MESHES AUGMENTED WITH
NONLOCAL LINKS
Because one of the major drawbacks of low-dimensional
meshes and tori is their rapidly
increasing diameters when the number of processors
becomes large, it has been suggested
that mechanisms be provided in order to spe
MESHES OF TREES
The mesh of trees architecture represents another attempt
at combining the advantages
of tree and mesh structures. Like the pyramid, an l-level
mesh of trees architecture has a
2 l 1 2 l l base whose processors are the leaves of 2l1
row tr
PYRAMID AND MULTIGRID
SYSTEMS
The pyramid architecture combines 2D mesh and tree
connectivities in order to gain
advantages from both schemes. Topologically, the pyramid
inherits low logarithmic diameter
from the tree and relatively wide bisection from th
MESHES WITH DYNAMIC LINKS
There are various ways of designing meshes so that node
connectivities can change
dynamically. For example, if buses are segmented through
the insertion of switches that can
be opened and closed under the control of a nearby
proc
HYPERCUBES AND THEIR
ALGORITHMS
DEFINITION AND MAIN PROPERTIES
The origins of the hypercube architecture can be traced
back to the early 1960s [Squi63].
Subsequently, both the direct (single-stage) version,
discussed in this chapter, and the indirect
or m
EMBEDDINGS AND THEIR
USEFULNESS
Given the architectures A and A' we embed A into A' by
specifying
A node mapping (indicating that Node v of A is mapped
onto Node v' of A'); the node
mapping can be many-to-one.
An edge mapping (indicating that Edge uv of
ROUTING PROBLEMS ON A
HYPERCUBE
Intuitively, hypercubes should perform better in dealing
with routing problems than 2D
meshes, because their larger node degree translates to the
availability of more alternate
(shortest) paths between processors, its logar
BITONIC SORTING ON A HYPERCUBE
If we sort the records in the lower (xq1 = 0) and upper
(xq1 = 1) subcubes in opposite
directions, the resulting sequence in the entire cube will be
bitonic. Now the shifting of the
upper half of the sequence to the lower ha
DEFINING THE SORTING PROBLEM
The general problem of sorting on a hypercube is as
follows: Given n records distributed
evenly among the p = 2q processors of a q-cube (with each
processor holding n/p records),
rearrange the records so that the key values ar
EMBEDDING OF ARRAYS AND TREES
In this section, we show how meshes, tori, and binary trees
can be embedded into
hypercubes in such a way as to allow a hypercube to run
mesh, torus, and tree algorithms
efficiently, i.e., with very small dilation and congest