Fixed Point Numbers
In a fixed point number system, each number has
exactly the same number of
digits, and the point is always in the same place.
Examples from the decimal
number system would be 0.23, 5.12, and 9.11. In these
examples each number
has 3 di
HYPERCUBES AND THEIR ALGORITHMS
261
13.1. 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 ind
9
Sorting on a 2D
Mesh or Torus
There are good reasons for beginning our discussion of 2D mesh and torus
architectures through algorithms for the seemingly difficult sorting problem. First,
sorting on the 2D mesh is nothing like its counterpart on the PRA
NUMERICAL 2D MESH ALGORITHMS
231
If each of the two merge phases can be done in O( )steps, then the running time of the
algorithm will be T ( n ) = T ( n/4) + O( ) = O( ). Take the horizontal merge phase in the
upper half of the mesh. There are at most
/2
INTRODUCTION TO PARALLEL PROCESSING
216
Figure 11.5. Lower/upper triangular square matrix.
.
.
.
Such a triangular system of linear equations can be easily solved by back substitution.
Compute x0 from the top equation, substitute into the next equation to
ROUTING ON A 2D MESH OR TORUS
201
Figure 10.7. Demonstrating the worst-case buffer requirement with row-first routing.
highest speed with a constant buffer space per node. This, while possible, leads to a very
complicated algorithm with fairly large buffe
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INTRODUCTION TO PARALLEL PROCESSING
Figure 12.10. Some processor states in a reconfigurable mesh.
logarithmic time combining phase in Column 0 then yields the final result. In fact, we can
do better than this by using more (but still fewer than 2 p 1
16
A Sampler of Other
Networks
In this chapter, we study several other classes of interconnection architectures,
focusing in particular on hybrid or hierarchical schemes that combine features
from two or more different architectures or are based on multil
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INTRODUCTION TO PARALLEL PROCESSING
b.
c.
Extend the result of part (a) by showing that any p -node h -D mesh is a subgraph of the
(log2 p +h 1)-cube.
Show that the result of part (b) is in general the best possible by providing an example of
a p -nod
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INTRODUCTION TO PARALLEL PROCESSING
that are (sub)logarithmic in terms of their sizes p' and p', the product graph G will also have
a (sub)logarithmic node degree or diameter in terms of its size p.
Given optimal or efficient routing algorithms for G'
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INTRODUCTION TO PARALLEL PROCESSING
Figure 15.5. Butterfly network with permuted dimensions.
fat tree is a treelike network specifically designed to remedy this problem. In a fat tree, the
link multiplicity or capacity increases as we approach the roo
SORTING AND ROUTING ON HYPERCUBES
291
The bit-reversal permutation is an example of a bad routing problem for dimension-order
routing. Bit-reversal permutation routing is when each Node x = x q1 x q2. . . x 1 x0 needs to
send a value to the node whose bin