David W. Krumme

Gossiping in minimal time

The gossip problem involves communicating a unique item from each node in a graph to every other node. This paper studies the minimum time required to do this under the weakest model of parallel communication, which allows each node to participate in just one communication at a time as either sender or receiver. A number of topologies are studied, including the omplete graph, grids, hypercubes, and rings. Definitive new optimal time algorithms are derived for complete graphs, rings, regular grids, and toroidal grids that significantly extend existing results. In particular, an open problem about minimum time gossiping in complete graphs is settled. Specifically, for a graph with N nodes, at least

Fast gossiping for the hypercube

\log _\rho N

Diameter-preserving orientations of the torus

communication steps, where the logarithm is in the base of the golden ratio

MULTI-SOURCE SPANNING TREE PROBLEMS

\rho

A practical method for code generation based on exhaustive search

, are required by any algorithm under the weakest model of communication. This bound, which is approximately

Reordered gossip schemes

1.44\log _2 N

Portable execution traces for parallel program debugging and performance visualization

, can be realized for some networks and so the result is optimal.

Integrated debugging and performance monitoring for parallel programs

The gossip problem involves communicating a unique item from every node in a graph to every other node. The minimum time required to do this for the binary hypercube under two models of communication is studied. In the first model, all communication links may be used concurrently but each may only carry information in one direction at a time. In the second, weaker model each node may be involved in only one communication at a time either as sender or receiver. In both cases, simple algorithms exist that are close to optimal. This paper shows that neither of these algorithms is optimal by exhibiting faster algorithms. In the first case, an optimal algorithm is obtained.

A flexible system call interface for interprocessor communication in a distributed memory multicomputer

The diameter of a directed graph is the maximum of the lengths of the shortest paths between all pairs of vertices. A directed graph is said to be tightly oriented if it has the same diameter as its undirected image graph. Our main result is tight orientations for all sufficiently large toroids, except those whose sizes in both dimensions are odd. We also prove the impossibility of tightly orienting all the toroids for which we do not present tight orientations, and we give partial results for dimensionality higher than two.

Representations of gossip schemes

We consider a model of multicast communication in a network whereby multiple sources have messages to disseminate among all sites of a network. We propose that the messages from all sources are disseminated along the same spanning tree of the network and consider the problem of constructing an optimal such tree. One measure for suitability of the construction is the sum of distances from all sources to all other vertices. We show that finding the exact solution in this case in -hard (in the strong sense). We then investigate solutions for some restricted classes of graphs and give efficient algorithms for those. We also consider an alternative measure of goodness for the spanning tree, being the maximum eccentricity of a source. We show that the problem of finding such a minimum eccentricity spanning tree is somewhat easier to solve and give a pseudo-polynomial solution algorithm.