Fast Hybrid Network Algorithms for Shortest Paths in Sparse Graphs

We consider the problem of computing shortest paths in hybrid networks, in which nodes can make use of different communication modes. For example, mobile phones may use ad-hoc connections via Bluetooth or Wi-Fi in addition to the cellular network to solve tasks more efficiently. Like in this case, the different communication modes may differ considerably in range, bandwidth, and flexibility. We build upon the model of Augustine et al. [SODA \u2720], which captures these differences by a local and a global mode. Specifically, the local edges model a fixed communication network in which O(1) messages of size O(log n) can be sent over every edge in each synchronous round. The global edges form a clique, but nodes are only allowed to send and receive a total of at most O(log n) messages over global edges, which restricts the nodes to use these edges only very sparsely. We demonstrate the power of hybrid networks by presenting algorithms to compute Single-Source Shortest Paths and the diameter very efficiently in sparse graphs. Specifically, we present exact O(log n) time algorithms for cactus graphs (i.e., graphs in which each edge is contained in at most one cycle), and 3-approximations for graphs that have at most n + O(n^{1/3}) edges and arboricity O(log n). For these graph classes, our algorithms provide exponentially faster solutions than the best known algorithms for general graphs in this model. Beyond shortest paths, we also provide a variety of useful tools and techniques for hybrid networks, which may be of independent interest

Distance Computations in the Hybrid Network Model via Oracle Simulations

The Hybrid network model was introduced in [Augustine et al., SODA '20] for laying down a theoretical foundation for networks which combine two possible modes of communication: One mode allows high-bandwidth communication with neighboring nodes, and the other allows low-bandwidth communication over few long-range connections at a time. This fundamentally abstracts networks such as hybrid data centers, and class-based software-defined networks. Our technical contribution is a \emph{density-aware} approach that allows us to simulate a set of \emph{oracles} for an overlay skeleton graph over a Hybrid network. As applications of our oracle simulations, with additional machinery that we provide, we derive fast algorithms for fundamental distance-related tasks. One of our core contributions is an algorithm in the Hybrid model for computing \emph{exact} weighted shortest paths from $\tilde O(n^{1/3})$ sources which completes in $\tilde O(n^{1/3})$ rounds w.h.p. This improves, in both the runtime and the number of sources, upon the algorithm of [Kuhn and Schneider, PODC '20], which computes shortest paths from a single source in $\tilde O(n^{2/5})$ rounds w.h.p. We additionally show a 2-approximation for weighted diameter and a $(1+\epsilon)$-approximation for unweighted diameter, both in $\tilde O(n^{1/3})$ rounds w.h.p., which is comparable to the $\tilde \Omega(n^{1/3})$ lower bound of [Kuhn and Schneider, PODC '20] for a $(2-\epsilon)$-approximation for weighted diameter and an exact unweighted diameter. We also provide fast distance \emph{approximations} from multiple sources and fast approximations for eccentricities.Comment: To appear in STACS 202

Low Cost Quality of Service Multicast Routing in High Speed Networks

Many of the services envisaged for high speed networks, such as B-ISDN/ATM, will support real-time applications with large numbers of users. Examples of these types of application range from those used by closed groups, such as private video meetings or conferences, where all participants must be known to the sender, to applications used by open groups, such as video lectures, where partcipants need not be known by the sender. These types of application will require high volumes of network resources in addition to the real-time delay constraints on data delivery. For these reasons, several multicast routing heuristics have been proposed to support both interactive and distribution multimedia services, in high speed networks. The objective of such heuristics is to minimise the multicast tree cost while maintaining a real-time bound on delay. Previous evaluation work has compared the relative average performance of some of these heuristics and concludes that they are generally efficient, although some perform better for small multicast groups and others perform better for larger groups. Firstly, we present a detailed analysis and evaluation of some of these heuristics which illustrates that in some situations their average performance is reversed; a heuristic that in general produces efficient solutions for small multicasts may sometimes produce a more efficient solution for a particular large multicast, in a specific network. Also, in a limited number of cases using Dijkstra's algorithm produces the best result. We conclude that the efficiency of a heuristic solution depends on the topology of both the network and the multicast, and that it is difficult to predict. Because of this unpredictability we propose the integration of two heuristics with Dijkstra's shortest path tree algorithm to produce a hybrid that consistently generates efficient multicast solutions for all possible multicast groups in any network. These heuristics are based on Dijkstra's algorithm which maintains acceptable time complexity for the hybrid, and they rarely produce inefficient solutions for the same network/multicast. The resulting performance attained is generally good and in the rare worst cases is that of the shortest path tree. The performance of our hybrid is supported by our evaluation results. Secondly, we examine the stability of multicast trees where multicast group membership is dynamic. We conclude that, in general, the more efficient the solution of a heuristic is, the less stable the multicast tree will be as multicast group membership changes. For this reason, while the hybrid solution we propose might be suitable for use with closed user group multicasts, which are likely to be stable, we need a different approach for open user group multicasting, where group membership may be highly volatile. We propose an extension to an existing heuristic that ensures multicast tree stability where multicast group membership is dynamic. Although this extension decreases the efficiency of the heuristics solutions, its performance is significantly better than that of the worst case, a shortest path tree. Finally, we consider how we might apply the hybrid and the extended heuristic in current and future multicast routing protocols for the Internet and for ATM Networks.

Shortest Path and Distance Queries on Road Networks: An Experimental Evaluation

Computing the shortest path between two given locations in a road network is an important problem that finds applications in various map services and commercial navigation products. The state-of-the-art solutions for the problem can be divided into two categories: spatial-coherence-based methods and vertex-importance-based approaches. The two categories of techniques, however, have not been compared systematically under the same experimental framework, as they were developed from two independent lines of research that do not refer to each other. This renders it difficult for a practitioner to decide which technique should be adopted for a specific application. Furthermore, the experimental evaluation of the existing techniques, as presented in previous work, falls short in several aspects. Some methods were tested only on small road networks with up to one hundred thousand vertices; some approaches were evaluated using distance queries (instead of shortest path queries), namely, queries that ask only for the length of the shortest path; a state-of-the-art technique was examined based on a faulty implementation that led to incorrect query results. To address the above issues, this paper presents a comprehensive comparison of the most advanced spatial-coherence-based and vertex-importance-based approaches. Using a variety of real road networks with up to twenty million vertices, we evaluated each technique in terms of its preprocessing time, space consumption, and query efficiency (for both shortest path and distance queries). Our experimental results reveal the characteristics of different techniques, based on which we provide guidelines on selecting appropriate methods for various scenarios.Comment: VLDB201