1. Efficient Algorithms for the Weighted 2-Center Problem in a Cactus Graph Qiaosheng Shi Joint work: Boaz Ben-moshe Binay Bhattacharya

2. Dec. 21, 2005Center problems in a cactus2 Introduction Definition: A cactus is a connected graph in which two simple cycles have at most one vertex in common. Hinge vertices

3. Dec. 21, 2005Center problems in a cactus3 Introduction Facility Location Problems: Given a set of clients in some metric space, locate facilities in this space to provide some kind of service to the clients such that some objective function is minimized. In our problem, G=(V,E) –Each client (vertex v) has a demand (weight) w(v). –Objective: the maximum weighted distance to the facilities is minimized.

4. Dec. 21, 2005Center problems in a cactus4 Some known results Unweighted 1-center problem O(n) –the weight of every vertex equals 1. –[Lan et al. 1999] Obnoxious center problem O(cn) –the weight of every vertex is negative. –c is the number of distinct vertex weights. –[Zmazek et al. 2004] In a tree graph –Weighted 1-center problem O(n) –Obnoxious center problem O(nlog 2 n)

5. Dec. 21, 2005Center problems in a cactus5 Our Contribution Continuous/discrete 1-center Obnoxious center Continuous 2-center Discrete 2-center O(nlogn) O(nlog 3 n) O(nlog 2 n) ProblemsOur result are constant; where and is original size. Binary-search:

6. Dec. 21, 2005Center problems in a cactus6 The weighted 1-center problem In a tree graphOne simple property

7. Dec. 21, 2005Center problems in a cactus7 The weighted 1-center problem In a tree graphBinary-search

8. Dec. 21, 2005Center problems in a cactus8 The weighted 1-center problem In a cactus graphOne similar property

9. Dec. 21, 2005Center problems in a cactus9 The weighted 1-center problem Binary-searchIn a cactus graph Lemma: It takes O(nlogn) time to locate A*.

10. Dec. 21, 2005Center problems in a cactus10 Locate an optimal center in A* A* is a subtreeEasy ! It’s similar to locate 1-center in a tree.

11. Dec. 21, 2005Center problems in a cactus11 Locate an optimal center in A* Theorem 1: The weighted 1-center problem in a cactus can be solved in O(nlogn) time. A* is a cycle It’s similar to locate 1-center in a cycle. Cut-edge

12. Dec. 21, 2005Center problems in a cactus12 The weighted 2-center problem Split-edges In a tree In a cactus, the split-edges lie in one block.

13. Dec. 21, 2005Center problems in a cactus13 The weighted 2-center problem Two steps –locate the block B* where an optimal split- edge set lies, called as split-block. –compute an optimal split-edge set R*.

14. Dec. 21, 2005Center problems in a cactus14 Locate the split-block B*

15. Dec. 21, 2005Center problems in a cactus15 Locate the split-block B*

16. Dec. 21, 2005Center problems in a cactus16 Locate the split-block B* Lemma 3: It’s O(nlog 2 n) time to locate the split-block.

17. Dec. 21, 2005Center problems in a cactus17 Compute R* in B* B* is a subtreeEasy ! Binary-search can be applied here.

18. Dec. 21, 2005Center problems in a cactus18 Compute R* in B* B* is a cycleR* contains two split-edges. The complexity of algorithm is determined by the comple- xity of computing the service cost of a given split-edge set.

19. Dec. 21, 2005Center problems in a cactus19 Compute service cost with a given split-edge set

20. Dec. 21, 2005Center problems in a cactus20 Compute service cost with a given split-edge set

21. Dec. 21, 2005Center problems in a cactus21 Compute service cost with a given split-edge set

22. Dec. 21, 2005Center problems in a cactus22 Two-level tree decomposition Theorem 2: The weighted 2-center problem in a cactus can be solved in O(nlog 3 n) time. Tree decomposition –Cactus graphs are partial 2-trees Centroid tree decomposition, top-tree decomposition, spine tree decomposition, … … –the height of the tree decomposition is logarithmic. Lemma: The service cost of a point in a cactus graph can be answered in O(log 2 n) by the two-level tree decomposition data structure.

23. Dec. 21, 2005Center problems in a cactus23 Future work An optimal algorithm for the weighted 1- center in a cactus graph. Whether the parametric search technique can be applied in the weighted p-center problem in a cactus graph for any p ? Another challenging work is to find efficient algorithms to solve the p-center problem in partial k-trees.

24. Dec. 21, 2005Center problems in a cactus24 Thanks.