Efficient Communication and Routing for Parallel Computing

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Lecture 4 Routing Algorithms - II
By Shietung Peng
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Turn Model

• The turn model is a systematic approach to
  develop partially adaptive routing algorithm for
  a given network. The fundamental concept behind
  the turn model is to prohibit the smallest number
  of turns such that cycles are prevented. Thus,
  deadlock can be avoided by prohibiting just
  enough turns to break the cycles. There are five
  steps for developing maximally adaptive routing
  algorithms for meshes and k-ary n-cubes.

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Turn Model

• Classify channels according to the directions in
  which they route packets.
• Identify the turns that occur between one
  direction and another.
• Identify the simple cycles that turns can form.
• Prohibit one turn in each cycle.
• Add 0-degree and 180-degree turns without
  reintroducing cycles.

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An Example

• Consider a 2-D mesh. There are 8 possible turns
  and two possible abstract cycles. XY routing
  prohibits 4 of them. However, to prevent cycles,
  only 2 cycles need to be prohibited.

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An Example (Conti.)

• The corresponding west-first routing algorithm
  routes a packet first west, if necessary, and
  then adaptively south, east, and north. The two
  turns prohibited are the turns to the west. The
  minimal west-first routing algorithm is shown in
  the next page. For a non-minimal version of this
  algorithm, see Exercise 4.5. For minimal routing,
  the algorithm is fully adaptive if the
  destination is on the right-hand side of the
  source.

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Minimal West-first Routing Algorithm
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West-first Routing

• Examples of west-first routing in 2-D mesh.

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Alternatives

• Of the 16 different ways to prohibit two turns,
  12 prevent deadlock (a counter-example is shown
  in the figure) and only 3 are unique if symmetry
  is taken into account. These 3 combinations
  correspond to the west-first, north first, and
  negative-first routing algorithms. The
  north-first does not allow turns from north to
  west or from north to east. The negative-first
  does not allow turns from north to west and from
  east to south.

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A Counter-example
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Turn Model in Hypercubes

• An adaptive routing algorithm (called P-cube
  routing) in an n-cube Let E be the set of all
  dimension numbers in which the source s and the
  destination t differ. Divide E into two disjoint
  subsets E0 and E1, where i in E0 (E1) if the ith
  bit of s is 0 (1),
• The fundamental concept of P-cube routing is to
  divide the routing selection into two phases
  route through dimensions in E0 first, and then
  route through dimensions in E1.

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Minimal P-cube Routing Algorithm
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Fully Adaptive Algorithms

• We first introduce the algorithms designed for
  SAF networks using central queues. Deadlocks are
  avoided by splitting buffers into several classes
  and restricting packets to move from one buffer
  to another in such a way that buffer class is
  never decremented. These algorithms are known as
  hop algorithms. The simplest hop algorithm
  (positive-hop algorithm) starts by injecting a
  packet into the buffer of class 0 at the current
  node. Every time a packet stored in a buffer of
  class i takes a hop to another node, it moves to
  a buffer of class i1.

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Fully Adaptive Algorithms in SAF

• The number of buffers per node can be reduced by
  allowing packets to move between buffers of the
  same class. In this case, classes must be defined
  such that packets moving between buffers of the
  same class cannot form cycles. In the
  negative-hop algorithm, network is partitioned
  into several subsets such that no subset contains
  two adjacent nodes.

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A Partition Scheme
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Fully Adaptive Algorithm for Wormhole Switching

• The methodology starts from a hop algorithm,
  replacing central buffers by virtual channels.
  The basic idea consists of splitting each
  physical channel into as many virtual channels as
  there were central buffers in the original hop
  algorithm, and assigning virtual channels the
  same way that central buffers were assigned. The
  situation is depicted in the figure in the next
  page.

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Extension of the SAF Algorithm
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Virtual Networks

• Idea splitting the network into several virtual
  networks, each of them is a subset of channels
  that are used to route packets toward a
  particular set of destinations. The channel sets
  corresponding to different virtual networks are
  disjoint. Depending on the destination, each
  packet is injected into a particular virtual
  network, where it is routed until it arrives at
  its destination.

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Virtual Networks for a 2-D mesh

• Idea splitting the network into several virtual
  networks such that packets injected into a
  virtual network can only move in one direction
  for each dimension. The figure in the next page
  shows the four virtual networks for a 2-D mesh.
  Packets are injected into a single virtual
  network depending on the destination. Once a
  packet is being routed in a given virtual
  network, the packet cannot be transferred to
  another virtual network.

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Virtual Networks for a 2-D mesh
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Reducing the Number of Virtual Networks for a 2-D
mesh

• Idea each virtual network has channels in both
  directions in the 0th dimension and only one
  direction in the remaining dimensions. The
  figures in the next pages show an example for a
  2-D mesh and the routing algorithms for 2-D
  meshes.

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Two Virtual Networks for a 2-D mesh
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Routing Algorithm for 2-D Meshes
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Routing Algorithm for 2-D Meshes
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Extension of Virtual Networks to k-ary n-cube
Networks

• IdeaEach virtual network is split into several
  levels, each level has its own virtual channels.
  Every time a packet crosses a wraparound channel,
  it moves to the next level. The figure in the
  next page shows the virtual network and their
  levels for a 2-D torus.
• The number of resources required can be reduced
  by using proper routing sub-function and extended
  channel dependency graphs.

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• Virtual
• network
• for 2-D torus