Juraj Hromkovic y , Ralf Klasing, Elena A. Stohr, Hubert Wagener z Department of Mathematics and Computer Science University of Paderborn, 33095 Paderborn, Germany Abstract The communication modes (one-way and two-way mode) used for sending messages to processors of interconnection networks via vertex-disjoint paths in one communication step are investigated. The complexity of communication algorithms is measured by the number of communication steps (rounds). Here, the complexity of gossiping in grids and in planar graphs is investigated. The main results are the following: 1. Effective one-way and two-way gossip algorithms for d-dimensional grids, d 2, are designed. 2. The lower bound 2 log 2 n \Gamma log 2 k \Gamma log 2 log 2 n \Gamma 2 is established on the number of rounds of every two-way gossip algorithm working on any graph of n nodes and vertex bisection k. This proves that the designed two-way gossip algorithms on d-dimensional grids, d 3, are almost optimal, and it al...

In this paper, we show that the multicast problem in trees can be expressed in term of arranging rows and columns of boolean matrices. Given a p \Theta q matrix M with 0-1 entries, the shadow of M is defined as a boolean vector x of q entries such that x i = 0 if and only if there is no 1-entry in the ith column of M , and x i = 1 otherwise. (The shadow x can also be seen as the binary expression of the integer x = P q i=1 x i 2 q\Gammai . Similarly, every row of M can be seen as the binary expression of an integer.) According to this formalism, the key for solving a multicast problem in trees is shown to be the following. Given a p \Theta q matrix M with 0-1 entries, finding a matrix M such that: 1. M has at most one 1-entry per column; 2. every row r of M (viewed as the binary expression of an integer) is larger than the corresponding row r of M , 1 r p; and 3. the shadow of M (viewed as an integer) is minimum. We show that there is an O(q(p + q)) algorithm that fin...

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...

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).

Given a node s of an n-node network G = (V; E), and a set of nodes D V, multicasting from s to D consists to transmit a piece of information from s to all the nodes in D. Broadcasting is the particular case in which D = V, and gossiping in D consists to perform multicast simultaneously from every node in D to all the other nodes in D. Motivated by switching technologies allowing distance-insensitive communications (e.g., circuit-switching, single-hop WDM optical routing, etc.), we consider networks in which nodes are allowed to place long-distance transmission calls, and we derive algorithms which return ecient multicast, broadcast, and gossip protocols for these networks. In particular, we give a polynomial-time O( log n log t opt)-approximation algorithm for the multicast problem where t opt is the completion-time of an optimal multicast protocol. This improves a previous O( log n log log n)-approximation algorithm by Kortsarz and Peleg [29], e.g., by a factor ( log n log log n) for networks of broadcast time t opt = n), > 0. We also give an O(1)-approximation algorithm for the gossip problem. All phases of our communication protocols are free of link and node contention between messages. Therefore our protocols limit the congestion inside the network. If we slightly relax the model by allowing node-contention but not linkcontention, then we show that there is an ln

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.

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.