The communication power of the one-way and two-way edgedisjoint path modes for broadcast and gossip is investigated. The complexity of communication algorithms is measured by the number of communication steps (rounds). The main results achieved are the following: 1. For each connected graph Gn of n nodes, the complexity of broadcast in Gn , Bmin(Gn ), satisfies dlog 2 ne Bmin(Gn) dlog 2 ne + 1. The complete binary trees meet the upper bound, and all graphs containing a Hamiltonian path meet the lower bound. 2. For each connected graph Gn of n nodes, the one-way (two-way) gossip complexity R(Gn ) (R 2 (Gn)) satisfies dlog 2 ne R 2 (Gn) 2 \Delta dlog 2 ne + 1, 1:44 : : : log 2 n R(Gn) 2 \Delta dlog 2 ne + 2. All these lower and upper bounds a...

This paper addresses the one-to-all broadcasting problem, and the one-to-many broadcasting problem, usually simply called broadcasting and multicasting, respectively. Broadcasting is the information dissemination problem in which a node of a network sends the same piece of information to all the other nodes. Multicasting is a partial broadcasting in the sense that only a subset of nodes forms the destination set. Both operations have many applications in parallel and distributed computing. In this paper, we study these problems in both line model, and cut-through model. The former assumes long distance calls between non-neighboring processors. The latter strengthens the line model by taking into account the use of a routing function. Long distance calls are possible in circuit-switched and wormhole routed networks, and also in many networks supporting optical facilities. In the line model, it is well-known that one can compute in polynomial time a dlog 2 ne-round broadcast o...

) Ralf Klasing Department of Mathematics and Computer Science University of Paderborn 33095 Paderborn, Germany Abstract. The two-way communication mode used for sending messages to processors of interconnection networks via vertex-disjoint paths in one communication step is investigated. The complexity of communication algorithms is measured by the number of communication steps (rounds). This paper establishes a direct relationship between the gossip complexity and the vertex bisection width. More precisely, the main results are the following: 1. The lower bound 2 log 2 n \Gamma log 2 k \Gamma log 2 log 2 k \Gamma 2 is proved on the number of rounds of every two-way gossip algorithm working on any graph Gn;k of n nodes and vertex bisection k. 2. A graph Gn;k of n nodes and vertex bisection k, and a two-way gossip algorithm for Gn;k is constructed working in 2 log 2 n \Gamma log 2 k \Gamma log 2 log 2 k + 2 rounds. The first result improves the lower bound of 2 log 2 n\Gammalog 2 k \...

We give a polynomial-time O( log n log OPT)-approximation algorithm for minimum-time broadcast and minimum-time multicast in n-node networks under the single-port vertex-disjoint paths mode. This improves a previous approximation algorithm by Kortsarz and Peleg. In contrast, we give an (log n) lower bound for the approximation ratio of the minimum-time multicast problem in directed networks. This lower bound holds unless NP Dtime(n log log n). An important consequence of this latter result is that the Steiner version of the Minimum Degree Spanning Tree (MDST) problem in digraphs cannot be approximated within a constant ratio, as opposed to the undirected version. Finally, we give a polynomial-time O(1)-approximation algorithm for minimumtime gossip (i.e., all-to-all broadcast).

This paper addresses the minimum-time broadcast problem under several modes of the line model, i.e., when long-distance calls can be placed along paths in the network.

Graphs are models of communication networks. This paper applies symbolic combinatorial techniques in order to characterize the interplay between two parameters of a random graph, namely its density (the number of edges in the graph) and its robustness to link failures. Here, robustness means multiple connectivity by short disjoint paths. We determine the expected number of ways to get from a source to a destination via two edgedisjoint paths of length ` in the classical random graph model G n;p . We then derive bounds on related threshold probabilities.

This paper is devoted to the gossip (or all-to-all) problem in the chordal ring under the one-port line model. The line model assumes long distance calls between non neighboring processors. In this sense, the line model is strongly related to circuit-switched networks, wormhole routing, optical networks supporting wavelength division multiplexing, ATM switching, and networks supporting connected mode routing protocols. Since the chordal rings are competitors of networks as meshes or tori because of theirs short diameter and bounded degree, it is of interest to ask whether they can support intensive communications (typically all-to-all) as e ciently as these networks. We propose polynomial algorithms to derive optimal or near optimal gossip protocols in the chordal ring.