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

## Presentation on theme: "The Within-Strip Discrete Unit Disk Cover Problem Bob Fraser (joint work with Alex López-Ortiz) University of Waterloo CCCG Aug. 8, 2012."— Presentation transcript:

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

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

Applications 3

Related Problems 4

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

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

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

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

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

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

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

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

13

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

Covering Intervals 15

4-approximate algorithm 16

4-approximate algorithm 17

3-approximate algorithm 18

3-approximate algorithm 19

Summary 20

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

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

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