Hubert Wagener

Optimal Algorithms for Broadcast and Gossip in the Edge-Disjoint Path Modes

The communication power of the one-way and two-way edge-disjoint 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 graphGnofnnodes, the complexity of broadcast inGn,Bmin(Gn), satisfies ?log2n??Bmin(Gn)??log2n?+1. The complete binary trees meet the upper bound, and all graphs containing a Hamiltonian path meet the lower bound. 2. For each connected graphGnofnnodes, the one-way (two-way) gossip complexityR(Gn) (R2(Gn)) satisfies?log2n??R2(Gn)?2·?log2n?+1,1.44...log2n?R(Gn)?2·?log2n?+2.All these lower and upper bounds are shown to be sharp up to 1. 3. All planar graphs ofnnodes and degreehhave a two-way gossip complexity of at least 1.5log2n?log2log2n?0.5log2h?8, and the two-dimensional grid ofnnodes has the gossip complexity 1.5log2n?log2log2n±O(1); i.e., two-dimensional grids are optimal gossip structures among planar graphs of bounded degree. Some upper bounds are also obtained for the one-way mode. 4. Thed-dimensional grid,d?3, ofnnodes has the two-way gossip complexity (1+1/d)·log2n?log2nlog2n±O(d).

Gossiping in Vertex-Disjoint Paths Mode in d-Dimensional Grids and Planar Graphs

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 log2n −log2k −log2 log2n −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 also shows that the 2-dimensional grid belongs to the best gossip graphs among all planar graphs. 3. Another lower bound proof is developed to get some tight lower bounds on one-way “well-structured” gossip algorithms on planar graphs (note that all gossip algorithms designed until now in vertexdisjoint paths mode are “well-structured”).

Effective systolic algorithms for gossiping in cycles and two-dimensional grids

The complexity of systolic dissemination of information in one-way (telegraph) and two-way (telephone) communication mode is investigated. The following main results are established: (i) tight lower and upper bounds on the complexity of one-way systolic gossip in cycles for any length of the systolic period, (ii) optimal one-way and two-way systolic gossip algorithms in 2-dimensional grids whose complexity (the number of rounds) meets the trivial lower bound (the sum of the sizes of its dimensions).

Optimal algorithms for broadcast and gossip in the edge-disjoint path modes

The communication power of the one-way and two-way edge-disjoint 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 [log2n]≤Bmin(Gn)≤[log2n]+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) (R2(Gn)) satisfies

The complexity of systolic dissemination of information in interconnection networks

\begin{gathered}\left\lceil {\log _2 n} \right\rceil \leqslant R^2 (G_n ) \leqslant 2 \cdot \left\lceil {\log _2 n} \right\rceil + 1, \hfill \\1.44...\log _2 n \leqslant R(G_n ) \leqslant 2 \cdot \left\lceil {\log _2 n} \right\rceil + 2. \hfill \\\end{gathered}

Optimal Algorithms for Broadcast and Gossip in the Edge-Disjoint Path Modes (Extended Abstract)

. All these lower and upper bounds are tight. 3. All planar graphs of n nodes and degree h have a two-way gossip complexity of at least 1.5log2n−log2log2n−0.5log2h−2, and the two-dimensional grid of n nodes has the gossip complexity 1.5log2n−log2log2n±O(1), i.e. two-dimensional grids are optimal gossip structures among planar graphs. Similar results are obtained for one-way mode too.

Effective Systolic Algorithms for Gossiping in Cycles and Two-Dimensional Grids (Extended Abstract)

The complexity of systolic dissemination of information in the rings of processors (cycles) in the one-way (telegraph) mode is investigated. Tight lower and upper bounds on the complexity of one-way systolic gossip are established.

The complexity of systolic dissemination of information in interconnection networks (extended abstract)

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