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”,Micha Sharir, Emo Welzl, Proceeding 12th ACM Symposium onComputational 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.

Topic

1. Introduction

2. The latest results

3. Axis-transformation way

4. Location domain way

5. My conception

6. Future work and open problems

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. (for fixed p)

related problems : p-center, piercing rings, piercing squares,…

p = 3*

* *

The latest results

1999

p = 2,3 => O(n) time

p = 4 => O(nlogn) time

p = 5 => O(nlogn) time

p 6 => O(n logn) time

p-4

1996

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

p-4

3

4

5

Axis-transformation way

( cx , cy , dx , dy )

centroid( cx , cy )** dx

dy

cx

dx

cy

dy

AB

C

D

x

y

*AC B

* D*

*A * B*C*

D

query point p

( Px , 0 , Py , 0 )

p*

*** *

Cone

p = 1

x-cone cover ally-cone cover all

*

*

*

** *

*

p = 2

C1 C2

* *

*

** *

X

* *

*

** *

C3 C4

Y

first,

second,

(C1 C3) (C2 C4) cover all rectangles.

or (C1 C4) (C2 C3) cover all rectangles.

p = 3

first,

second,

C1 , C2 , C3 cover all in X-axis.C4 , C5 , C6 cover all in Y-axis.

assume C1, C3, and C4, C6 are constrained cones

(since there exists at least one constrained-cones pair)

4 possibilities => C1C4, C1C6, C3C4, C3C6.

the rest of rectangles => 2-piercing

p = 4

first,

second, similar to p = 3

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

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)

insert or delete

6 combinations => independent

each need O(n) steps each step need O(logn) updates

=> O(nlogn) time

C1 C4*

* *

*

*

*

*

*

**

*

*

C3 before C3 after

Location domain way

if an axis-parallel line traverses all rectangles

else

*

T

B

L R

*

** Observation

p = 2

p = 3

2-piercable => diagonal pair

3-piercable=> at least one vertex in corner

the rest of rectangles => similar to 2-piercing way

4 possibilities

p = 4try each intersection part

=> O(n) time

the rest of rectangles => 3-piercable

=> O(log n) time3

=> O(nlog n) time3

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)

case3. there exist 4 points in boundary (not in any corner),

and 1 point inside.

case2,3 O(logn) time

However, it can’t workas p = 6.

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

TT0

(2) the intersection of the rectangles that intersect only one boundary.

the set of the rectangles that intersect the boundary T called Tr, similar to Br, Lr, Rr.

e.g. Tr \ (Br Lr Rr)

call the line segments T1,B1,L1,R1

T T1

consider the intersection of T0 and T1

if there are at least two segment, not 4-piercable.else …………….

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 T

L R

B

call the line segmentsT2a,B2a,L2a,R2a

consider the intersection of T0 and T2a

T2a T2a

Note in (a) the boundaries are adjacent

T

B

L R

T2a

consider the intersection of T0 and T2a in above case=> O(n) time

(b) the boundaries are opposite

e.g. For Tr,BrT

B

call the line segments

T2b,B2b,L2b,R2b

consider the intersection of T0 and T2b

T2bT2b

T2b T2b

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.

Future work

p = 2~5, special case

p 6 , how ?

In graph theory, transformation to intersection graph

The properties of the intersection graphs

The clique cover problem in this graphs

Open problems

1. By my concept, p = 4,5 => O(n) time ?

2. As p 6, can we reduce the complexity ?