Newsgroups: comp.parallel
From: Louxin Zhang <lzhang@neumann.uwaterloo.ca>
Subject: Ph.D thesis: Emulations and Embeddings of Parallel Networks
Organization: University of Waterloo
Date: Mon, 3 Apr 1995 13:35:56 -0400
Message-ID: <3lrki0$amq@usenet.srv.cis.pitt.edu>

 The following ph.d thesis is available at University of Waterloo
 in  tech. report from. Please send a msg to 
 Louxin Zhang (lzhang@neumann.uwaterloo.ca)
 for a copy if you are interested in it. 
-----------------------------------------------------------------------------

    Emualtions  and Embeddings  of Meshes of Trees 
    and Hypercubes of cliques

          Louxin Zhang

           Abstract

     Embedding and emulation are two important problems in the study of
   the fixed-connection network model of parallel computation. The
   goal of emulation is to emulate $T$ steps of any computation in the
   guest network $G$ on  the host network $H$ in a  minimum number of
   steps.  Embedding is a topological concept, which is used to characterize 
   the similarity of  networks viewed as graphs. Good embeddings
   imply  efficient emulations.  Here embeddings and emulations  for   
   meshes of trees and hypercubes of cliques are studied.

      The mesh of trees,  a hybrid based on the complete binary tree and
   the mesh, has highly recursive structure,  small diameter and large
   bisection width. As a consequence, it supports many efficient parallel
   algorithms for various computational problems. The  mesh cannot be  
   embedded in the  mesh of trees with constant load and dilation.
   However, we prove that  the mesh of trees can emulate the mesh  in 
   real-time.  Conversely, we show that the mesh cannot emulate the 
   mesh of trees in real-time under the pebbling process model of emulation.  
   We also discover  a one-to-one embedding of a  mesh of trees in  a
   like-sized butterfly with constant dilation and congestion and
   that the mesh of trees cannot  emulate the butterfly in
   real-time.

     The hypercube of cliques is a variation of the hypercube network.
   Roughly speaking, a hypercube of cliques is defined as a product of 
   a hypercube and a clique of certain size. Aiello and Leighton
   discovered that  the bases of  Hamming codes induce a one-to-one
   embedding  of a hypercube of cliques in a same-sized  hypercube. By
   constructing  a class of bases for  Hamming codes with height
   3 and width 4, we obtain a one-to-one embedding of a hypercube of cliques 
   in a same-sized hypercube with dilation 3 and congestion 10.  
   Such an embedding is very useful in studying  embeddings of binary trees 
   in  hypercubes and the reconfiguration of the hypercube around  faults.

