1 / 37

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.

dian
Download Presentation

Computational Geometry with imprecise data

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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?

More Related