No abstract

Given a planar graph G and a partition of the neighbors of each vertex v in four sets v, v, v, and v, the problem Windrose Planarity asks to decide whether G admits a windrose-planar drawing, that is, a planar drawing in which (i) each neighbor u ∈ v is above and to the right of v, (ii) each neighbor u ∈ v is above and to the left of v, (iii) each neighbor u ∈ v is below and to the left of v, (iv) each neighbor u ∈ v is below and to the right of v, and (v) edges are represented by curves that are monotone with respect to each axis. By exploiting both the horizontal and the vertical relationship among vertices, windrose-planar drawings allow to simultaneously visualize two partial orders defined by means of the edges of the graph.Although the problem is N P-hard in the general case, we give a polynomial-time algorithm for testing whether there exists a windrose-planar drawing that respects a given combinatorial embedding. This algorithm is based on a characterization of the plane triangulations admitting a windrose-planar drawing. Furthermore, for any embedded graph with n vertices that has a windrose-planar drawing, we can construct one with at most one bend per edge and with at most 2n − 5 bends in total, which lies on the 3n × 3n grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area.

With the increasing diffusion of Internet probing technologies, a large amount of regularly collected traceroutes are available for Internet Service Providers (ISPs) at low cost. We show how it is possible, given solely an arbitrary set of traceroutes, to spot routing paths that change similarly over time and aggregate them into inferred routing events. With respect to previous works, our approach does not require any knowledge of the network, does not need complex integration of several data sources, and exploits the asynchronicity of measurements to accurately position events in time. The formal model at the basis of our methodology revolves around the notion of empathy, a relation that binds similarly behaving traceroutes. The correctness and completeness of our approach are based on structural properties that are easily expressed in terms of empathic measurements. We perform experiments with data from public measurement infrastructures like RIPE Atlas, showing the effectiveness of our algorithm in distilling significant events from a large amount of traceroute data. We also validate the accuracy of the inferred events against ground-truth knowledge of routing changes originating from induced and spontaneous routing events. Given these promising results, we believe our methodology can be an effective aid for troubleshooting at the ISPs level. The source code of our algorithm is publicly available at https://github.com/empadig.

We introduce L-drawings, a novel paradigm for representing directed graphs aiming at combining the readability features of orthogonal drawings with the expressive power of adjacency matrix representations. In an L-drawing, vertices have exclusive [Formula: see text]- and [Formula: see text]-coordinates and edges consist of two segments, one exiting the source vertically and one entering the destination horizontally. We study the problem of computing L-drawings using minimum ink. We prove its NP-hardness and provide a heuristic based on a polynomial-time algorithm that adds a vertex to a drawing using the minimum additional ink. We performed an experimental analysis of the heuristic which confirms its effectiveness.

We introduce a hybrid metaphor for the visualization of the reconciliations of co-phylogenetic trees, that are mappings among the nodes of two trees. The typical application is the visualization of the co-evolution of hosts and parasites in biology. Our strategy combines a space-filling and a node-link approach. Differently from traditional methods, it guarantees an unambiguous and 'downward' representation whenever the reconciliation is time-consistent (i.e., meaningful). We address the problem of the minimization of the number of crossings in the representation, by giving a characterization of planar instances and by establishing the complexity of the problem. Finally, we propose heuristics for computing representations with few crossings.