1. Half-Space Intersections
Randomized algorithm and expected run time analysis

2. Problem Definition – 2D
Given n half-planes as linear inequalities, we wish to find a boundary where they all intersect

3. Problem Definition – 3D We extend our problem to 3D
The intersection is a convex polyhedron
This polyhedron is represented as a graph with degree 3
inter(S): the intersection of a set S of half-spaces

4. Further Assumptions
inter(S) is bounded. If not – there’s a point at infinity which bounds it
No 3 vertices are collinear
Intersection of planes and spaces can be computed in constant time
Deciding if a point is inside a half-space can be computed in constant time

5. Algorithm Outline Randomize a permutation of the set S of half spaces
For :
Add half space hi to inter(Si-1) by
removing all the vertices in
adding new vertices on the boundary of

6. Addition of a Half Space
At step i we have a bidirectional pointer from every to some
When we add hi we start at the conflicting vertex and search the graph until we enter hi
We destroy all the vertices we encounter and create new ones on the face of

7. Addition of a Half Space

8. Pointer Update Recall: the pointer is bidirectional
When we delete a vertex v pointing to a half-space h ∈ S \ Si we have to find a new vertex w, such that
We perform BFS along until we encounter vertices inside inter(Si). This is our w.

9. Number of Vertices Created
Backwards analysis: suppose we delete vertices from inter(Si) and get inter(Si-1)
The convex polyhedron inter(Si) has i faces and O(i) vertices (because the degree is bounded by 3)
If we choose one of i half-spaces to delete, the expected number of vertices we delete is
Hence: the expected number of vertices created at the addition of hi is constant

10. Search Cost Analysis
When adding a half-space, we perform a BFS starting from the pointed vertex
Time – linear in # of deleted vertices
Each deleted vertex was created once: we only count creations
How many vertices are created in total?

11. Total # of Vertices Created
Total vertices created when adding hi:
c(H,h):# of conflict vertices between inter(H) and h
Backwards analysis – the expectation over all hi is bounded by:
since the degree of each vertex is bounded by 3

12. Total # of Vertices Created
hi+1 is chosen randomly, so:
The expected # of vertices is bounded by:
We saw the algorithm runs in time linear to the number of vertices created, so this expression also bounds the run time of the algorithm

13. Bounding the Total Run Time
c(Si,hi+1) is also the expected number of vertices deleted by the addition of hi+1

14. Bounding the Total Run Time
Let tc(v) be the time v is created and td(v) be the time v is destroyed
Then:
for all the vertices v created in the algorithm

15. Bounding the Total Run Time
Obviously, tc(v) ≤ td(v) – 1. Therefore:
Recall that the expected number of vertices created at each iteration is constant!
Constant

16. Bounding the Total Run Time
The total run time of the algorithm is:

17. Convex Hull in 3D
Using duality we can reduce the problem of convex hull in 3D to finding half-space intersection (in 3D)
We just saw how to compute the half-space intersection in O(nlogn) time
3D Convex Hull problem takes also O(nlogn) time to compute

18. Thank You!