1. at University of Texas at Dallas
A Localized Algorithms for Coverage Problems in Heterogeneous Sensor Networks
My T. Thai, Yingshu Li, Feng Wang, and Ding-Zhu Du, "O(log n)-Localized Algorithms on the Coverage Problem in Heterogeneous Sensor Networks," 26th IEEE International Performance Computing and Communications Conference (IPCCC 2007), New Orleans, LA, April 11-13, 2007.
Presented By Donghyun Kim July 23, Mobile Computing and Wireless Networking Research Group
at University of Texas at Dallas

2. Sensing and Communication Range
Sensor Node
Sensing Range
Communication Range
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

3. Sensor Coverage Problem
Basically, it is a scheduling problem. T
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

4. Area vs. Target Coverage Problem
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

5. Target Coverage vs. Set Cover Problem
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

6. Notations We have sensors .
A sensor’s state can be transmit, receive, idle, or sleep.
The lifetime of is unit time.
is the sensing range of .
is the transmission range of .
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

7. Definitions Definition 1: Maximum Lifetime Coverage
Given a network consisting of sensors and an interest region, find a family of ordered pairs , where is a set of sensors that completely cover the interest region and is the time duration for to be active such that to maximize Each sensor appears in with a total time at most .
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

8. Definitions – cont’
For example, suppose four sensors and 2 targets Given , we can find the family of ordered pairs
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

9. Definitions – cont’
Some people found that organizing sensors into a maximum number of non- disjoint set covers may obtain a longer network lifetime.
Example
Disjoint set cover and Then total lifetime is 1.0 unit time.
Non-disjoint set cover – , and each set is activated for 0.5 unit time. Total lifetime is 1.5 unit time.
Find a set of set covers first and decide their activation time.
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

10. Definitions Definition 2: Domatic Number
The domatic number of a directed graph is the maximum number of disjoint dominating sets of graph .
Maximum lifetime Dominating Sets Problem (MDS)
Given a directed graph , where each node has a maximum lifetime , find a family of ordered pairs , where is a dominating set and is the time duration for to be active, such that the total active time of all dominating sets is maximized.
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

11. Overall Idea
Maximum Lifetime
Target Coverage Problem
Maximum Disjoint Set Cover Problem in Bipartite Graph
Maximum Lifetime Dominating Sets Problem in Directed Graph
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

12. Graph Transformation
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

13. Graph Transformation – cont’
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

14. Correctness Lemma 1. A dominating set of is a cover set of .
Lemma 2. A cover set in can be transformed to a dominating set of .
Theorem 1. A feasible solution to the MDS problem is a feasible solution of the MTC problem.
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

15. Lower and Upper Bounds of Domatic Number in Directed Graphs
Theorem 2. Given a directed graph , the domatic number is bounded as
where and the term “ ” goes to zero as goes to , is the minimum indegree.
Proof
Upper bound: for any node , it can be dominated by at most nodes.
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

16. Lower and Upper Bounds of Domatic Number in Directed Graphs
Lower bound
For each vertex in the graph, randomly assign a color in the range
Let be the event that there is no vertex of color in , which is incoming neighbors of .
Let denote the probability of the event .
Let be the event that vertex does not have all color in
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

17. Lower and Upper Bounds of Domatic Number in Directed Graphs
Then, the expected number of bad event is :
And the expected number of colors that forms dominating sets is at least:
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

18. Lower and Upper Bounds of Domatic Number in Directed Graphs
Hence,
When
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

19. Two localized approximation algorithms
Case 1: for all and
Case 2: Otherwise
Presented by Donghyun Kim on July 23, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas