Efficient Algorithms for the Weighted 2 Center Problem

Efficient Algorithms for the Weighted 2 -Center Problem in a Cactus Graph Qiaosheng Shi Joint work: Boaz Ben-moshe Binay Bhattacharya Dec. 21, 2005 Center problems in a cactus

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

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. Dec. 21, 2005 Center problems in a cactus 3

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) Dec. 21, 2005 Center problems in a cactus 4

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

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

The weighted 1 -center problem In a tree graph Dec. 21, 2005 Binary-search Center problems in a cactus 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Two-level tree decomposition • 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 Theorem The weighted 2 -center problem in 2 n) by the graph can be 2: answered in O(log two-level 3 n) time. a cactus can be solved in O(nlog tree decomposition data structure. Dec. 21, 2005 Center problems in a cactus 22

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. Dec. 21, 2005 Center problems in a cactus 23

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