Computational Geometry with imprecise data

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Computational Geometry with imprecise data. Bob Fraser University of Manitoba fraser@cs.umanitoba.ca Ljubljana, Slovenia Oct. 29, 2013. Brief Bio Minimum Spanning Trees on Imprecise Data Other Research Interests * Approximation algorithms using disks*. Biography. Winnipeg. Vancouver.

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### Computational Geometry with imprecise data

Bob Fraser

University of Manitoba

fraser@cs.umanitoba.ca

Ljubljana, Slovenia

Oct. 29, 2013

Brief Bio

• Minimum Spanning Trees on Imprecise Data
• Other Research Interests
• *Approximation algorithms using disks*
Biography

Winnipeg

Vancouver

Sault Sainte Marie

Ottawa

Kingston

Waterloo

manitoba
• http://www.cs.umanitoba.ca/~compgeom/people.html
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?
Imprecise Data
• Traditionally in computational geometry, we assume that the input is precise.
• Abandoning this assumption, one must choose a model for the imprecision:

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°C

km/h

Let’s choose this one!

www.ccg-gcc.gc.ca

Max-MSTN is not these other things

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max-MSTN

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max-maxST

max-planar-maxST

tOday’s Results
• Parameterized algorithm for max-MSTN
• NP-hardness of MSTN
Parameterized Algorithms
• = separabilityof the instance
• min distance between any two disks

WAOA 2012

Parameterized max-MSTN Algorithm
• – factor approximation by choosing disk centres

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Tc

Tc’

Topt

Approximation algorithm:

Parameterized max-MSTN Algorithm
• – factor approximation by choosing disk centres

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Consider this edge

weight

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weight =

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Tc

Tc’

Topt

WAOA 2012

Hardness of MSTN

(with spinal path)

Reduce from planar 3-SAT

Need wires

e.g.

Hardness of MSTN

clause

(with spinal path)

Reduce from planar 3-SAT

• 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

variable

variable

variable

clause

clause

clause

variable

variable

Hardness of MSTN

Wires

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All wires are part of an optimal solution

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Only one wire from the clause gadget is connected to a variable gadget

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Hardness of MSTN

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Spinal Path

Spinal Path

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HARDNESS OF MSTN

Shortest path touching 2 disks

path weight

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unit distance

Hardness of MSTN

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“true” configuration

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Spinal Path

Spinal Path

Spinal Path

Spinal Path

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Hardness of MSTN

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• 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.

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Spinal Path

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Spinal Path

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Discrete Unit Disk cover

IJCGA 2012

DMAA 2010

WALCOM 2011

ISAAC 2009

• unit disks , points .
• Select a minimum subset of which covers .
Discrete Unit Disk cover

IJCGA 2012

DMAA 2010

WALCOM 2011

ISAAC 2009

• unit disks , points .
• Select a minimum subset of which covers .

OPEN: Add points to this plot!

CCCG 2012

Submitted to TCS

Within-Strip Discrete Unit Disk cover
• unit disks with centre points , points .
• Strip , defined by and , of height which contains and .

}

OPEN:Is there a nice PTAS for this problem?

CCCG 2010

Submitted to JoCG

The Hausdorff Core Problem
• Given a simple polygon P, a HausdorffCore of P is a convex polygon Q contained in P that minimizes the Hausdorff distance between P and Q.

OPEN: For what kinds of polygons is finding the Hausdorff Core easy?

CCCG 2013

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)

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OPEN: Design a data structure to quickly provide solutions to a query.

Submitted to LATIN 2014

Guarding Orthogonal Art Galleries with Sliding Cameras
• Choose axis aligned lines to guard the polygon:

OPEN: Is this problem (NP-) hard?

FWCG 2013

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)…

OPEN: What characterizes 2-admissible instances that can be dualized?

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!
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.
• StephaneDurocher, U. of Manitoba
• ArashFarzan, MPI fur Informatik
• OmritFiltser, Ben-Gurion U. of the Negev
• MengHe, Dalhouse U.
• FerranHurtado, U. Politecnica de Catalunya
• ShahinKamali, U. of Waterloo
• Akitoshi Kawamura, U. of Tokyo
• Alejandro López-Ortiz, U. of Waterloo
• Ali Mehrabi, Eindhoven U. of Tech.
• SaeedMehrabi, U. of Manitoba
• DebajyotiMondal, 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.

Computational Geometry with imprecise data

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Thanks!

Bob Fraser

fraser@cs.umanitoba.ca

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ISAAC 2013

4-Sector of Two Points

3-sector:

OPEN: Is the solution unique if P and Q are not points?