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Section#5: Direct interconnection networks I+II
(CS838: Topics in parallel computing, CS1221, Tue+Thu, Feb
2+4, 1999, 8:009:15 a.m.)
The contents
1.
Basic notions and terminology
2.
Requirements on interconnection networks
3.
Meshbased topologies
4.
Hypercubic topologies
5.
Treebased topologies
6.
Shufflebased topologies
A
direct interconnection network
(IN) of a multiprocessor system is represented by a
connected graph
whose vertices represent
processing nodes
and edges represent
communication links
. A processing node
(PN) usually consists of one or more processors, local memory, and communication router. This section is
devoted to the description and analysis of topologies and properties of important INs.
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beginning of the
page
Back to the
CS838 class
schedule
Basic notions and terminology
Alphabets and strings
d
ary alphabet
is denoted by
Z
d
={0,1,.
.,d1}
. The operation
+
modulo
n
+
n
.
n
letter
d
ary strings
Z
d
n
={x
n1
... x
0
; x
i
in Z
d
}
,
n>= 1
. The length of string
x
len(x)
. The
empty
string (i.e., of
length 0) is denoted by
e
. The
i

fold concatenation
of string
x
x
i
.
Binary alphabet
B
. The
inversion
of bit
b
i
is
\non(b
i
)=1b
1
. If
b=b
n1
.. b
i+1
b
i
b
i1
.. b
0
in B
n
, then
\non
i
(b)=b
n1
.. b
\non(b
i
)b
i1
.. b
0
.
Graph theory
Vertex and edge set
of graph
G
V(G)
and
E(G)
, respectively. Two
adjacent
vertices
u
and
v
form edge
( u,v)
.
They are
incident
with the edge. Edges
( u,v)
and
( v,w)
are
adjacent
.
H
= subgraph
of
G
,
H\subset G
, if
V(H)\subset V(G)
and
E(H)\subset E(G)
.
5/13/2011
Direct interconnection networks I+II
pages.cs.wisc.edu/…/Section5.html
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View Full Document H
= induced subgraph
of
G
if it is a
maximal
subgraph of
G
with vertices
V(H)
.
H
= spanning subgraph
G
if
V(H)=V(G)
.
Union
of two disjoint graphs
G
1
and
G
2
,
G
1
\cup G
2
, is a graph with vertices
V(G
1
)\cup V(G
2
)
and edges
E(G
1
)\cup E(G
2
)
.
Cartesian product
G
1
and
G
2
is graph
G=G
1
x G
2
with
V(G)={(x,y); x in V(G
1
), y in V(G
2
)}
and
E(G)={( (x
1
,y),
(x
2
,y)); ( x
1
,x
2
) in E(G
1
)}\cup {( (x,y
1
),(x,y
2
)); ( y
1
,y
2
) in E(G
2
.
CAPTION: An example of a cartesian product
Degree of vertex
u
,
deg
G
(u)
, is the number of neighbors of
u
.
Degree set of graph
G
,
deg(G)
, is the set
{deg
G
(u); u in V(G)}
.
Maximum degree
G
is
\triangle(G)=max(deg(G))
.
Minimum degree
G
\delta(G)=min(deg(G))
.
k
regular graph
G
has
\triangle(G)=\delta(G)=k
.
Connected graph
Every pair
u
,
v
of vertices is joined by a
path
P(u,v)
, a sequence of adjacent edges.
Path length
len(P(u,v))
, is the number of edges
P(u,v)
.
Distance
between vertices
u
and
v
,
dist
G
(u,v)
, is the length of a shortest path joining
u
and
v
.
Average distance
in
G
,
dist
G
(u,v)
, is
\Sigma
u,v
dist
G
(u,v)/(N(N1))
.
Diameter
G
,
diam(G)
, is the maximum distance between any two vertices of
G
.
Cycle
is a closed path.
Vertexdisjoint paths
P
1
(u,v)
and
P
2
(u,v)
have no vertices in common except for
u
and
v
Edgedisjoint paths
P
1
(u,v)
and
P
2
(u,v)
have no edges in common.
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This note was uploaded on 09/10/2011 for the course CS 6143 taught by Professor Hadimioglu during the Spring '10 term at NYU Poly.
 Spring '10
 HADIMIOGLU

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