Graphs are commonly used to model the topological structure of internetworks, to study problems ranging from routing to resource reservation. A variety of graphs are found in the literature, including fixed topologies such as rings or stars, "well-known" topologies such as the ARPAnet, and randomly generated topologies. While many researchers rely upon graphs for analytic and simulation studies, there has been little analysis of the implications of using a particular model, or how the graph generation method may a ect the results of such studies. Further, the selection of one generation method over another is often arbitrary, since the differences and similarities between methods are not well understood. This paper considers the problem of generating and selecting graph models that reflect the properties of real internetworks. We review generation methods in common use, and also propose several new methods. We consider a set of metrics that characterize the graphs produced by a method, and we quantify similarities and differences amongst several generation methods with respect to these metrics. We also consider the effect of the graph model in the context of a speciffic problem, namely multicast routing.

A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2-connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be NP -hard. We consider the problem of finding a better approximation to the smallest 2-connected subgraph, by an efficient algorithm. For 2-edge connectivity our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2-vertex connectivity our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP -hard as well. We also consider the case where the graph has edge weigh...

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Contents 1 Introduction 2 1.1 Outline of Chapter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Edge-Connectivity Problems 3 2.1 Weighted Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Unweighted Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.1 2 Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.2 Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3 Vertex-Connectivity Problems 11 3.1 Weighted Vertex-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3.2 Unweighted Vertex-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3.2.1 2 Vertex-Connectivity : : : : : : : : : : : : : : : : :

We present an approximation algorithm for solving graph problems in which a low-cost set of edges must be selected that has certain vertex-connectivity properties. In the survivable network design problem, one is given a value r ij for each pair of vertices i and j, and must find a minimum-cost set of edges such that there are r ij vertex-disjoint paths between vertices i and j. In the case for which r ij 2 f0; 1; 2g for all i; j, we can find a solution of cost no more than 3 times the optimal cost in polynomial time. In the case in which r ij = k for all i; j, we can find a solution of cost no more than 2H(k) times optimal, where H(n) = 1 + 1 2 + \Delta \Delta \Delta + 1 n . No approximation algorithms were previously known for these problems. Our algorithms rely on a primal-dual approach which has recently led to approximation algorithms for many edge-connectivity problems. 1 Introduction Let G = (V; E) be an undirected graph with non-negative costs c e 0 on all edges e 2 E. In...

Rumor mongering (also known as gossip) is an epidemiological protocol that implements broadcasting with a reliability that can be very high. Rumor mongering is attractive because it is generic, scalable, adapts well to failures and recoveries, and has a reliability that gracefully degrades with the number of failures in a run. However, rumor mongering uses random selection for communications. We study the impact of using random selection in this paper. We present a protocol that superficially resembles rumor mongering but is deterministic. We show that this new protocol has most of the same attractions as rumor mongering. The one attraction that rumor mongering has---namely graceful degradation---comes at a high cost in terms of the number of messages sent. We compare the two approaches both at an abstract level and in terms of how they perform in an Ethernet and small wide area network of Ethernets.

Ensemble is a Group Communication System built at Cornell and the Hebrew universities. It allows processes to create process groups within which scalable reliable fifo-ordered multicast and point-to-point communication are supported. The system also supports other communication properties, such as causal and total multicast ordering, flow control, etc. This paper describes the security protocols and infrastructure of Ensemble. Applications using Ensemble with the extensions described here benefit from strong security properties. Under the assumption that trusted processes will not be corrupted, all communication is secured from tampering by outsiders. Our work extends previous work performed in the Horus system (Ensemble’s predecessor) by adding support for multiple partitions, efficient rekeying, and application defined security policies. Unlike Horus, which used its own security infrastructure with non-standard key distribution and timing services, Ensemble’s security mechanism is based on off-the shelf authentication systems, such as PGP and Kerberos. We extend previous results on group rekeying, with a novel protocol that makes use of diamond-like data structures. Our Diamond protocol allows the removal of untrusted members within milliseconds.

This paper presents Araneola 1, a scalable reliable application-level multicast system for highly dynamic wide-area environments. Araneola supports multi-point to multi-point reliable communication in a fully distributed manner while incurring constant load (in terms of message and space complexity) on each node. For a tunable parameter k ≥ 3, Araneola constructs and dynamically maintains a basic overlay structure in which each node’s degree is either k or k +1, and roughly 90 % of the nodes have degree k. Empirical evaluation shows that Araneola’s basic overlay achieves three important mathematical properties of k-regular random graphs (i.e., random graphs in which each node has exactly k neighbors) with N nodes: (i) its diameter grows logarithmically with N; (ii) it is generally k-connected; and (iii) it remains highly connected following random removal of linear-size subsets of edges or nodes. The overlay is constructed and maintained at a low cost: each join, leave, or failure is handled locally, and entails the sending of only about 3k messages in total, independent of N. Moreover, this cost decreases as the churn rate increases. The low degree of Araneola’s basic overlay structure allows for allocating plenty of additional bandwidth for specific application needs. In this paper, we give an example for such a need — communicating with nearby nodes; we enhance the basic overlay with additional links chosen according to geographic

This paper describes and evaluates Fireflies, a scalable protocol for supporting intrusion-tolerant network overlays. While such a protocol cannot distinguish Byzantine nodes from correct nodes in general, Fireflies provides correct nodes with a reasonably current view of which nodes are live, as well as a pseudo-random mesh for communication. The amount of data sent by correct nodes grows linearly with the aggregate rate of failures and recoveries, even if provoked by Byzantine nodes. The set of correct nodes form a connected submesh; correct nodes cannot be eclipsed by Byzantine nodes. Fireflies is deployed and evaluated on PlanetLab. 1.

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