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p -Piercing Problems

p -Piercing Problems. Liao Chung-Shou. 2001. 8.29. “On Piercing Sets of Axis-Parallel Rectangles and Rings” , Michael Segal, International Journal of Computational Geometry and Applications, vol 9(3), ( 1999 ), 219-233. “Rectilinear and Polygonal p-Piercing and p-Center Problems” ,

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p -Piercing Problems

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  1. p-Piercing Problems Liao Chung-Shou 2001. 8.29

  2. “On Piercing Sets of Axis-Parallel Rectangles and Rings”, Michael Segal, International Journal of Computational Geometry and Applications, vol 9(3), (1999), 219-233. “Rectilinear and Polygonal p-Piercing and p-Center Problems”, Micha Sharir, Emo Welzl, Proceeding 12th ACM Symposium on Computational Geometry, (1996), 122-132. “Obnoxious Facility Location: Complete Service with Minimal Harm”, Michael Segal,…etc, International Journal of Computational Geometry and Applications, vol 10(6), (2000), 219-233.

  3. Topic • Introduction • The latest results • Axis-transformation way • Location domain way • My conception • Future work and open problems

  4. Introduction Given a collection of axis-parallel rectangles in the plane. Determine whether there exists a set of p points whose union intersects all the given rectangles. (forfixed p) * p = 3 * * related problems : p-center, piercing rings, piercing squares,…

  5. 1996 1999 p = 2,3 => O(n) time p = 4 => O(nlogn) time p = 5 => O(nlogn) time p 6 => O(n logn) time p = 2,3 => O(n) time p = 4 => O(nlog n) time p = 5 => O(nlog n) time p 6 => O(n log n) time 3 4 p-4 p-4 5 The latest results

  6. y C B centroid ( cx , cy ) * A D x dy dx D * B * * B C D A A * * * * cy cx Axis-transformation way dy * dx ( cx , cy , dx , dy ) C

  7. * * * Cone * * p * * * * * * * query point p ( Px , 0 , Py , 0 ) p = 1 x-cone cover all y-cone cover all

  8. C2 C1 C4 C3 * * * * * * * * * * * * X Y p = 2 first, second, (C1  C3)  (C2  C4) cover all rectangles. or (C1  C4)  (C2  C3) cover all rectangles.

  9. p = 3 C1 , C2 , C3 cover all in X-axis. C4 , C5 , C6 cover all in Y-axis. first, assume C1, C3, and C4, C6 are constrained cones (since there exists at least one constrained-cones pair) second, 4 possibilities => C1C4, C1C6, C3C4, C3C6. the rest of rectangles => 2-piercing

  10. p = 4 first, C1 , C2 , C3 , C4 cover all in X-axis. C5 , C6 , C7 , C8 cover all in Y-axis. assume C1, C4, and C5, C8 are constrained cones second, similar to p = 3 find a pair Ci  Cj such that the rest of rectangles are 3-piercable, where i {1,2,3,4} and j {5,6,7,8}. (a) i {1,4} and j {5,8} (b) i {2,3} and j {6,7} imply i' {1,4} and j' {5,8} (c) each constrained cone maps to an unconstrained cone (Worst)

  11. C3 before C1 C4 * * * * * * * * * * * * C3 after insert or delete 6 combinations => independent each need O(n) steps each step need O(logn) updates => O(nlogn) time

  12. T B L R Location domain way if an axis-parallel line traverses all rectangles else * * Observation * *

  13. p = 2 2-piercable => diagonal pair p = 3 3-piercable => at least one vertex in corner 4 possibilities the rest of rectangles => similar to 2-piercing way

  14. => O(log n) time 3 3 => O(nlog n) time p = 4 try each intersection part => O(n) time the rest of rectangles => 3-piercable

  15. p = 5 case1. there exists some point in corner 4 possibilities the rest of rectangles => 4-piercable case2. there exist 5 points in boundary (not in any corner)

  16. case3. there exist 4 points in boundary (not in any corner), and 1 point inside. case2,3 O(logn) time However, it can’t work as p = 6.

  17. T T0 My conception(as p = 4) Step1. pre-check whether there exists one point in corner. 4 possibilities, and the rest of rectangles => 3-piercable. Step2. there exists 4 points in boundary (not in any corner). (1) the intersection of outside rectangles in each side call the line segments T0, B0, L0, R0

  18. T T1 the set of the rectangles that intersect the boundary T called Tr, similar to Br, Lr, Rr. (2) the intersection of the rectangles that intersect only one boundary. e.g. Tr \ (Br  Lr  Rr) call the line segments T1,B1,L1,R1 consider the intersection of T0 and T1 if there are at least two segment, not 4-piercable. else …………….

  19. L R T T2a T2a B consider no rectangles that intersect three or four of T,B,L,R. (3) the intersection of the rectangles that intersect exact two boundaries (a) the boundaries are adjacent e.g. For Tr,Lr,Rr call the line segments T2a,B2a,L2a,R2a consider the intersection of T0 and T2a

  20. L R T T2a B Note in (a) the boundaries are adjacent consider the intersection of T0 and T2a in above case => O(n) time

  21. T T2b T2b B T2b T2b (b) the boundaries are opposite e.g. For Tr,Br call the line segments T2b,B2b,L2b,R2b consider the intersection of T0 and T2b

  22. We have to estimate the time complexity only in Step2. (1). Scan from L to R or from B to T => O(n) time (2)and(3). Similar to (1) => O(n) time • Now, we calculate the combination of the line segments we get • except (3)(a)note, the number of other line segments is fixed. • except (3)(a)note, the time complexity of combination is O(1) goal : O(n)O(1) = O(n) time and extend to p = 5.

  23. Future work p = 2~5, special case p  6 , how ? In graph theory, transformation to intersection graph

  24. The properties of the intersection graphs The clique cover problem in this graphs

  25. Open problems • By my concept, p = 4,5 => O(n) time ? • As p  6, can we reduce the complexity ?

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