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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

Computational Geometry with imprecise data

Bob Fraser

University of Manitoba

fraser@cs.umanitoba.ca

Ljubljana, Slovenia

Oct. 29, 2013

slide2

Brief Bio

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

Winnipeg

Vancouver

Sault Sainte Marie

Ottawa

Kingston

Waterloo

manitoba
manitoba
  • http://www.cs.umanitoba.ca/~compgeom/people.html
minimum spanning tree on imprecise data
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
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
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
tOday’s Results
  • Parameterized algorithm for max-MSTN
  • NP-hardness of MSTN
parameterized algorithms
Parameterized Algorithms
  • = separabilityof the instance
    • min distance between any two disks
parameterized max mstn algorithm

WAOA 2012

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

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Tc

Tc’

Topt

Approximation algorithm:

parameterized max mstn algorithm1
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

hardness of mstn

WAOA 2012

Hardness of MSTN

Need clause gadgets

(with spinal path)

Reduce from planar 3-SAT

Need wires

Need variable gadgets

e.g.

hardness of mstn1
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 mstn2
Hardness of MSTN

Wires

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Clause gadget

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To variable gadgets

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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 mstn3
Hardness of MSTN

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Variable Gadget

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Spinal Path

Spinal Path

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hardness of mstn4
HARDNESS OF MSTN

Shortest path touching 2 disks

path weight

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unit distance

hardness of mstn5
Hardness of MSTN

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Variable Gadget

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“true” configuration

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Spinal Path

Spinal Path

Spinal Path

Spinal Path

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hardness of mstn8
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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To variable gadgets

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Spinal Path

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Spinal Path

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discrete unit disk cover
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 cover1
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!

within strip discrete unit disk cover

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?

the hausdorff core problem

WADS 2009

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?

k enclosing objects in a coloured point set

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)

How about c=(0,1,3)?

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

guarding orthogonal art galleries with sliding cameras

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?

geometric duality for set cover and hitting set problems1

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
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
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 data1
Computational Geometry with imprecise data

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Thanks!

Bob Fraser

fraser@cs.umanitoba.ca

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4 sector of two points

ISAAC 2013

4-Sector of Two Points

3-sector:

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