Communication algorithms for isotropic tasks in hypercubes and wraparound meshes

Cover title.Includes bibliographical references (p. 29-30).Research supported by the NSF. NSF-ECS-8519058 Research supported by the ARO. DAAL03-86-K-0171by Emmanouel A. Varvarigos and Dimitri P. Bertsekas

Mixing times of lozenge tiling and card shuffling Markov chains

We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an L X L region we bound the mixing time by O(L^4 log L), which improves on the previous bound of O(L^7), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste, by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov chain for linear extensions.Comment: 39 pages, 8 figure

Transposition of banded matrices in hypercubes : a "nearly isotropic" task

Includes bibliographical references (p. 19).Supported by NSF. NSF-DDM-8903385 Supported by the ARO. DAAL03-92-G-0115by Emmanouel A. Varvarigos, Dimitri P. Bertsekas

Permutation Routing in the Hypercube and Grid Topologies

The problem of edge disjoint path routing arises from applications in distributed memory parallel computing. We examine this problem in both the directed hypercube and two-dimensional grid topologies. Complexity results are obtained for these problems where the routing must consist entirely of shortest length paths. Additionally, approximation algorithms are presented for the case when the routing request is of a special form known as a permutation. Permutations simply require that no vertex in the graph may be used more than once as either a source or target for a routing request. Szymanski conjectured that permutations are always routable in the directed hypercube, and this remains an open problem