A Survey on Small-Area Planar Graph Drawing
We survey algorithms and bounds for constructing planar drawings of graphs in small area.
cs.CG↗
arXiv subjects
Giuseppe Di Battista
Publications and source records attributed to Giuseppe Di Battista.
Advances on Testing C-Planarity of Embedded Flat Clustered Graphs
We show a polynomial-time algorithm for testing c-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face.
cs.DS↗
On the Complexity of Clustered-Level Planarity and T-Level Planarity
In this paper we study two problems related to the drawing of level graphs, that is, T-LEVEL PLANARITY and CLUSTERED-LEVEL PLANARITY. We show that both problems are NP-complete in the general case and that they become polynomial-time solvable when restricted to proper instances.
cs.DS↗
Morphing Planar Graph Drawings Optimally
We provide an algorithm for computing a planar morph between any two planar straight-line drawings of any $n$-vertex plane graph in $O(n)$ morphing steps, thus improving upon the previously best known $O(n^2)$ upper bound. Further, we prove that our algorithm is optimal, that is, we show that there exist two planar straight-line drawings $Γ_s$ and $Γ_t$ of an $n$-vertex plane graph $G$ such that any planar morph between $Γ_s$ and $Γ_t$ requires $Ω(n)$ morphing steps.
cs.DS↗
Relaxing the Constraints of Clustered Planarity
In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region, or region-region crossings. Determining the complexity of testing whether a clustered graph admits a c-planar drawing is a long-standing open problem in the Graph Drawing research area. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, such a condition is not sufficient and the consequences on the problem due to the requirement of not having edge-region and region-region crossings are not yet fully understood. In order to shed light on the c-planarity problem, we consider a relaxed version of it, where some kinds of crossings (either edge-edge, edge-region, or region-region) are allowed even if the underlying graph is planar. We investigate the relationships among the minimum number of edge-edge, edge-region, and region-region crossings for drawings of the same clustered graph. Also, we consider drawings in which only crossings of one kind are admitted. In this setting, we prove that drawings with only edge-edge or with only edge-region crossings always exist, while drawings with only region-region crossings may not. Further, we provide upper and lower bounds for the number of such crossings. Finally, we give a polynomial-time algorithm to test whether a drawing with only region-region crossings exist for biconnected graphs, hence identifying a first non-trivial necessary condition for c-planarity that can be tested in polynomial time for a noticeable class of graphs.
cs.DM↗
From BGP to RTT and Beyond: Matching BGP Routing Changes and Network Delay Variations with an Eye on Traceroute Paths
Many organizations have the mission of assessing the quality of broadband access services offered by Internet Service Providers (ISPs). They deploy network probes that periodically perform network measures towards selected Internet services. By analyzing the data collected by the probes it is often possible to gain a reasonable estimate of the bandwidth made available by the ISP. However, it is much more difficult to use such data to explain who is responsible of the fluctuations of other network qualities. This is especially true for latency, that is fundamental for several nowadays network services. On the other hand, there are many publicly accessible BGP routers that collect the history of routing changes and that are good candidates to be used for understanding if latency fluctuations depend on interdomain routing. In this paper we provide a methodology that, given a probe that is located inside the network of an ISP and that executes latency measures and given a set of publicly accessible BGP routers located inside the same ISP, decides which routers are best candidates (if any) for studying the relationship between variations of network performance recorded by the probe and interdomain routing changes. We validate the methodology with experimental studies based on data gathered by the RIPE NCC, an organization that is well-known to be independent and that publishes both BGP data within the Routing Information Service (RIS) and probe measurement data within the Atlas project.
cs.NI↗
Strip Planarity Testing of Embedded Planar Graphs
In this paper we introduce and study the strip planarity testing problem, which takes as an input a planar graph $G(V,E)$ and a function $γ:V \rightarrow \{1,2,\dots,k\}$ and asks whether a planar drawing of $G$ exists such that each edge is monotone in the $y$-direction and, for any $u,v\in V$ with $γ(u)<γ(v)$, it holds $y(u)<y(v)$. The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, upward planarity, and level planarity. We show that the problem is polynomial-time solvable if $G$ has a fixed planar embedding.
cs.CG↗
Intra-Domain Pathlet Routing
Internal routing inside an ISP network is the foundation for lots of services that generate revenue from the ISP's customers. A fine-grained control of paths taken by network traffic once it enters the ISP's network is therefore a crucial means to achieve a top-quality offer and, equally important, to enforce SLAs. Many widespread network technologies and approaches (most notably, MPLS) offer limited (e.g., with RSVP-TE), tricky (e.g., with OSPF metrics), or no control on internal routing paths. On the other hand, recent advances in the research community are a good starting point to address this shortcoming, but miss elements that would enable their applicability in an ISP's network. We extend pathlet routing by introducing a new control plane for internal routing that has the following qualities: it is designed to operate in the internal network of an ISP; it enables fine-grained management of network paths with suitable configuration primitives; it is scalable because routing changes are only propagated to the network portion that is affected by the changes; it supports independent configuration of specific network portions without the need to know the configuration of the whole network; it is robust thanks to the adoption of multipath routing; it supports the enforcement of QoS levels; it is independent of the specific data plane used in the ISP's network; it can be incrementally deployed and it can nicely coexist with other control planes. Besides formally introducing the algorithms and messages of our control plane, we propose an experimental validation in the simulation framework OMNeT++ that we use to assess the effectiveness and scalability of our approach.
cs.NI↗
Implementing a Partitioned 2-page Book Embedding Testing Algorithm
In a book embedding the vertices of a graph are placed on the "spine" of a "book" and the edges are assigned to "pages" so that edges on the same page do not cross. In the Partitioned 2-page Book Embedding problem egdes are partitioned into two sets E_1 and E_2, the pages are two, the edges of E_1 are assigned to page 1, and the edges of E_2 are assigned to page 2. The problem consists of checking if an ordering of the vertices exists along the spine so that the edges of each page do not cross. Hong and Nagamochi give an interesting and complex linear time algorithm for tackling Partitioned 2-page Book Embedding based on SPQR-trees. We show an efficient implementation of this algorithm and show its effectiveness by performing a number of experimental tests. Because of the relationships between Partitioned 2-page Book Embedding and clustered planarity we yield as a side effect an implementation of a clustered planarity testing where the graph has exactly two clusters.
cs.DS↗
Computational Complexity of Traffic Hijacking under BGP and S-BGP
Harmful Internet hijacking incidents put in evidence how fragile the Border Gateway Protocol (BGP) is, which is used to exchange routing information between Autonomous Systems (ASes). As proved by recent research contributions, even S-BGP, the secure variant of BGP that is being deployed, is not fully able to blunt traffic attraction attacks. Given a traffic flow between two ASes, we study how difficult it is for a malicious AS to devise a strategy for hijacking or intercepting that flow. We show that this problem marks a sharp difference between BGP and S-BGP. Namely, while it is solvable, under reasonable assumptions, in polynomial time for the type of attacks that are usually performed in BGP, it is NP-hard for S-BGP. Our study has several by-products. E.g., we solve a problem left open in the literature, stating when performing a hijacking in S-BGP is equivalent to performing an interception.
cs.NI↗
Object-Oriented Design of Graph Oriented Data Structures
Applied research in graph algorithms and combinatorial structures needs comprehensive and versatile software libraries. However, the design and the implementation of flexible libraries are challenging activities. Among the other problems involved in such a difficult field, a very special role is played by graph classification issues. We propose new techniques devised to help the designer and the programmer in the development activities. Such techniques are especially suited for dealing with graph classification problems and rely on an extension of the usual object-oriented paradigm. In order to support the usage of our approach, we devised an extension of the C++ programming language and implemented the corresponding pre-compiler.
cs.SE↗