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Bob Fraser University of Manitoba Ljubljana, Slovenia Oct. 29, 2013 COMPUTATIONAL GEOMETRY WITH IMPRECISE DATA

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

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BIOGRAPHY 3 Sault Sainte Marie Ottawa Vancouver Kingston Waterloo Winnipeg

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MANITOBA 4

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RESEARCH 5

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MINIMUM SPANNING TREE ON IMPRECISE DATA 6 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?

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IMPRECISE DATA 7 Traditionally in computational geometry, we assume that the input is precise. Abandoning this assumption, one must choose a model for the imprecision:.... Lets choose this one! °C km/h

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... MST – MINIMUM SPANNING TREE

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(MIN WEIGHT) MST WITH NEIGHBORHOODS MSTN 9 WAOA 2012, Invited to TOCS special issue Steiner Points......

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MAX WEIGHT MST WITH NEIGHBORHOODS max-MSTN 10 WAOA 2012

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MAX-MSTN IS NOT THESE OTHER THINGS max-MSTN max-maxST max-planar-maxST 11

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TODAYS RESULTS Parameterized algorithm for max-MSTN NP-hardness of MSTN 12

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PARAMETERIZED ALGORITHMS 13

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PARAMETERIZED MAX-MSTN ALGORITHM T opt TcTc T c 14 WAOA 2012

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PARAMETERIZED MAX-MSTN ALGORITHM T opt TcTc T c Consider this edge 15

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HARDNESS OF MSTN Reduce from planar 3-SAT (with spinal path) Need variable gadgets Need clause gadgets Need wires 16 WAOA 2012

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HARDNESS OF MSTN Reduce from planar 3-SAT clause variable clause 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 17

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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 18

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HARDNESS OF MSTN.. Spinal Path..... Variable Gadget 19

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

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HARDNESS OF MSTN Variable Gadget.. Spinal Path true configuration 21

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HARDNESS OF MSTN 22

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HARDNESS OF MSTN 23

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. HARDNESS OF MSTN To variable gadgets Spinal Path... 24

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OTHER RESEARCH 25

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DISCRETE UNIT DISK COVER 26 IJCGA 2012 DMAA 2010 WALCOM 2011 ISAAC 2009

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DISCRETE UNIT DISK COVER 27 IJCGA 2012 DMAA 2010 WALCOM 2011 ISAAC 2009 OPEN: Add points to this plot!

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WITHIN-STRIP DISCRETE UNIT DISK COVER 28 CCCG 2012 Submitted to TCS } OPEN: Is there a nice PTAS for this problem?

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THE HAUSDORFF CORE PROBLEM 29 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 2009 CCCG 2010 Submitted to JoCG OPEN: For what kinds of polygons is finding the Hausdorff Core easy?

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K -ENCLOSING OBJECTS IN A COLOURED POINT SET 30 Given a coloured point set and a query c =(c 1,…,c t ). 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.

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GUARDING ORTHOGONAL ART GALLERIES WITH SLIDING CAMERAS 31 Choose axis aligned lines to guard the polygon: Submitted to LATIN 2014 OPEN: Is this problem (NP-) hard?

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GEOMETRIC DUALITY FOR SET COVER AND HITTING SET PROBLEMS 32 Dualizing unit disks is beautiful! FWCG 2013

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GEOMETRIC DUALITY FOR SET COVER AND HITTING SET PROBLEMS 33 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 34 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 35 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.

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COMPUTATIONAL GEOMETRY WITH IMPRECISE DATA Thanks! Bob Fraser 36

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4-SECTOR OF TWO POINTS 37 ISAAC sector: OPEN: Is the solution unique if P and Q are not points?

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