An O(log⁡n)O(\log n)-approximation for the Set Cover Problem with Set Ownership

In highly distributed Internet measurement systems distributed agents periodically measure the Internet using a tool called {\tt traceroute}, which discovers a path in the network graph. Each agent performs many traceroute measurement to a set of destinations in the network, and thus reveals a portion of the Internet graph as it is seen from the agent locations. In every period we need to check whether previously discovered edges still exist in this period, a process termed {\em validation}. For this end we maintain a database of all the different measurements performed by each agent. Our aim is to be able to {\em validate} the existence of all previously discovered edges in the minimum possible time. In this work we formulate the validation problem as a generalization of the well know set cover problem. We reduce the set cover problem to the validation problem, thus proving that the validation problem is ${\cal NP}$-hard. We present a $O(\log n)$-approximation algorithm to the validation problem, where $n$ in the number of edges that need to be validated. We also show that unless ${\cal P = NP}$ the approximation ratio of the validation problem is $\Omega(\log n)$

On the Tomography of Networks and Multicast Trees

In this paper we model the tomography of scale free networks by studying the structure of layers around an arbitrary network node. We find, both analytically and empirically, that the distance distribution of all nodes from a specific network node consists of two regimes. The first is characterized by rapid growth, and the second decays exponentially. We also show that the nodes degree distribution at each layer is a power law with an exponential cut-off. We obtain similar results for the layers surrounding the root of multicast trees cut from such networks, as well as the Internet. All of our results were obtained both analytically and on empirical Interenet data

Placing Servers for Session-Oriented Services

The provisioning of dynamic forms of services is becoming the main stream of today\u27s network. In this paper, we focus on services assisted by network servers and different forms of associated sessions. We identify two types of services: transparent, where the session is unaware of the server location, and configurable, where the sessions need to be configured to use their closest server. For both types we formalize the problem of optimally placing network servers and introduce approximated solutions. We present simulation result of approximations and heuristics. We also solve the location problem optimally for a special topology. We show, through a series of examples, that our approaches can be applied to a variety of different services