1. Optimal Point Movement for Covering Circular Regions
Haitao Wang
Utah State University
ISAAC 2012
Joint work with Danny Z. Chen, Xuehou Tan, and Gangshan Wu

2. Motivation
Mobile sensor barrier coverage
A mobile sensor r
a sensor

3. Motivation
Protect a region by sensors
Full coverage of region

4. Motivation Barrier coverage: only covering its border
The border is the barrier
intruder

5. Mobile sensors
How to move sensors to minimize the movement?

6. Our problem: a simplified version
The region is a circle C
Sensors are given inside C
Move all sensors to C to form a regular n-gon
The sensor range is explicitly determined
Objective: min-max:
minimize the maximum
sensor movement

7. Previous work Our result O(n3.5log n) time, Bhattacharya et al. 2009
O(n2.5log n) time, Tan and Wu, 2010
Our result
O(nlog3 n) time
Dynamically maintain the maximum matching of a circular convex bipartite graph
Support vertex insertions/deletions O(log2 n) time each

8. The difficulty

9. The difficulty
The optimal solution (OPT) happens at some moment during the rotation

10. The decision version
Given a distance d, determine whether it is possible to move the sensors to form a regular n-gon on C such that the maximum sensor movement is ≤ d
If yes, d is a feasible distance, and the n-gon is a feasible solution d

11. An overview of our approach
d* : the maximum sensor movement in OPT
The key idea:
Find a set S of distances, such that d* ∈ S
d* must be the smallest feasible distance in S
A straightforward solution:
Sort all distances in S
Find d* in S by binary search
need an algorithm for the decision version
Two challenges:
How to find S (as small as possible)?
Design an efficient algorithm for the decision version

12. Dealing with the first challenge
For each red point pi,
xi : is the point on C closest to pi
pi’ : the position of pi on C in an OPT
center of C
pi’ pi pi xi

13. Observation (previous work)
Suppose |pip’i|=d* in an OPT, then
p’i is xi , or
there is another sensor pj such that
|pip’i |=|pjp’j|, and
if we rotate all blue points in either direction, one of |pip’i| and |pjp’j| increases and the other one decreases pj pi
xi = p’i
p’j pi
pj increases
pi decreases
pi’

14. Observation (previous work)
Suppose in an OPT, pi moves the largest distance, then
p’i is xi , or
S1 : the set of all such distances |pip’i |
there is another sensor pj such that
|pip’i = pjp’j|, and
if we rotate the solution in either direction, one of |pip’i| and |pjp’j| increases and the other one decreases
S2 : the set of all such distances |pip’i |=|pjp’j|
Let S = S1 ∪ S2 , and then d* ∈ S
What is the size of S?
|S1|= n
|S2|= O(n3) (previous work)

15. Our approach First challenge: Find a smaller set S2 with |S2|=O(n2)
S2 can be implicitly determined and sorted
Correspond to the intersections of n pseudo-lines
the y-coordinates of the intersections
Only need to implicitly search the pseudo-line arrangement
Second challenge: a new algorithm for the decision version
Previous work: O(n3.5)
Our result: O(nlog2n)

16. The decision algorithm
Given a distance d, whether d is a feasible distance?
If d is feasible, a feasible solution happens at some moment during the rotation

17. Two difficulties How to discretize the rotation?
For each candidate position, how to determine whether there is a feasible solution?

18. Tackle the second difficulty
For each candidate position, whether it is possible to move all red points to blue points within maximum moving distance d?
Observation: qi must be on the
brown circular segment qi d pi

19. Tackle the second difficulty
Build a bipartite graph G
connect each red point to all blue points on its circular segment on C
Observation: It is feasible solution if and only if there is a perfect matching in G qi pi

20. Tackle the first difficulty
How to discretize the rotation?
key: determine when the graph G changes
Totally 2n vertex insertions and deletions
Need to dynamically maintain the maximum matching pi

21. G is a circular convex bipartite graph
We build a data structure in O(nlog2n) time to dynamically maintain the maximum matching in G
O(log2 n) time for each vertex insertion/deletion
The decision problem is solved
in O(nlog2 n) time pj pi

22. Conclusion An O(nlog3 n) time new algorithm
Is further improvement possible?
Improve the O(nlog2 n) time decision algorithm
Improve the O(log2 n) time maximum matching update

23. Thank you
Questions?