Optimal Point Removal in Closed-2PM Labeling

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Optimal Point Removal in Closed-

2PM Labeling

Farshad Rostamabadi , Iman Sadeghi ,Mohammad Ghodsi , Ramtin Khosravi

Information Processing Letters 105 (2008) 108113

Advisor: Yue-Li Wang

As presented by Ying-Jhih Chen

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Outline

Introduction 2PM labeling

Algorithm

Conflict graph

Adjacent graph

Constraint pair 3-Rule in algorithm

Example Remove point

Relabeling Conclusion

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2PM Labeling

disjoint axis-parallel equal-size squares maximum length

attached to its corresponding point on themiddle of one of its horizontal edges.

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Closed-2PM labeling

no intersect

Optimal closed-2PM labeling:

Not Optimality

Optimality!

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Algorithm

Initiation: Optimality Preprocessing Conflict graph

Adjacent graph

Point Removal Optimality Check

Update conflict graph and adjacent graph

Relabel

O(nlgn)

O(lgn)

O(lgn)

O(n)

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Example of Conflict Graph G

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Example of Adjacency Graph H

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: The length of label, isthe (same) length of all

labels.

Assume = 23

23

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0

8

0

0

Max{20,8}=20

20

Max{15,20}=20

0

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0

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20

15

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Constraint pair

Both are not relaxing label

relaxing label

Not relaxing label

Both are relaxing label

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Rule 1

Rule 1: e.g.1

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45

Fail!

Label 2, 5:

Relaxing label

Delete label 4

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Rule 2

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Rule 3

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23

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Rule 1

Rule 3

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Remove Example

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3 4

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765

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A

B

C

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3 4

5 76

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A

B

C

Not optimal, relabeling!

Optimal

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Relabeling

B

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C

Sweep line

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765

B

C

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Sweep Line Upward

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B

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6 Sweep line

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B

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765

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Relabeling Solution

76

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B

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3 4

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Sweep line6

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3 4

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765

B

C

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References [1] S. Doddi,M.V.Marathe, A.Mirzaian, B.M.E.Moret, B. Zhu, Map labeling and its

generalizations, in: Proc. 8th ACMSIAM Symposium on Discrete Algorithms(SODA97), 47 January 1997, pp. 148157.

[2] C. Iturriaga, Map labeling problems, PhD thesis, University of Waterloo, 1999.

[3] C.K. Poon, B. Zhu, F. Chin, A polynomial time solution for labeling a rectilinearmap, Information Processing Letters 65 (4) (1998) 201207.

[4] F. Rostamabadi, M. Ghodsi, A fast algorithm for updating a labeling to avoid amoving point, in: Proceeding of the 16th Canadian Conference on ComputationalGeometry (CCCG04), 2004, pp. 204208.

[5] F. Rostamabadi, M. Ghodsi, Label updating in 2pm to avoid a moving point, in: The21st EuropeanWorkshop on Computational Geometry (EWCG05), 2005, pp. 131134.

[6] F. Rostamabadi, M. Ghodsi, Incremental labeling in 2pm model, in: The 11thInternational CSI Computer Conference (CSICC06), 2006, pp. 913.

[7] F. Rostamabadi, M. Ghodsi, Label updating to avoid point-shaped obstacles in

fixed model, Theoretical Computer Science 369 (13) (2006) 197210. [8] T. Strijk, M. van Kreveld, Labeling a rectilinear map more efficiently, Information

Processing Letters 69 (1) (1999) 2530.

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Optimal Point Removal in Closed-2PM Labeling