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The Within-Strip Discrete Unit Disk Cover Problem Bob Fraser (joint work with Alex López-Ortiz) University of Waterloo CCCG Aug. 8, 2012

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Within-Strip Discrete Unit Disk Cover (WSDUDC) 2 }

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Applications 3

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Related Problems 4

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Strip-Separated DUDC 5 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8 [Ambühl et al., 2006]

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Within-Strip DUDC 6 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8

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Results 7 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8

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Hardness (1/4) Reduce from Vertex Cover on planar graphs of max degree 3. 8

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Convert to disk piercing problem: Adding extra points to edges: Making Wires 9................

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Hardness (2/4) Reduce from Vertex Cover on planar graphs of max degree 3. Remove cis-3 vertices 10

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Hardness (3/4) Add vertices at intersections with vertical lines: Consecutive arrays differ in size by at most one: 11

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Hardness (4/4) Each edge has even number of added points: Convert to instance of WSDUDP: 12

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Covering Gaps 14 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8

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Covering Intervals 15

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4-approximate algorithm 16

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4-approximate algorithm 17

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3-approximate algorithm 18

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3-approximate algorithm 19

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Summary 20

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Thanks! 21 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8

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Outline Within-Strip Discrete Unit Disk Cover (WSDUDC) Related Problems (DUDC, SSDUDC) Hardness of WSDUDC Approximating the Optimal Solution 22

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Strip-based Approach to DUDC 23 [Das et al., 2011]

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Trade-off in DUDC algorithms 24

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Applications 25........................... Image from http://www.marum.de/Method.html

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References C. Ambühl, T. Erlebach, M. Mihal’´ak, and M. Nunkesser. Constant factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. In APPROX, pages 3–14, 2006. G. Das, R. Fraser, A. Lopez-Ortiz, and B. Nickerson. On the discrete unit disk cover problem. In WALCOM: Algorithms and Computation, volume 6552 of LNCS, pages 146–157, 2011. D. S. Hochbaum and W. Maass. Approximation schemes for covering and packing problems inimage processing and VLSI. Journal of the ACM, 32:130–136, 1985. D. Johnson. The NP-completeness column: An ongoing guide. Journal of Algorithms, 3(2):182–195, 1982. N. Mustafa and S. Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44:883–895, 2010. 26

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