1. Weak Visibility Queries of Line Segments in Simple Polygons
Haitao Wang
Utah State University
ISAAC 2012
Joint work with Danny Z. Chen

2. Visibility
p is visible to q p q

3. Visibility of line segmentsq p
Both p and q are (weakly) visible to s

4. (Weak) visibility polygons
The visibility polygon of s: the set of points visible to s s

5. Visibility polygons (cont.)

6. Our problem: visibility queries
Given a simple polygon P, design a data structure (or do preprocessing)
For any query segment s in P,
compute the visibility polygon of s
Notation
n: the size of P
k: the size of the visibility polygon of s

7. Previous work (no preprocessing)
A linear time algorithm (GHLMT 87’)

8. Previous work (with preprocessing)
Data structures
preprocessing time
space
Query time
Bose et al. 02’
O(n3 log n)
O(n3)
O(klog n)
Aronov et al. 02’
O(n2 log n)
O(n2)
O(klog2 n)
Bygi and Ghodsi 11’
O(k+log n)

9. The visibility graph of P
Connecting all visible pairs of vertices
The size of the graph is O(n2)

10. Our results K=O(n2): the size of the visibility graph of P
Data structures
preprocessing time
space
Query time
Bose et al. 02’
O(n3 log n)
O(n3)
O(klog n)
Aronov et al. 02’
O(n2 log n)
O(n2)
O(klog2 n)
Bygi and Ghodsi 11’
O(k+log n)
Our Result 1
O(K)
Our Result 2
Our recent result: O(n) O(n) O(k log n)
K=O(n2): the size of the visibility graph of P

11. Computing the visibility polygon

12. Computing the visibility polygonq v u

13. Computing the visibility polygonq v u

14. Computing the visibility polygon
critical constraints q v u

15. Computing the visibility polygon
critical constraints
The purple segments q v u
Question: How to determine the critical constraints dynamically?

16. Observations: finding the critical constraints dynamicallyq v u

17. Our two approaches Determine critical constraints dynamically
in the query algorithm
Pre-compute the critical constraint arrangement
in the preprocessing

18. The first approach: determine critical constraints dynamically
Using visibility graphs (in the paper)
Preprocessing time and space: O(K)
Using a ray-rotating data structure (our recent result)
Preprocessing time and space: O(n)
Query
determine each critical constraint in O(log n) time
Total query time: O(klog n)
The total number of critical constraints intersecting s is O(k)

19. The ray-rotating queries
Given a ray r with origin z in P, rotate r clockwise
return the first vertex of P visible to z that will be hit by the ray r z
Our result: with O(n) time preprocessing,
each query can be answered in O(log n) time
Interesting in its own right?

20. The second approach: pre-compute the critical constraint arrangement
Preprocessing
Compute all critical constraints explicitly
O(n3) time and space
Query
Follow the query segment in the arrangement
Query time: O(k+logn)

21. Critical constraint arrangement
Each critical constraint is an
extension of a visibility graph edge
The number of critical constraints is O(n2)
The size of the arrangement is O(n3)

22. Critical constraint arrangement
Given a query segment s
The complexity of all yellow cells are O(k)
k: the size of the visibility polygon of s

23. The zone theorem of line segment arrangements in the plane
A: the arrangement of m line segments
for any other line segment s
The zone Z(s): the set of all faces of A intersecting s
The zone theorem:
The complexity of Z(s): O(mα(m)) (Edelsbrunner et al. 92)

24. The zone theorem of line segment arrangements in simple polygons
A new question: What is the complexity of Z(s), if
All segments of A are in a simple polygon P
The endpoints of each segment in A are on the boundary of P
Our answer: O(M)
M: the number of segments of A intersecting Z(s)

25. Open problem Open problem: are there sub-cubic data structures for
preprocessing time
space
Query time
Bose et al. 02’
O(n3 log n)
O(n3)
O(klog n)
Aronov et al. 02’
O(n2 log n)
O(n2)
O(klog2 n)
Bygi and Ghodsi 11’
O(k+log n)
Our Recent Result
O(n)
Our Result 2
Open problem: are there sub-cubic data structures for
O(k + log n) time query?

26. Thank you
Questions?