Back-Up 2-Center on a Path/Tree/Cycle/Unicycle

An Optimal Algorithm for the Weighted Backup 2-Center Problem on a Tree

“…The algorithm is asymptotically optimal, and therefore improves the current best result on trees, given by Bhattacharya et al [6]. The strategy of our algorithm is prune-and-search, which is widely applied in solving distance-related problems [10,14].…”

“…The algorithm of Bhattacharya et al depends on the following observations. Lemma 1 (See [6]). Let {c 1 , c 2 } be any 2-center.…”

“…While processing x 1 from left to right, the corresponding x 2 moves monotonically to the left, and therefore a one-pass scan is sufficient to find the optimal solution. Readers can refer to [6] for details. To improve the time complexity, we elaborate on some properties of the objective function below.…”

“…When the edges are of identical length, Hong and Kang [8] proposed a linear time algorithm on interval graphs. Recently, Bhattacharya et al [6] consider a weighted formulation of the backup 2-center problem, in which each vertex is associated with a nonnegative weight, and the distance from vertex u to v is weighted by the weight of u. They proposed O(n)-, O(n log n)-, O(n 2 )-, and O(n 2 log n)-time algorithms on paths, trees, cycles, and unicycles, respectively, where n is the number of vertices.…”

“…The rest of this paper is organized as follows. In Section 2, we formally define the problem and briefly review the result given by Bhattacharya et al [6]. Based on their observations, a further elaboration on the objective function is given.…”