1. Locating an Obnoxious Line among Planar Objects
Danny Z. Chen and Haitao Wang
Computer Science and Engineering
University of Notre Dame
Indiana, USA

2. Outline Problem definitions Motivations Previous work and our results
Our algorithms

3. Obnoxious Line among Points
Input: A set of points in the plane P={p1,p2,…,pn}, where each pi has a weight wi>0
Objective: Find a line L which intersects the convex hull of P and maximizes the minimum weighted Euclidean distance of L to all points in P, i.e., min{wid(L,pi) | 1≤i≤n}. L is called an obnoxious line of P.
obnoxious line

4. The non-weighted case: The widest empty corridor problem

5. Obnoxious Line among Polygons
Given: m polygons {P1,P2,…,Pm }
Totally n vertices
Each polygon has a weight wi>0
Objective: Find an obnoxious line L which intersects the convex hull of all vertices and maximize the minimum weighted distance from L to all polygons, i.e., min{wid(L,Pi) | 1≤i≤m}.

6. Obnoxious Line among Polygons
an obnoxious line
Every polygon has a weight

7. Motivations
A linear route through existing facilities such that the route is hazardous to its surroundings
The maximum clearance of a route with respect to the existing facilities
The weights may represent various importance of the facilities

8. Previous Work and Our Results
Obnoxious line among points
The first algorithm: O(n3) time [DW 89’]
The best known: O(n2log3n) time [DRS 07’]
Our result: O(n2logn) time
The non-weighted case: O(n2)
Obnoxious line among polygons
Best known: O(mn+nlog2nlogm+m2lognlog2m)[DRS 07’]
Our result: O(mn+nlog2n m2logn)
The non-weighted case:
Best known: O((m2+nlogm)logn) [DRS 07’]
Our result: O(m2+nlogm)

9. Outline Algorithm for obnoxious line among points
Extend it to the polygon case

10. Our Approach Based on parametric search
Reduce the problem to a sorting problem
Apply Cole’s parametric search
Solve the decision problem first: The disks separability problem

11. The Disks Separability Problem
Given n disks in the plane
Determine whether there exists a separation line such that the line does not intersect the interior of any disk and there is at least one disk on each side of the line.
No previous work has been found
a separation line

12. An Observation
There exists a separation line if and only if there exists a separation line which is a common tangent of two disks.

13. Our Algorithm There are O(n2) common tangents Check every one of them
A straightforward method gives an O(n3) time solution
Our improved solution: O(n2) time

14. An Observation
engaging arc a di dj b
A tangent of di intersects the interior of dj if and only if
the tangent point on di is on the engaging arc

15. An Observation (cont.) di dj

16. An Observation (cont.) For each disk di, there are O(n) engaging arcs
A tangent of di does not intersect the interior of any other disk if and only if the corresponding tangent point on di does not belong to any engaging arc of di

17. The Algorithm for the Disks Separability Problem
Ai : Common tangent points on disk di with size O(n)
If Ai is sorted cyclically, then we can check them in O(n) time by a tangent sweeping algorithm
Sort all Ai’s: Duality and curve arrangement construction
Time complexity: O(n2) di

18. Summary The disks separability problem is solvable in O(n2) time
Works as a decision procedure in the parametric search
Parametric search
Another view on the obnoxious line problem

19. Another Problem View

20. Another Problem View

21. The Parametric Search For any t>0, let D(t) be the set of all disks
Assume t* is the stop time
There is a separation line for D(t) if and only if t≤t*

22. Two Possible Cases di dk dj Case 1 Easy Case 2 Difficult
an obnoxious line di dk dj
Case 1
Easy
Case 2
Difficult

23. Two Inner Common Tangents
rij
lij di dj
aij
bij

24. An Observation
If we sort all aij(t*)’s and bij(t*)’s, if case 1 occurs, then we have aij(t*)=bij(t*); otherwise, we have aij(t*)=bik(t*) for a suitably labeled disks di, dj and dk as follows: di dj dk

25. The Parametric Search
We reduce the original problem to a sorting problem
Sort all aij(t*)’s and bij(t*)’s by parametric search, where Cole’s scheme can be applied and our disks separability algorithm is utilized to make decisions at every comparison of the sorting
Total running time: O(n2logn)

26. Summary An obnoxious line among points can be found in O(n2logn) time
Extend the algorithm to solving the polygon case

27. Obnoxious Line among Polygons

28. Obnoxious Line among Polygons
Each region is the Minkowski sum of the original polygon and a disk
Call the region the M-region

29. Decision Problem M-region separability problem: Preprocessing
Given m M-regions MR
Determine whether there is a separation line
Preprocessing
Find all connected components of those M-regions and compute their convex hulls MH
A line is a separation line for MR if and only if it is a separation line for MH
Use an extension of the disks separability algorithm on MH
Time complexity: O(m2+nlogn)

30. Parametric Search View each M-region as a “deformed” disk
Extend the algorithm for the point case
Use the M-region separability algorithm to make decisions
Time complexity: O(mn+nlog2n+m2logn)

31. Non-weighted Version Previous solution: O((m2+nlogm)logn) [DRS 07’]
A procedure to merge two planar subdivisions
Replace the procedure by a linear time algorithm for computing the overlay of simply connected planar subdivisions [Finke, Hinrich, 95’]
Improved result: O(m2+nlogm)

32. Thank you
Questions?