Maintaining Centdians in a Fully Dynamic Forest

In this paper, we consider the problem of maintaining the centdians in a fully dynamic forest. A centdian is a location in the given network which minimizes a convex combination of the longest distance and the sum of distances from all clients to it. A fully dynamic forest is a forest, i.e., a set of trees, in which the following operations are allowed: to insert an edge between different trees in the forest, to delete an edge in the forest, and to change a vertex weight in the forest. We propose an algorithm to maintain a centdian of each tree in the given fully dynamic forest. It preprocesses the input forest with time and space linear to the size of the forest and uses O(logn) update time per edge insertion, edge deletion, or vertex weight change, where n is the number of vertices of the trees involved in the update.