Dynamic and Static Algorithms for Optimal Placement of Resources in a Tree

We consider the problem of optimally placing identical resources at the nodes of a weighted tree-shaped network of size N. The resources satisfy requests issued from the nodes of the network. The cost of a placement of the resources is the sum over all nodes of the network of the product of the node weight times the distance from the closest resource. The static problem consists in determining the minimal cost placement of the resources. The dynamic version consists in recomputing such a cost when a node weight has changed. We present a linear time algorithm that computes the optimal placement of two resources in a tree. For the dynamic version we give an O(log N) time algorithm for placing one resource and, for the case of complete binary trees, an O(log N 3) time algorithm for two resources. The static algorithm is faster than the algorithms found in literature, while the dynamic algorithms are the rst for this problem.