Computational Geometry with imprecise data
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Transcript of Computational Geometry with imprecise data
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Bob Fraser
University of Manitoba
Ljubljana, Slovenia
Oct. 29, 2013
COMPUTATIONAL GEOMETRY WITH IMPRECISE DATA
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• Brief Bio
• Minimum Spanning Trees on Imprecise Data
• Other Research Interests
• *Approximation algorithms using disks*
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BIOGRAPHY
Sault Sainte MarieOttawa
Vancouver
KingstonWaterloo
Winnipeg
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MANITOBA• http://www.cs.umanitoba.ca/~compgeom/people.html
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RESEARCH
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MINIMUM SPANNING TREE ON IMPRECISE DATA
• What is imprecise data?
• What does it mean to solve problems in this setting?
• Given data imprecision modelled with disks, how well can the minimum spanning tree problem be solved?
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www.ccg-gcc.gc.ca
IMPRECISE DATA
• Traditionally in computational geometry, we assume that the input is precise.
• Abandoning this assumption, one must choose a model for the imprecision:
. . ..
Let’s choose this one!
°C
km/h
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MST – MINIMUM SPANNING TREE
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(MIN WEIGHT) MST WITH NEIGHBORHOODS
...
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. ..MSTN
WAOA 2012, Invited to TOCS special issue
Steiner Points
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MAX WEIGHT MST WITH NEIGHBORHOODS
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max-MSTN
WAOA 2012
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MAX-MSTN IS NOT THESE OTHER THINGS
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max-MSTN
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max-maxST
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max-planar-maxST
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TODAY’S RESULTS
• Parameterized algorithm for max-MSTN• NP-hardness of MSTN
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PARAMETERIZED ALGORITHMS
• = separability of the instance
• min distance between any two disks
𝑟𝑚
𝑟𝑚4, so𝑘=0.25
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PARAMETERIZED MAX-MSTN ALGORITHM
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• – factor approximation by choosing disk centres
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. .
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Topt Tc Tc’
Approximation algorithm:
WAOA 2012
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PARAMETERIZED MAX-MSTN ALGORITHM
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..
• – factor approximation by choosing disk centres
..
. ..
..
. .
. ..
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Topt Tc Tc’
Consider this edge
𝑟 𝑖
𝑟 𝑗
weight = weight
𝑑+𝑟 𝑖+𝑟 𝑗
𝑑+2𝑟 𝑖+2𝑟 𝑗≥…¿
𝑘+2𝑘+4¿1−
2𝑘+4
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HARDNESS OF MSTN
Reduce from planar 3-SAT
(𝑥1 , 𝑥2 ,𝑥3)
(𝑥2 , 𝑥3 , 𝑥5) (𝑥2 , 𝑥4 ,𝑥5)
(𝑥2, 𝑥4 ,𝑥5)
𝑥2
𝑥3
(with spinal path)
Need variable gadgets
Need clause gadgets
Need wires
e.g.
WAOA 2012
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HARDNESS OF MSTN
Reduce from planar 3-SAT
clause
variable
clause clause
clause
variable
variablevariable
variable
(with spinal path)
Create instance of MSTN so that:- Clause gadgets join to only one variable- Weight of optimal solution for a
satisfiable instance may be precomputed- Weight of solution corresponding to a
non-satisfiable instance is greater than a satisfiable one by a significant amount
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HARDNESS OF MSTN
Wires
. .. . . . . . . . . . . . . . . . . . . . . . ..
..
. . .. . .
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.....Clause gadget To variable gadgets
All wires are part of an optimal solution
Only one wire from the clause gadget is connected to a variable gadget
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HARDNESS OF MSTN
..
Spinal Path
−
+¿
..
Spinal Path
−
+¿
.𝐵 ¿.𝐴(𝑥𝑖−)
.𝐶 ¿
Variable Gadget
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HARDNESS OF MSTN
Shortest path touching 2 disks
unit distance
.path weight
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HARDNESS OF MSTNVariable Gadget
..
Spinal Path
−
+¿
..
Spinal Path
−
+¿
.𝐵 ¿.𝐴(𝑥𝑖−)
.𝐶 ¿
......
.
......
...
. .. .......
...
.
.
Spinal PathSpinal Path
− −
+¿ +¿
𝐵 ¿𝐴(𝑥𝑖−)
𝐶 ¿
“true” configuration
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HARDNESS OF MSTN
(𝑥1 , 𝑥2 ,𝑥3)
(𝑥2 , 𝑥3 , 𝑥5) (𝑥2 , 𝑥4 ,𝑥5)
(𝑥2, 𝑥4 ,𝑥5)
𝑥2
𝑥3
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HARDNESS OF MSTN
𝑥2
𝑥3
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HARDNESS OF MSTN• Weight of an optimal solution:
• weight of all wires, including clause gadgets
• weight of joining to all but m pairs in variable gadgets
• weight of joining to m clause gadgets
• What if the instance of 3SAT is not satisfiable?
• At least one clause gadget is joined suboptimally.
. . .. . .
....
..... To variable gadgets
......
.
......
...
. .. .......
....
Spinal Path
− −
+¿ +¿
𝐵 ¿𝐴(𝑥𝑖−)
...𝐵 ¿
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OTHER RESEARCH
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DISCRETE UNIT DISK COVER
• unit disks , points .
• Select a minimum subset of which covers .
IJCGA 2012DMAA 2010WALCOM 2011ISAAC 2009
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DISCRETE UNIT DISK COVER
• unit disks , points .
• Select a minimum subset of which covers .
IJCGA 2012DMAA 2010WALCOM 2011ISAAC 2009
OPEN: Add points to this plot!
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WITHIN-STRIP DISCRETE UNIT DISK COVER
• unit disks with centre points , points .
• Strip , defined by and , of height which contains and .
CCCG 2012Submitted to TCS
𝑠h}
ℓ2
ℓ1
OPEN: Is there a nice PTAS for this problem?
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THE HAUSDORFF CORE PROBLEM• Given a simple polygon P, a Hausdorff Core of P is a convex polygon Q contained in
P that minimizes the Hausdorff distance between P and Q.
WADS 2009CCCG 2010 Submitted to JoCG
OPEN: For what kinds of polygons is finding the Hausdorff Core easy?
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K-ENCLOSING OBJECTS IN A COLOURED POINT SET• Given a coloured point set and a query c=(c1,…,ct).
• Does there exist an axis aligned rectangle containing a set of points satisfying the query exactly?
Say colours are (red,orange,grey)
c=(1,1,3)
How about c=(0,1,3)?
...
.. ..
. .
CCCG 2013
OPEN: Design a data structure to quickly provide solutions to a query.
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GUARDING ORTHOGONAL ART GALLERIES WITH SLIDING CAMERAS• Choose axis aligned lines to guard the polygon:
Submitted to LATIN 2014
OPEN: Is this problem (NP-) hard?
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GEOMETRIC DUALITY FOR SET COVER AND HITTING SET PROBLEMS
• Dualizing unit disks is beautiful!
FWCG 2013
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GEOMETRIC DUALITY FOR SET COVER AND HITTING SET PROBLEMS
• 2-admissibility: boundaries pairwise intersect at most twice.
• It seems like dualizing these sets should work (to me)…
FWCG 2013
OPEN: What characterizes 2-admissible instances that can be dualized?
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THE STORY
• Disks are useful for modelling imprecision, and they crop up in all sorts of problems in computational geometry.
• Disks may be used to model imprecise data if a precise location is unknown.
• Simple problems may become hard when imprecise data is a factor.
• There are lots of directions to go from here: new problems, new models of imprecision, and new applications!
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ACKNOWLEDGEMENTS
Collaborators on the discussed results• Luis Barba, Carleton U./U.L. Bruxelles
• Francisco Claude, U. of Waterloo
• Gautam K. Das, Indian Inst. of Tech. Guwahati
• Reza Dorrigiv, Dalhousie U.
• Stephane Durocher, U. of Manitoba
• Arash Farzan, MPI fur Informatik
• Omrit Filtser, Ben-Gurion U. of the Negev
• Meng He, Dalhouse U.
• Ferran Hurtado, U. Politecnica de Catalunya
• Shahin Kamali, U. of Waterloo
• Akitoshi Kawamura, U. of Tokyo
• Alejandro López-Ortiz, U. of Waterloo
• Ali Mehrabi, Eindhoven U. of Tech.
• Saeed Mehrabi, U. of Manitoba
• Debajyoti Mondal, U. of Manitoba
• Jason Morrison, U. of Manitoba
• J. Ian Munro, U. of Waterloo
• Patrick K. Nicholson, MPI fur Informatik
• Bradford G. Nickerson, U. of New Brunswick
• Alejandro Salinger, U. of Saarland
• Diego Seco, U. of Concepcion
• Matthew Skala, U. of Manitoba
• Mohammad Abdul Wahid, U. of Manitoba
Research supported by various grants from NSERC and the University of Waterloo.
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4-SECTOR OF TWO POINTS
ISAAC 2013
3-sector:
OPEN: Is the solution unique if P and Q are not points?