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)

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unit disks with centre points , points . Strip , defined by and , of height which

contains and .Select a minimum subset of which

covers .

𝑠

h}ℓ2

ℓ1

Applications

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Related ProblemsDiscrete Unit Disk Cover: no strip

◦NP-hard (Johnson, 1982).◦PTAS (Mustafa & Ray, 2010).◦18-approximate, time algorithm

(Das et al., 2011).Minimum Geometric Disk Cover:

disk centres are not restricted.◦NP-hard, PTAS using shifting grids

(Hochbaum & Maass, 1985).

Strip-Separated DUDC

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p1p2

p4

p3

p5

q1 q2q4

q3

q5 q6 q7q9

q8

ℓ2

ℓ1

All points contained in strip, and disks centred outside strip.

Exact DP algorithm in time.

[Ambühl et al. , 2006]

Within-Strip DUDC All points and disk centres in strip. If easy, implies a simple PTAS for DUDC based on shifting grids… MAX independent set of unit disk graph is easy in strips of fixed height! Within-Strip Discrete Unit Square Cover is easy in strips of fixed height!

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p1p2

p4

p3

p5

q1

q2q4

q3

q5

q6

q7

q9q8

ℓ2

ℓ1

Results NP-hard for strips of any height -approximate algorithm for strips of height 4-approximate algorithm for strips of height 3-approximate algorithm for strips of height

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p1p2

p4

p3

p5

q1

q2q4

q3

q5

q6

q7

q9q8

ℓ2

ℓ1

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Hardness (1/4) Reduce from Vertex Cover on

planar graphs of max degree 3.

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Convert to disk piercing problem:

Adding extra points to edges:

Making Wires

. ..

. . . . .

. . .

. . ...

𝑝1 𝑝2 𝑝3

𝑝4𝑝5𝑝1 𝑝2 𝑝3

𝑝1 𝑝2 𝑝3

𝑝4 𝑝5𝑝1 𝑝2 𝑝3

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Hardness (2/4) Reduce from Vertex Cover on

planar graphs of max degree 3.

Remove cis-3 vertices

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Hardness (3/4)Add vertices at intersections with

vertical lines:

Consecutive arrays differ in size by at most one:

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Hardness (4/4)Each edge has even number of

added points:

Convert to instance of WSDUDP:

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-approximate algorithmFind the rectangle of height

circumscribed by each disk, then take the intersection with the strip:

Rectangles define intervals, the remainder are gaps.

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Covering GapsAll disk centres are

outside of the gap. Instance of SSDUDC, so

use time algorithm to cover gap optimally.

Optimality is lost when combining solutions from multiple gaps.

p1p2

p4

p3

p5

q1 q2q4

q3

q5 q6 q7q9

q8

ℓ2

ℓ1

ℓ2

ℓ1

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Covering IntervalsSimply add rectangles to the solution set

greedily.Approximation factor depends on the width

of the rectangles (and hence the height of the strip).

Produces factor approximation for WSDUDC. Running time dominated by SSDUDC, time.

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4-approximate algorithmCheck for simple redundancy in a

strip of height Rectangles have width If any two consecutive rectangles in

the solution set may be replaced by a single disk, then do so.

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4-approximate algorithmAny consecutive disks in solution now

require two disks to cover the same set of points.

One disk covers the points of at most 4 rectangles in the solution.

2-approximation for covering intervals!A disk can only be used to cover points

in 2 gaps (rectangles are wide).The running time of the check is , so

the running time is the same as previously.

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3-approximate algorithmApplies to strips where height An optimal solution for covering

the intervals has mutually spanning sets of maximum degree 3:

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3-approximate algorithmConsider all single disks, as well

as pairs and triples.Run a dynamic program from left

to right on the strip, finding the minimum weight set which covers all points to the left of the current disks under consideration.

Runs in time.

Summary

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WSDUDC is NP-hard.A general -approximate algorithm

exists for strips of height , running in time.

4-approximate algorithm exists for strips of height , running in time.

3-approximate algorithm exists for strips of height , running in time.

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Thanks!

p1p2

p4

p3

p5

q1

q2q4

q3

q5

q6

q7

q9q8

ℓ2

ℓ1

OutlineWithin-Strip Discrete Unit Disk

Cover (WSDUDC)

Related Problems (DUDC, SSDUDC)

Hardness of WSDUDC

Approximating the Optimal Solution 22

Strip-based Approach to DUDC

The approximation ratio was improved to 18.

6-approximation for within-strip problem.12-approximation for outside-strip

problem.Running time improved to .Recent advances in WSDUDC improve

the approximation factor to 15.23

1

√2

[Das et al., 2011]

Trade-off in DUDC algorithms

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Applications

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

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.

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