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PARALLEL AND DISTRIBUTED COMPUTING
INTERCONNECTION TOPOLOGIES
NETWORK TOPOLOGIES
Two types of topologies
 Shared Network: Communicate at most one message at a
time that’s why limited scalability
 Switched Network: Simultaneously transfer several messages
between different pairs of nodes.
NETWORK TOPOLOGIES
 Degree: Maximum number of neighbors of any node
 Diameter: length of longest of all shortest path
 BisectionWidth: minimum number of edges(links) to be
removed to disconnect the network into two halves. In case of
odd number of nodes one half can include one more node.
NETWORK TOPOLOGIES
 Constant Degree: Should be constant i.e. independent of
network size.This property allows scale large number of nodes
without adding excessive number of connections.
 Low Diameter: Should be minimum in order to provide
efficient communication between any pair of nodes
 High BisectionWidth: low bisection width can slow down
communication and thus limit the performance of applications.
Parallel and Distributed Computing Chapter 11
NETWORK TOPOLOGIES
 Linear Array
 2D Mesh/torus
 3D Mesh/torus
 BinaryTree
 Hypercube
LINEAR ARRAY
 Every node is connected to left and its right neighbor and thus
degree is 2 i.e. deg(Ln)=2
 Longest distance between any two nodes is between leftmost
node and the right most node i.e. Diam(Ln)=n-1
 Bisection width is one i.e. BW(Ln)=1, Since only one link
between P(n-1)/2 and P(n/2) needs to be removed in order to split
into two disconnected halves.
2D MESHTOPOLOGY
 Nodes are arranged in a (usually
square) grid
 Each node has most four neighbors i.e.
deg(M4,4)=√n=4
 Longest distance between P0,0 and P3,3
Leading to diameter of 6 i.e.
Daim(Mk,k)=2(√n-1)=6.
 Removing all links between second
and third row/column disconnects
M4,4 into equal halves.Thus bisection
width is bw(Mk,k)=(√n =4)
2D TORUSTOPOLOGY
 A frequently used extension of mesh is torus
 2D Torus Tk,k is extends Mk,k by adding wrap-around
edges to directly connect left and right node of each
raw as well the bottom and top node of each
column.
 Each node has most four neighbors i.e. deg(M4,4)=4
 Longest distance between P0,0 and P3,3 Leading to
diameter of 6 i.e. Daim(Mk,k)=(√n-1)=3.
 Removing all links between second and third
row/column disconnects M4,4 into equal halves.Thus
bisection width is bw(Mk,k)=(2√n = 8)
3D MESHTOPOLOGY
 Nodes are arranged in a (usually
square) grid
 Each node has most six neighbors
i.e. degree is Deg(Mk,k,k)=6
 Longest distance between P0,0 and
P3,3 Leading to diameter of 6 i.e.
Daim(Mk,k,k)=3(3√n-1).
 Bisection width can be calculated as
BW(Mk,k)=( n2/3)
3D MESHTOPOLOGY
 3D Mesh topology has many desirable features such as a
constant degree, relatively low diameter, and relatively high
bisection width.
 Thus it has been used as an interconnection network in top
supercomputers
3D TORUSTOPOLOGY
 Nodes are arranged in a (usually
square) grid
 Similar to 3d-Mesh topology just by
adding wrap-around edges
BINARY TREE TOPOLOGY
 In order to reduce network diameter, a tree based structure can be
used.
 Each node is connected to its parent and two Childs hence degree
is deg(BTd)=3
 The longest distance is when travelling between left most node to
right most half of tree, which requires growing up to the root and
then down again so Diam(BTd) = 2(d-1) = 2log2 (n+1).
 In-spite of constant degree and diameter is low disadvantage of
binary tree is its bisection width.
BINARY TREE TOPOLOGY
 Removing only a single link
with the root, we can split
the network into two
disconnected components
i.e. bitwise width bw(BTd)=1
HYPERCUBE NETWORK TOPOLOGY
 Hypercube network can be
represented as a graph with
n=2d nodes that has each
vertex labeled with a distinct
bit of length d
 Vertices that are connected iff
bits string differ in exactly one
bit.
Parallel and Distributed Computing Chapter 11
HYPERCUBE NETWORK TOPOLOGY
 Each node is connected to d
other nodes so degree will
deg(Qd)=d=log2(n)
 Longest distance between two
nodes diam(Qd)=d=log2(n)
 Removing all links between nodes
starting with label O and all
nodes starting with 1
disconnects Hd into two halves
i.e. it holds bw(Qd)=n/2
NETWORK TOPOLOGY
HYPERCUBE NETWORK TOPOLOGY
 Overall hypercube has the highest bisection width
 Furthermore diameter is very low (logarithmic)
 But disadvantage is non constant degree. i.e. number of required
links per node is logarithmic
 Making it difficult to scale up to large number of nodes.

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Parallel and Distributed Computing Chapter 11

  • 1. PARALLEL AND DISTRIBUTED COMPUTING INTERCONNECTION TOPOLOGIES
  • 2. NETWORK TOPOLOGIES Two types of topologies  Shared Network: Communicate at most one message at a time that’s why limited scalability  Switched Network: Simultaneously transfer several messages between different pairs of nodes.
  • 3. NETWORK TOPOLOGIES  Degree: Maximum number of neighbors of any node  Diameter: length of longest of all shortest path  BisectionWidth: minimum number of edges(links) to be removed to disconnect the network into two halves. In case of odd number of nodes one half can include one more node.
  • 4. NETWORK TOPOLOGIES  Constant Degree: Should be constant i.e. independent of network size.This property allows scale large number of nodes without adding excessive number of connections.  Low Diameter: Should be minimum in order to provide efficient communication between any pair of nodes  High BisectionWidth: low bisection width can slow down communication and thus limit the performance of applications.
  • 6. NETWORK TOPOLOGIES  Linear Array  2D Mesh/torus  3D Mesh/torus  BinaryTree  Hypercube
  • 7. LINEAR ARRAY  Every node is connected to left and its right neighbor and thus degree is 2 i.e. deg(Ln)=2  Longest distance between any two nodes is between leftmost node and the right most node i.e. Diam(Ln)=n-1  Bisection width is one i.e. BW(Ln)=1, Since only one link between P(n-1)/2 and P(n/2) needs to be removed in order to split into two disconnected halves.
  • 8. 2D MESHTOPOLOGY  Nodes are arranged in a (usually square) grid  Each node has most four neighbors i.e. deg(M4,4)=√n=4  Longest distance between P0,0 and P3,3 Leading to diameter of 6 i.e. Daim(Mk,k)=2(√n-1)=6.  Removing all links between second and third row/column disconnects M4,4 into equal halves.Thus bisection width is bw(Mk,k)=(√n =4)
  • 9. 2D TORUSTOPOLOGY  A frequently used extension of mesh is torus  2D Torus Tk,k is extends Mk,k by adding wrap-around edges to directly connect left and right node of each raw as well the bottom and top node of each column.  Each node has most four neighbors i.e. deg(M4,4)=4  Longest distance between P0,0 and P3,3 Leading to diameter of 6 i.e. Daim(Mk,k)=(√n-1)=3.  Removing all links between second and third row/column disconnects M4,4 into equal halves.Thus bisection width is bw(Mk,k)=(2√n = 8)
  • 10. 3D MESHTOPOLOGY  Nodes are arranged in a (usually square) grid  Each node has most six neighbors i.e. degree is Deg(Mk,k,k)=6  Longest distance between P0,0 and P3,3 Leading to diameter of 6 i.e. Daim(Mk,k,k)=3(3√n-1).  Bisection width can be calculated as BW(Mk,k)=( n2/3)
  • 11. 3D MESHTOPOLOGY  3D Mesh topology has many desirable features such as a constant degree, relatively low diameter, and relatively high bisection width.  Thus it has been used as an interconnection network in top supercomputers
  • 12. 3D TORUSTOPOLOGY  Nodes are arranged in a (usually square) grid  Similar to 3d-Mesh topology just by adding wrap-around edges
  • 13. BINARY TREE TOPOLOGY  In order to reduce network diameter, a tree based structure can be used.  Each node is connected to its parent and two Childs hence degree is deg(BTd)=3  The longest distance is when travelling between left most node to right most half of tree, which requires growing up to the root and then down again so Diam(BTd) = 2(d-1) = 2log2 (n+1).  In-spite of constant degree and diameter is low disadvantage of binary tree is its bisection width.
  • 14. BINARY TREE TOPOLOGY  Removing only a single link with the root, we can split the network into two disconnected components i.e. bitwise width bw(BTd)=1
  • 15. HYPERCUBE NETWORK TOPOLOGY  Hypercube network can be represented as a graph with n=2d nodes that has each vertex labeled with a distinct bit of length d  Vertices that are connected iff bits string differ in exactly one bit.
  • 17. HYPERCUBE NETWORK TOPOLOGY  Each node is connected to d other nodes so degree will deg(Qd)=d=log2(n)  Longest distance between two nodes diam(Qd)=d=log2(n)  Removing all links between nodes starting with label O and all nodes starting with 1 disconnects Hd into two halves i.e. it holds bw(Qd)=n/2
  • 19. HYPERCUBE NETWORK TOPOLOGY  Overall hypercube has the highest bisection width  Furthermore diameter is very low (logarithmic)  But disadvantage is non constant degree. i.e. number of required links per node is logarithmic  Making it difficult to scale up to large number of nodes.