We consider the k most vital edges (nodes) and min edge (node) blocker versions of the 1-median and 1-center location problems. Given a weighted connected graph with distances on edges and weights on nodes, the k most vital edges (nodes) 1-median (respectively 1-center) problem consists of finding a subset of k edges (nodes) whose removal from the graph leads to an optimal solution for the 1-me...

In this paper we consider a generalization of the classical k-center problem with capacities. Our goal is to select k centers in a graph, and assign each node to a nearby center, so that we respect the capacity constraints on centers. The objective is to minimize the maximum distance a node has to travel to get to its assigned center. This problem is NP -hard, even when centers have no capacity...

We consider the problem of placing sensors in a network to detect and identify the source of any contamination. We consider two variants of this problem: 1) sensorconstrained : we are allowed a fixed number of sensors and want to minimize contamination detection time; and 2) time-constrained : we must detect contamination within a given time limit and want to minimimize the number of sensors re...

The geodesic k-center problem in a simple polygon with n vertices consists in the following. Find a set S of k points in the polygon that minimizes the maximum geodesic distance from any point of the polygon to its closest point in S. In this paper, we focus on the case where k = 2 and present an exact algorithm that returns a geodesic 2-center in O(n log n) time.

We consider a classical k-center problem in trees. Let T be a tree of n vertices and every vertex has a nonnegative weight. The problem is to find k centers on the edges of T such that the maximum weighted distance from all vertices to their closest centers is minimized. Megiddo and Tamir (SIAM J. Comput., 1983) gave an algorithm that can solve the problem in O(n log n) time by using Cole’s par...

1.1 2-Approximation Algorithm for Symmetric k-Center There are many (essentially identical) 2-approximations known for the k-center problem. Here is one of them: letR be our “guess” for the optimal value—we run the following procedure with our guess for R, and return the smallest value R for which this procedure returns at most k facilities. Algorithm I (Symmetric k-center) 1. Label all vertice...

The k-center problem is a classic facility location problem, where given an edge-weighted graph G = (V,E) one is to find a subset of k vertices S, such that each vertex in V is “close” to some vertex in S. The approximation status of this basic problem is well understood, as a simple 2-approximation algorithm is known to be tight. Consequently different extensions were studied. In the capacitat...

We give an algorithm for determining an optimal step function approximation of weighted data, where the error is measured with respect to the L∞ norm. The algorithm takes Θ(n+ log n · b(1 + log n/b)) time and Θ(n) space, where b is the number of steps. Thus the time is Θ(n log n) in the worst case and Θ(n) when b = O(n/ log n log log n). A minor change determines the optimal reduced isotonic re...

Core-based multicast trees use less router state, but have significant drawbacks when compared to shortest-path trees, namely higher delay and poor fault tolerance. We evaluate the feasibility of using multiple independent cores within a shared multicast tree. We consider several basic designs and discuss how using multiple cores improves fault tolerance without sacrificing router state. We exa...

Given a set of n weighted points on the x-y plane, we want to find a step function consisting of k horizontal steps such that the maximum vertical weighted distance from any point to a step is minimized. We solve this problem in O(n) time when k is a constant. Our approach relies on the prune-and-search technique, and can be adapted to design similar linear time algorithms to solve the line-con...