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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. 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

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  1. Computational Geometry with imprecise data Bob Fraser University of Manitoba fraser@cs.umanitoba.ca Ljubljana, Slovenia Oct. 29, 2013

  2. Brief Bio • Minimum Spanning Trees on Imprecise Data • Other Research Interests • *Approximation algorithms using disks*

  3. Biography Winnipeg Vancouver Sault Sainte Marie Ottawa Kingston Waterloo

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

  5. research

  6. 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?

  7. Imprecise Data • Traditionally in computational geometry, we assume that the input is precise. • Abandoning this assumption, one must choose a model for the imprecision: . . . . °C km/h Let’s choose this one! www.ccg-gcc.gc.ca

  8. MST – Minimum Spanning Tree . . . . . . . .

  9. WAOA 2012, Invited to TOCS special issue (Min weight) MST with neighborhoods . . Steiner Points . . . . . . . . . . . . . . . . . MSTN . . .

  10. WAOA 2012 Max weight MST with NEIGHBORHOODS . . . . . . max-MSTN .

  11. Max-MSTN is not these other things . . . . . . . . . . . . . . . . . . . max-MSTN . . max-maxST max-planar-maxST

  12. tOday’s Results • Parameterized algorithm for max-MSTN • NP-hardness of MSTN

  13. Parameterized Algorithms • = separabilityof the instance • min distance between any two disks

  14. WAOA 2012 Parameterized max-MSTN Algorithm • – factor approximation by choosing disk centres . . . . . . . . . . . . . . . . . . . . . . . . Tc Tc’ Topt Approximation algorithm:

  15. Parameterized max-MSTN Algorithm • – factor approximation by choosing disk centres . . . Consider this edge weight . . weight = . . . . . . . . . . . . . . . . . . . Tc Tc’ Topt

  16. WAOA 2012 Hardness of MSTN Need clause gadgets (with spinal path) Reduce from planar 3-SAT Need wires Need variable gadgets e.g.

  17. 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

  18. 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 . . .

  19. Hardness of MSTN . Variable Gadget . . . . . Spinal Path Spinal Path .

  20. HARDNESS OF MSTN Shortest path touching 2 disks path weight . unit distance

  21. Hardness of MSTN . . Variable Gadget . . . . . . . . . . “true” configuration . . . . . . . . . . . . . . . . . . . . . . . . Spinal Path Spinal Path Spinal Path Spinal Path . .

  22. Hardness of MSTN

  23. Hardness of MSTN

  24. 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 . Spinal Path . .

  25. Other research

  26. Discrete Unit Disk cover IJCGA 2012 DMAA 2010 WALCOM 2011 ISAAC 2009 • unit disks , points . • Select a minimum subset of which covers .

  27. 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!

  28. 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?

  29. 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?

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

  31. 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?

  32. FWCG 2013 Geometric Duality for Set Cover and Hitting Set Problems • Dualizing unit disks is beautiful!

  33. 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?

  34. 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!

  35. 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.

  36. Computational Geometry with imprecise data . Thanks! Bob Fraser fraser@cs.umanitoba.ca . . . . . .

  37. ISAAC 2013 4-Sector of Two Points 3-sector: OPEN: Is the solution unique if P and Q are not points?

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