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

Visibility p is visible to q p q

Visibility of line segments q p Both p and q are (weakly) visible to s

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

Visibility polygons (cont.)

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

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

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)

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

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

Computing the visibility polygon

Computing the visibility polygon q v u

Computing the visibility polygon q v u

Computing the visibility polygon critical constraints q v u

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

Observations: finding the critical constraints dynamically q v u

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

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)

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?

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)

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)

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

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)

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)

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?

Thank you Questions?