Maurizio Patrignani

Computing the types of the relationships between autonomous systems

We investigate the problem of computing the types of the relationships between Internet Autonomous Systems. We refer to the model introduced by Gao [IEEE/ACM Transactions on Networking, 9(6):733-645, 2001] and Subramanian (IEEE Infocom, 2002) that bases the discovery of such relationships on the analysis of the AS paths extracted from the BGP routing tables. We characterize the time complexity of the above problem, showing both NP-completeness results and efficient algorithms for solving specific cases. Motivated by the hardness of the general problem, we propose approximation algorithms and heuristics based on a novel paradigm and show their effectiveness against publicly available data sets. The experiments provide evidence that our algorithms perform significantly better than state-of-the-art heuristics

Visual analysis of large graphs using (X,Y)-clustering and hybrid visualizations

Many different approaches have been proposed for the challenging problem of visually analyzing large networks. Clustering is one of the most promising. In this paper, we propose a new clustering technique whose goal is that of producing both intracluster graphs and intercluster graph with desired topological properties. We formalize this concept in the (X,Y) -clustering framework, where Y is the class that defines the desired topological properties of intracluster graphs and X is the class that defines the desired topological properties of the intercluster graph. By exploiting this approach, hybrid visualization tools can effectively combine different node-link and matrix-based representations, allowing users to interactively explore the graph by expansion/contraction of clusters without loosing their mental map. As a proof of concept, we describe the system Visual Hybrid (X,Y)-clustering (VHYXY) that implements our approach and we present the results of case studies to the visual analysis of social networks.

Visualizing Interdomain Routing with BGPlay

In this paper we describe the architecture and the visual interface of BGPlay, an on-line service for the visualization of the behavior and of the instabilities of Internet routing at the autonomous system level. A graph showing only the connections among autonomous systems is not enough to convey all the information needed to fully understand routing and its changes. BGPlay uses specifically tailored techniques and algorithms to display the state of routing at specific points in time and to animate its changes. The system obtains routing data from well known on-line archives of routing information which are constantly kept up-todate.

On the complexity of orthogonal compaction

Abstract We consider three closely related optimization problems, arising from the graph drawing and the VLSI research areas, and conjectured to be NP-hard, and we prove that, in fact, they are NP-complete. Starting from an orthogonal representation of a graph, i.e., a description of the shape of the edges that does not specify segment lengths or vertex positions, the three problems consist of providing an orthogonal grid drawing of it, while minimizing the area, the total edge length, or the maximum edge length, respectively. This result confirms a long surviving conjecture of NP-hardness, justifies the research about applying sophisticated, yet possibly time consuming, techniques to obtain optimally compacted orthogonal grid drawings, and discourages the quest for an optimally compacting polynomial-time algorithm.

Orthogonal and Quasi-upward Drawings with Vertices of Prescribed Size

We consider the problem of computing orthogonal drawings and quasi-upward drawings with vertices of prescribed size. For both types of drawings we present algorithms based on network flow techniques and show that the produced drawings are optimal within a wide class. Further, we present the results of an experimentation conducted on the algorithms that we propose for orthogonal drawings. The experiments show the effectiveness of the approach.

C-Planarity of C-Connected Clustered Graphs

We present the first characterization of c-planarity for c-connected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we provide a linear-time c-planarity testing and embedding algorithm for c-connected clustered graphs. The algorithm is reasonably easy to implement, since it exploits as building blocks simple algorithmic tools like the computation of lowest common ancestors, minimum and maximum spanning trees, and counting sorts. It also makes use of well-known data structures as SPQR-trees and BC-trees. If the test fails, the algorithm identifies a structural element responsible for the non-cplanarity of the input clustered graph.

ON EXTENDING A PARTIAL STRAIGHT-LINE DRAWING

We investigate the computational complexity of the following problem. Given a planar graph in which some vertices have already been placed in the plane, place the remaining vertices to form a plana...

The Complexity of the Matching-Cut Problem

Finding a cut or finding a matching in a graph are so simple problems that they are hardly considered problems at all. In this paper, by means of a reduction from the NAE3SAT problem, we prove that combining these two problems together, i.e., finding a cut whose split edges are a matching is an NP-complete problem. It remains intractable even if we impose the graph to be simple (no multiple edges allowed) or its maximum degree to be k, with k ? 4. On the contrary, we give a linear time algorithm that computes a matching-cut of a series-parallel graph. Its open whether the problem is tractable or not for planar graphs.

Topographic Visualization of Prefix Propagation in the Internet

We propose a new metaphor for the visualization of prefixes propagation in the Internet. Such a metaphor is based on the concept of topographic map and allows to put in evidence the relative importance of the Internet Service Providers (ISPs) involved in the routing of the prefix. Based on the new metaphor we propose an algorithm for computing layouts and experiment with such algorithm on a test suite taken from the real Internet. The paper extends the visualization approach of the BGPlay service, which is an Internet routing monitoring tool widely used by ISP operators

Fan-planarity

In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt 35], who proved that every n-vertex fan-planar drawing has at most 5 n - 10 edges, and that this bound is tight for n ? 20 . We extend their result from both the combinatorial and the algorithmic point of view. We prove tight bounds on the density of constrained versions of fan-planar drawings and study the relationship between fan-planarity and k-planarity. Also, we prove that testing fan-planarity in the variable embedding setting is NP-complete.In a \emph{fan-planar drawing} of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt, who proved that every