Non-crossing shortest paths in planar graphs with applications to max flow, and path graphs

This thesis is concerned with non-crossing shortest paths in planar graphs with applications to st-max flow vitality and path graphs. In the first part we deal with non-crossing shortest paths in a plane graph G, i.e., a planar graph with a fixed planar embedding, whose extremal vertices lie on the same face of G. The first two results are the computation of the lengths of the non-crossing shortest paths knowing their union, and the computation of the union in the unweighted case. Both results require linear time and we use them to describe an efficient algorithm able to give an additive guaranteed approximation of edge and vertex vitalities with respect to the st-max flow in undirected planar graphs, that is the max flow decrease when the edge/vertex is removed from the graph. Indeed, it is well-known that the st-max flow in an undirected planar graph can be reduced to a problem of non-crossing shortest paths in the dual graph. We conclude this part by showing that the union of non-crossing shortest paths in a plane graph can be covered with four forests so that each path is contained in at least one forest. In the second part of the thesis we deal with path graphs and directed path graphs, where a (directed) path graph is the intersection graph of paths in a (directed) tree. We introduce a new characterization of path graphs that simplifies the existing ones in the literature. This characterization leads to a new list of local forbidden subgraphs of path graphs and to a new algorithm able to recognize path graphs and directed path graphs. This algorithm is more intuitive than the existing ones and does not require sophisticated data structures

OutFlank Routing: Increasing Throughput in Toroidal Interconnection Networks

We present a new, deadlock-free, routing scheme for toroidal interconnection networks, called OutFlank Routing (OFR). OFR is an adaptive strategy which exploits non-minimal links, both in the source and in the destination nodes. When minimal links are congested, OFR deroutes packets to carefully chosen intermediate destinations, in order to obtain travel paths which are only an additive constant longer than the shortest ones. Since routing performance is very sensitive to changes in the traffic model or in the router parameters, an accurate discrete-event simulator of the toroidal network has been developed to empirically validate OFR, by comparing it against other relevant routing strategies, over a range of typical real-world traffic patterns. On the 16x16x16 (4096 nodes) simulated network OFR exhibits improvements of the maximum sustained throughput between 14% and 114%, with respect to Adaptive Bubble Routing.Comment: 9 pages, 5 figures, to be presented at ICPADS 201

Fault-tolerant additive weighted geometric spanners

Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance d_w(p, q) between two points p,q belonging to S is defined as w(p) + d(p, q) + w(q) if p \ne q and it is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance between p and q. A graph G(S, E) is called a t-spanner for the additive weighted set S of points if for any two points p and q in S the distance between p and q in graph G is at most t.d_w(p, q) for a real number t > 1. Here, d_w(p,q) is the additive weighted distance between p and q. For some integer k \geq 1, a t-spanner G for the set S is a (k, t)-vertex fault-tolerant additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set S' \subset S with cardinality at most k, the graph G \ S' is a t-spanner for the points in S \ S'. For any given real number \epsilon > 0, we obtain the following results: - When the points in S belong to Euclidean space R^d, an algorithm to compute a (k,(2 + \epsilon))-VFTAWS with O(kn) edges for the metric space (S, d_w). Here, for any two points p, q \in S, d(p, q) is the Euclidean distance between p and q in R^d. - When the points in S belong to a simple polygon P, for the metric space (S, d_w), one algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n}) edges and another algorithm to compute a geodesic (k, (\sqrt{10} + \epsilon))-VFTAWS with O(kn(\lg{n})^2) edges. Here, for any two points p, q \in S, d(p, q) is the geodesic Euclidean distance along the shortest path between p and q in P. - When the points in $S$ lie on a terrain T, an algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n}) edges.Comment: a few update

Improved Purely Additive Fault-Tolerant Spanners

Let $G$ be an unweighted $n$-node undirected graph. A \emph{$\beta$-additive spanner} of $G$ is a spanning subgraph $H$ of $G$ such that distances in $H$ are stretched at most by an additive term $\beta$ w.r.t. the corresponding distances in $G$. A natural research goal related with spanners is that of designing \emph{sparse} spanners with \emph{low} stretch. In this paper, we focus on \emph{fault-tolerant} additive spanners, namely additive spanners which are able to preserve their additive stretch even when one edge fails. We are able to improve all known such spanners, in terms of either sparsity or stretch. In particular, we consider the sparsest known spanners with stretch $6$, $28$, and $38$, and reduce the stretch to $4$, $10$, and $14$, respectively (while keeping the same sparsity). Our results are based on two different constructions. On one hand, we show how to augment (by adding a \emph{small} number of edges) a fault-tolerant additive \emph{sourcewise spanner} (that approximately preserves distances only from a given set of source nodes) into one such spanner that preserves all pairwise distances. On the other hand, we show how to augment some known fault-tolerant additive spanners, based on clustering techniques. This way we decrease the additive stretch without any asymptotic increase in their size. We also obtain improved fault-tolerant additive spanners for the case of one vertex failure, and for the case of $f$ edge failures.Comment: 17 pages, 4 figures, ESA 201

Path computation in multi-layer networks: Complexity and algorithms

Carrier-grade networks comprise several layers where different protocols coexist. Nowadays, most of these networks have different control planes to manage routing on different layers, leading to a suboptimal use of the network resources and additional operational costs. However, some routers are able to encapsulate, decapsulate and convert protocols and act as a liaison between these layers. A unified control plane would be useful to optimize the use of the network resources and automate the routing configurations. Software-Defined Networking (SDN) based architectures, such as OpenFlow, offer a chance to design such a control plane. One of the most important problems to deal with in this design is the path computation process. Classical path computation algorithms cannot resolve the problem as they do not take into account encapsulations and conversions of protocols. In this paper, we propose algorithms to solve this problem and study several cases: Path computation without bandwidth constraint, under bandwidth constraint and under other Quality of Service constraints. We study the complexity and the scalability of our algorithms and evaluate their performances on real topologies. The results show that they outperform the previous ones proposed in the literature.Comment: IEEE INFOCOM 2016, Apr 2016, San Francisco, United States. To be published in IEEE INFOCOM 2016, \<http://infocom2016.ieee-infocom.org/\&g