1. An Introduction to Computational Geometry
Joseph S. B. Mitchell
Stony Brook University

2. Point Location
Point-in-polygon test: Is point q inside simple polygon P ?
Naïve: O(n) per test
CG: O(log n) q
P n-gon
5 crossings  q inside

3. Point Location
More generally: Search for which face of a planar map contains query point q
Example: Map of Manhattan Applet q
Trapezoid under the mouse is highlighted

4. One Method: Partition the Plane into Slabs
Query time : O(log n)
binary search in x and then
binary search in y direction.
Storage space O(n2)
n/2
n/2
Computational Geometry, WS 2007/08
Prof. Dr. Thomas Ottmann

5. Valuable Resources Preprocessing time
Memory/storage: how big is the data structure?
Query time

6. Point Location in 1D
Given a partition of the x-axis by a set of points {x1 , x2 , …}.
Preprocess for location of a query point q: Sort the xi ‘s O(n log n) time, O(n) space, O(log n) query time
Can we achieve these same bounds (optimal) in 2D?

7. An Optimal Method: Kirkpatrick Hierarchical Triangulation
Begin with full triangulation
Remove a “large” set of low-degree ( 8) vertices that are independent
(such a set always exists in a planar graph, by Euler)
Retriangulate the resulting “holes”
Repeat until only the big outer triangle remains
Build DAG for searching
(no two adjacent)
O(n)
“large” :  Cn  only O(log n) iterations

8. [O’Rourke, page 277]
Proof uses Euler’s formula (f-e+v=2) and the fact that each vertex has degree at least 3 in our graph G.

9. Proof of Theorem 7.10.1 Sum of vertex degrees = ∑= 2E
Recall: in a planar n-vertex graph: E≤3n-6
Thus, ∑ ≤ 6n-12
Claim: There are ≥n/2 vertices of degree ≤8
Pf: Assume there are > n/2 vertices of deg ≥9
Then, sum of degrees of these is ≥ 9n/2, and the other vertices have deg ≥3, so the total degree sum, ∑, would satisfy: ∑ ≥ 9n/2+3n/2=6n
Contradicts ∑ ≤ 6n-12

10. Independent Set Algorithm
Input: planar graph G
Output: an independent set I of vertices of low degree (≤ 8)
I ←Ø
Mark vertices of G of degree ≥ 9
While some nodes are unmarked, do
Choose an unmarked vertex v (lowest index)
Mark v and all neighbors of v
I ←I {v}

11. Analysis of Algorithm
Each time we mark v, and the neighbors of v, the number of unmarked nodes goes down by at most 9
Thus, at least 1/9-th of the m unmarked (deg ≤8) vertices are added to I
We know m ≥ n/2
Thus, I ≥ n/18
Thus, the Algorithm always gives an independent set of deg ≤8 vertices of size ≥ n/18
Initially: the unmarked vertices are those with deg≤8.

12. Kirkpatrick Example
Full triangulation, T0
Triangles in T0

13. T0
Independent sets:
{2,5}
{3,7}
{4}
{6} T1

14. T2 T3
Independent sets:
{2,5}
{3,7}
{4}
{6}
Final DAG hierarchy and search path
for query point p T4

15. Another Example 4 Build Kirkpatrick hierarchy, for extra practice.
(note that vertex 2 has “high” degree) 10 5 2 8 9 3 6 1 7

16. Detecting if a unit disk contains a site: Why locating in Delaunay is not enough
From class discussion/review, 12/4/12

17. Example 4 cocircular points, but Delaunay diagram is a triangulation
From class discussion/review, 12/4/12

18. Examples: Recognizing a Voronoi Diagram
From class discussion/review, 12/4/12