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

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

The Visibility Graph of P Connect all pairs of visible vertices

Visibility Angles for Every Vertex

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

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

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)

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

Thank you Questions?