COMPUTATIONAL GEOMETRY WITH IMPRECISE DATA

Transcription

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

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

3 BIOGRAPHY Vancouver Winnipeg Sault Sainte Marie Waterloo Ottawa Kingston 3

4 MANITOBA 4

5 RESEARCH 5

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? 6

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! wwwccg-gccgcca 7

8 MST MINIMUM SPANNING TREE 8

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

10 WAOA 2012 MAX WEIGHT MST WITH NEIGHBORHOODS max-mstn 10

11 MAX-MSTN IS NOT THESE OTHER THINGS max-mstn max-planar-maxst max-maxst 11

12 TODAY S RESULTS Parameterized algorithm for max-mstn NP-hardness of MSTN 12

13 PARAMETERIZED ALGORITHMS k = separability of the instance min distance between any two disks kr m r m 4, so k = 025 r m 13

14 WAOA 2012 PARAMETERIZED MAX-MSTN ALGORITHM 1 2 k+4 factor approximation by choosing disk centres T opt T c T c Approximation algorithm: w(t c ) < w(t opt ), w(t c ) > r w(t opt ) 14

15 PARAMETERIZED MAX-MSTN ALGORITHM 1 2 k+4 factor approximation by choosing disk centres weight = d + r i + r j Consider weight this d + edge 2r i + 2r j r i r j T opt T c T c d + r i + r j d + 2r i + 2r j = k + 2 k + 4 = 1 2 k

16 WAOA 2012 HARDNESS OF MSTN Need clause gadgets Reduce from planar 3-SAT (with spinal path) (x 2, x 4, x 5 ) x 1 x 2 Need wires x 4 (x 1, x 2, x 3 ) Need variable gadgets (x 2, x 3, x 5 ) (x 2, x 4, x 5 ) x 3 x 5 eg (x 1 x 2 x 3 ) (x 2 x 4 x 5 ) (x 2 x 3 x 5 ) (x 2 x 4 x 5 ) 16

17 HARDNESS OF MSTN Reduce from planar 3-SAT (with spinal path) clause Create instance of MSTN so that: - Clause variable gadgets join to only one variable - Weight of optimal solution for a variable satisfiable instance may be precomputed - Weight of solution corresponding to a non-satisfiable instance is greater than a satisfiable one clause by a significant amount clause variable clause variable variable 17

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 18

19 HARDNESS OF MSTN Variable Gadget B(x i + ) A(x i ) Spinal Path + C(x i + ) + Spinal Path 19

20 HARDNESS OF MSTN Shortest path touching 2 disks path weight 0755 unit distance 20

21 HARDNESS OF MSTN Variable Gadget true configuration B(x + i ) A(x i ) Spinal Path + + C(x + i ) Spinal Path 21

22 HARDNESS OF MSTN (x 2, x 4, x 5 ) x 1 x 2 x 4 (x 1, x 2, x 3 ) (x 2, x 3, x 5 ) (x 2, x 4, x 5 ) x 3 x 5 22

23 HARDNESS OF MSTN x 1 x 2 x 4 x 3 x 5 23

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 B(x + i ) of 3SAT is not satisfiable? At least one clause gadget is joined suboptimally A(x i ) To variable gadgets 24

25 OTHER RESEARCH 25

26 DISCRETE UNIT DISK COVER m unit disks D, n points P Select a minimum subset of D which covers P IJCGA 2012 DMAA 2010 WALCOM 2011 ISAAC

27 DISCRETE UNIT DISK COVER m unit disks D, n points P Select a minimum subset of D which covers P IJCGA 2012 DMAA 2010 WALCOM 2011 ISAAC 2009 OPEN: Add points to this plot! 27

28 WITHIN-STRIP DISCRETE UNIT DISK COVER m unit disks D with centre points Q, n points P Strip s, defined by l 1 and l 2, of height h which contains Q and P CCCG 2012 Submitted to TCS s l 1 } h l 2 OPEN: Is there a nice PTAS for this problem? 28

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

30 K-ENCLOSING OBJECTS IN A COLOURED POINT SET CCCG 2013 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)? OPEN: Design a data structure to quickly provide solutions to a query 30

31 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? 31

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

33 GEOMETRIC DUALITY FOR SET COVER AND HITTING SET PROBLEMS FWCG 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? 33

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! 34

35 ACKNOWLEDGEMENTS Collaborators on the discussed results Luis Barba, Carleton U/UL 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 35

36 COMPUTATIONAL GEOMETRY WITH IMPRECISE DATA Thanks! Bob Fraser 36

37 ISAAC SECTOR OF TWO POINTS 3-sector: OPEN: Is the solution unique if P and Q are not points? 37