A linear-time algorithm for the center problem in weighted cycle graphs

Abstract

We study the problem of computing the center of cycle graphs whose vertices are weighted. The distance from a vertex to a point of the graph is defined as the weight of the vertex times the length of the shortest path between the vertex and the point. The weighted center of the graph is a point of the graph such that the maximum distance of the vertices of the graph to the point is minimum among all points of the graph. We present an O(n)-time algorithm for the discrete and continuous weighted center problem on cycle graphs with n vertices. Our algorithm improves upon the best known algorithm that takes O(nlog⁡n) time. Moreover, it is optimal for the weighted center problem of cycle graphs.