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...

We study deterministic gossiping in synchronous systems with dynamic crash failures. Each processor is initialized with an input value called rumor. In the standard gossip problem, the goal of every processor is to learn all the rumors. When processors may crash, then this goal needs to be revised, since it is possible, at a point in an execution, that certain rumors are known only to processors that have already crashed. We define gossiping to be completed, for a system with crashes, when every processor knows either the rumor of processor v or that v has already crashed, for any processor v. We design gossiping algorithms that are efficient with respect to both time and communication. Let t < n be the number of failures, where n is the number of processors. If n − t = Ω(n/polylog n), then one of our algorithms completes gossiping in O(log 2 t) time and with O(n polylog n) messages. We develop an algorithm that performs gossiping with O(n 1.77) messages and in O(log 2 n) time, in any execution in which at least one processor remains non-faulty. We show a trade-off between time and communication in gossiping algorithms: if the number of messages is at most O(n polylog n), then the time has to be at least Ω ( log n. By way of application, we show that if n − t = Ω(n), then log(n log n)−log t consensus can be solved in O(t) time and with O(n log 2 t) messages.

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...

A concept of systolic dissemination of information in interconnection networks is presented, and the complexity of systolic gossip and broadcast in one-way (telegraph) and two-way (telephone) communication mode is investigated. The following main results are established: (i) a general relation between systolic broadcast and systolic gossip, (ii) optimal systolic gossip algorithms on paths in both communication modes, and (iii) optimal systolic gossip algorithms for complete k-ary trees in both communication modes.

In the gossiping problem, each node in a network starts with a unique piece of information and must acquire the information of all other nodes using two-way communications between pairs of nodes. In this paper we investigate gossiping in n-node networks with n odd. We use a linear cost model in which the cost of communication is proportional to the amount of information transmitted. In synchronous gossiping, the pairwise communications are organized into rounds, and all communications in a round start at the same time. We present optimal synchronous algorithms for all odd values of n. In asynchronous gossiping, a pair of nodes can start communicating while communications between other pairs are in progress. We provide a short intuitive proof that an asynchronous lower bound due to Peters, Raabe, and Xu is not tight.

Given a network of processors, a structured communication problem consists to route a communication pattern known in advance. Structured communication problems appear frequently in parallel computing. Hence, communication libraries (e.g, PVM or MPI) generally include a specific access to procedures solving the most common problems of this type. A standard communication model assumes that information proceeds by a sequence of calls between neighboring nodes of the network, and that each node is allowed to call at most one neighbor at a time. In this context, most of the decision problems corresponding to the usual structured communication problems have been shown to be NP-complete. Therefore, several approximation algorithms have been proposed to solve specific problems. Each of these algorithms is dedicated to a particular problem. In this paper, we present a high level method which can be used to derive approximation algorithms for many different structured communication problems on ...

. This article focuses on principles for the design of e#cient parallel algorithms for distributed memory computing systems. We describe the general trend in the development of architectural properties and evaluate the state-of-the-art in a number of basic primitives like graph embedding, partitioning, dynamic load distribution, and communication which are used, to some extent, within all parallel applications. We discuss possible directions for future work on the design of universal basic primitives, able to perform e#ciently on a broad range of parallel systems and applications, and we also give certain examples of speci#c applications which demand specialized basic primitives in order to obtain e#cient parallel implementations. Finally,we show that programming frames can o#er a convenientway to encapsulate algorithmic know-how on applications and basic primitives and to o#er this knowledge to nonspecialist users in a very e#ectiveway. 1 Introduction Parallel processi...

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...

Motivated by applications in surveillance sensor networks, we present a distributed algorithm for the automatic, external, metric calibration of a network of cameras with no centralized processor. We model the set of uncalibrated cameras as nodes in a communication network, and propose a distributed algorithm in which each camera only communicates with other cameras that image some of the same scene points. Each node independently forms a neighborhood cluster on which the local calibration takes place, and calibrated nodes and scene points are incrementally merged into a common coordinate frame. The accurate performance of the algorithm is illustrated using examples that model real-world sensor networking situations.

We consider the broadcasting problem in the shouting communication mode in which any node of a network can inform all its neighbours in one time step. In addition, during any time step a number of links less than the edge-connectivity of the network can be faulty. The problem is to find an upper bound on the number of time steps necessary to complete broadcasting under this additional assumption. Fraigniaud and Peyrat proved for the n-dimensional hypercube that n + O(log n) time steps are sufficient. De Marco and Vaccaro decreased the upper bound to n+7 and showed a worst case lower bound n + 2 for n 3. We prove that n + 2 time steps are sufficient. Our method is related to the isoperimetric problem in graphs and can be applied to other networks.