Computational Geometry with imprecise data

Bob Fraser

University of Manitoba

[email protected]

Ljubljana, Slovenia

Oct. 29, 2013

COMPUTATIONAL GEOMETRY WITH IMPRECISE DATA

• Brief Bio

• Minimum Spanning Trees on Imprecise Data

• Other Research Interests

• *Approximation algorithms using disks*

BIOGRAPHY

Sault Sainte MarieOttawa

Vancouver

KingstonWaterloo

Winnipeg

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MANITOBA• http://www.cs.umanitoba.ca/~compgeom/people.html

RESEARCH

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?

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

...

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

MAX-MSTN IS NOT THESE OTHER THINGS

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max-MSTN

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...

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max-maxST

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max-planar-maxST

TODAY’S RESULTS

• Parameterized algorithm for max-MSTN• NP-hardness of MSTN

PARAMETERIZED ALGORITHMS

• = separability of the instance

• min distance between any two disks

𝑟𝑚

𝑟𝑚4, so𝑘=0.25

PARAMETERIZED MAX-MSTN ALGORITHM

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• – factor approximation by choosing disk centres

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Topt Tc Tc’

Approximation algorithm:

WAOA 2012

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

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

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

HARDNESS OF MSTN

Wires

. .. . . . . . . . . . . . . . . . . . . . . . ..

..

. . .. . .

....

.....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

HARDNESS OF MSTN

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Spinal Path

−

+¿

..

Spinal Path

−

+¿

.𝐵 ¿.𝐴(𝑥𝑖−)

.𝐶 ¿

Variable Gadget

HARDNESS OF MSTN

Shortest path touching 2 disks

unit distance

.path weight

HARDNESS OF MSTNVariable Gadget

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Spinal Path

−

+¿

..

Spinal Path

−

+¿

.𝐵 ¿.𝐴(𝑥𝑖−)

.𝐶 ¿

......

.

......

...

. .. .......

...

.

.

Spinal PathSpinal Path

− −

+¿ +¿

𝐵 ¿𝐴(𝑥𝑖−)

𝐶 ¿

“true” configuration

HARDNESS OF MSTN

(𝑥1 , 𝑥2 ,𝑥3)

(𝑥2 , 𝑥3 , 𝑥5) (𝑥2 , 𝑥4 ,𝑥5)

(𝑥2, 𝑥4 ,𝑥5)

𝑥2

𝑥3

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.

. . .. . .

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..... To variable gadgets

......

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......

...

. .. .......

....

Spinal Path

− −

+¿ +¿

𝐵 ¿𝐴(𝑥𝑖−)

...𝐵 ¿

OTHER RESEARCH

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!

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?

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?

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.

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?

GEOMETRIC DUALITY FOR SET COVER AND HITTING SET PROBLEMS

• Dualizing unit disks is beautiful!

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)…

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.

COMPUTATIONAL GEOMETRY WITH IMPRECISE DATA

Thanks!

Bob Fraser

[email protected]

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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?