it.

• The resulting is called a polygonal domain.

• The origins and destinations as points on the

boundary of the polygonal domain.


7

Modeling

• A thick path is the minkowski sum of a path

with a unit disk.

• Each airplane route is a thick path.

• Given k pairs of terminals on a polygonal

domain, the problem is to find k thick non

crossing paths between pairs of terminals.


8

An example

• The objective is to find k paths such that the total length is minimum.


9

History

• The problem is NP- hard even for k=2 (Kim et.al)

• (Thin) path problem is NP-hard (1998, Bastert)

• Minimizing the length of the longest path is NP-hard (2007, polishuk)


10

Feasibility

• Mapping on a cycle


K+1 regions

(K+1)h Threadings for h holes

11

The main idea for simple polygon without holes

• Inflate the boundary of the polygon by 1 unit.• For each path, route the shortest path leaving

enough space for other fat paths.


13

Bridging to remove holes

• A bridge is a shortest path from a hole to a point on the boundary of the polygon.


14

The main idea for simple polygon with holes

• Specify the threading,• Bridge the holes,• Change the problem to

polygons without holes,• Route the path• Repeat for other pairs

of terminals.


15

Time complexity


Bridging time# of pathsTotal complexity of the polygonal domain

# of holes

Michell & Polischuk

16

Comparing the results


17

Terminals on hole boundaries


18

Inflating the holes


19

Finding feasible paths


Bridging time

# of pathsTotal complexity of the polygonal domain

# of holes with terminal# of holes

without terminal

20

Solving the problem with GA

• Why GA:

We need to consider all threadings and find the one that routes shorter paths.

Finding paths need specifying a threading, so, local approaches do not work!


21


• Thick paths:

• We use MathLab• Evaluation function is total length of the paths• Initial paths are generated


22

• Find a random placement of the holes and random order of the pairs of terminals.

• Bridge the holes with shortest path algorithm.• Inflate the boundary of the polygon.• Find a set of paths using visibility graph.



23

Finging paths – visibility graph


24

New generation:

• 20 % mutation• 70 % cross over• 10% best answers from last generation

• In mutation, random number of paths are inherited from the first, and the rest are inherited from the second parent.


Threading

Threading:above/below holes

For every threadings

For k = 1…K

Inflate correspondingly

Shortest reference path

of given threading

25Thick path planning - M.TahmasbiShahid Beheshti Uni.

Thick path planning - M.Tahmasbi 26Shahid Beheshti Uni.


5 hours, 42 minutes 34 seconds

•E. M. Arkin, J. S. B. Mitchell, and V. Polishchuk, Maximum thick paths in static and dynamic environments, Computational Geometry Theory and Applications, (2010) 43(3): 279–294.

•O. Bastert and S. P. Fekete, Geometric wire routing, Technical Report 332, Zentrum für Angewandte Informatik (1998).

•J. Kim, J. S. B. Mitchell, V. Polishchuk, S. Yang, J. Zou, Routing multi-class traffic flows in the plane. Computational Geometry: Theory and Applications, (2012) 45(3): 99-114.

Thick path planning - M.Tahmasbi 32

Main references

Shahid Beheshti Uni.

•D. T. Lee, Non crossing path problems, Manuscript ,Dept. of EECS, Northwestern University (1991)

•J.S.B. Mitchell, V. Polishchuk,Thick non-crossing paths and minimum-cost flows in polygonal domains, 23rd ACM Symposium on Computational Geometry, (2007) pp. 56–65.

•M.Tahmasbi, N.Mirehi, Thick non-crossing paths in a polygon with one hole, Proceedings of 3nd Contemporary Issues in computer and Information Science 2012.

Thick path planning - M.Tahmasbi 33

Main references

Shahid Beheshti Uni.

34

Thank you


thick non crossing paths in polygonal domain joint work with n. mirehii shahid beheshti uni. 1 thick...

Documents

thick path planning

thick path shahid beheshti

path problem

tahmasbi slide

shortest path

longest path

holes shahid beheshti

mirehii shahid beheshti