1. Danny Z. Chen and Haitao Wang University of Notre Dame Indiana, USA
An Improved Algorithm for Reconstructing a Simple Polygon from the Visibility Angles
Danny Z. Chen and Haitao Wang
University of Notre Dame
Indiana, USA

2. Visibility Angles
Consider a simple polygon P: For vertex v, four visibility angles around v a3 a2 a4 a1 v

3. The Visibility Graph of P
Connect all pairs of visible vertices

4. Visibility Angles for Every Vertex

5. What is the Problem? The cyclically ordered vertex list is given v6 v4
The vertex list: v1, v2, v3, v4, v5, v6, v7

6. What is the Problem? All visibility angles are given v6 v4 v7 v5 v2 v1

7. What is the Problem? Input: Goal:
A cyclically ordered vertex list of P
Visibility angles around every vertex of P
Goal:
Reconstruct the simple polygon P (up to similarity)

8. Previous Work and Our Result
Disser, Mihalak, and Widmayer [SWAT 2010]:
The polygon P can be uniquely determined, up to similarity.
An O(n3log n) time algorithm for reconstructing P is given.
Our result:
An O(n2) time algorithm for reconstructing P
Worst case optimal
Based on new geometric observations

9. What is the Difficulty for the Reconstruction?
Every vertex v sees some vertices, but cannot identify them, i.e., their vertex labels are not known to v. v6 v4 v7 v5 v2 v1 v3

10. The Reconstruction Algorithm
Construct the visibility graph G
The key part
Reconstructing P from G
Generally, a long-standing open problem (in PSPACE),
but easy in our setting (O(n2) time) with
angle data information
the ordered vertex list of P

11. Constructing the Visibility Graph G
For every two vertices vi and vj, determine whether they are visible to each other
An observation: vi is visible to vj if and only if there exists a vertex v in the red part such that Δvivjv does not intersect P\{vi,vj,v} vj vj v v vi vi

12. An Observation (cont.)
vi’: The last visible vertex to vj in the blue part counterclockwise
vj’: The first visible vertex to vi in the blue part counterclockwise vj vi vj v
vj’
triangle witness
vi’
vj’ v
vi’ vi
vi is visible to vj if and only if the sum of the three angles is 180 degree

13. A Preliminary Algorithm
For every pair of vertices vi and vj, to determine whether they are visible to each other:
Check every vertex to see whether there exists a triangle witness
Running time:
There are O(n2) pairs of vertices
O(n) time is needed for each pair
O(n3) time is needed

14. A New Observation Do not check every vertex in the red part
It is sufficient to check one particular vertex (to see whether it is a triangle witness)
Specifically, check the last visible vertex to vi counterclockwise in the red part

15. A New Observation (cont.)vj v vi
vi is visible to vj if and only if v is a triangle witness

16. The Proof
Goal: vi is visible to vj if and only if v is a triangle witness
If v is a triangle witness, then vi is visible to vj by the previous observation
If vi is visible to vj, then v is a triangle witness
The key is to prove v is also visible to vj vj v vi

17. The Proof
Goal: vi is visible to vj if and only if v is a triangle witness
If v is a triangle witness, then vi is visible to vj by the previous observation
If vi is visible to vj, then v is a triangle witness
The key is to prove v is also visible to vj vj v vi

18. The Algorithm Implementation
There are n/2 iterations
In the k-th iteration (k=1…n/2), we check for each i=1…n whether vi is visible to vi+k
It takes O(1) time to check the visibility of every pair of vertices
By using some arrays and auxiliary variables
Overall running time: O(n2)

19. Preprocess The visibility array for v: A[1,2,3,4]={a1,a2,a3,a4}
Preprocessing:
Given any i and j with i≤j, the value A[i,j] = A[i]+A[i+1]+…+A[j] can be computed in O(1) time, a3 a2 a4 a1 v

20. An implementation detail
vj’: The first visible vertex to vi in the blue part counterclockwise
Determine the angle x between viv and vivj’
Suppose v is the t-th visible vertex of vi along the red part
Suppose b is the number of visibility vertices of vi that have been identified so far (in the red part)
Both t and b are already known
The angle x is A[t,b+1], where A is the visibility array for vi vj vi vj v
vj’ y y
vj’ v x x vi

21. Conclusions
Given the visibility angles around every vertex of a simple polygon P and its cyclically ordered vertex list,
we give an algorithm for reconstructing P in O(n2) time

22. Thank you
Questions?