Hypercubes

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Hypercubes
by Shietung Peng
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Introduction

• The hypercube architecture has played an
  important role in the development of parallel
  processing and is still quite popular and
  influential. The logarithmic diameter, linear
  bisection, and highly symmetric recursive
  structure of the hypercube support a variety of
  elegant and efficient parallel algorithms that
  often serve as benchmarks for evaluating
  algorithms on other architectures.

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Definition

• Hypercubes are also called binary q-cubes, or
  q-cubes, where q indicates the number of
  dimensions.
• A q-cube can be defined recursively as depicted
  below

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Main Properties

• A node x in a q-cube has a unique label, its
  binary ID, that is a q-bit binary number.
• The labels of any two neighboring nodes differ in
  exactly 1 bit.
• Two nodes whose labels differ in k bits are
  connected by a shortest path of length k (have a
  Hamming distance k).
• Hypercube is both node- and edge- symmetric.
• Any two nodes with Hamming distance k have k
  node-disjoint paths that connect them.
• Simple routing algorithms available.

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Embeddings

• Given the architectures A and A, embedding A
  into A includes a node mapping and an edge
  mapping (an edge (u,v) in A is mapped into a path
  u ? v in A, where u and v are given by the
  node mapping).
• Embedding a 7-node binary tree into 2D meshes of
  various sizes is depicted below.

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Embeddings

• In order to gauge the effectiveness of an
  embedding with regard to algorithm performance,
  various measures has been defined.
• The most important ones are listed below with the
  numerical values of the examples in the previous
  page.

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Embedding of Arrays and Rings

• We show how meshes, tori, and binary trees can be
  embedded into hypercube in such a way as to allow
  a hypercube to run mesh, torus, and tree
  algorithms efficiently, i.e., with very small
  dilation and congestion.
• Any p-node graph that can embed a p-ring with
  dilation and load factor of 1 is said to be
  Hamiltonian.
• To prove the q-cube is Hamiltonian is equivalent
  to construct a q-bit Gray code. A q-bit Gray code
  can be built by induction as follows

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Cross-product Graphs

• The nodes of the product graph G are labeled with
  k-tuples, where the ith element of the k-tuple is
  chosen from the node set of the ith component
  graph. The edges of the product graph connect
  pairs of nodes whose labels are identical in all
  but the j-th element, and the two nodes
  corresponding to the j-th elements are connected
  by an edge in the j-th component graph.

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Cross-product Graphs

• Examples of product graphs

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Embedding of Meshes

• An h-D mesh, where the number of nodes in each
  dimension is a power of 2, is a sub-graph of the
  q-cube.
• The proof is based on product graph and is stated
  as follows

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Embedding of Meshes

• The 4-by-4 mesh is a sub-graph of the 4-cube as
  depicted below.

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Exercise 3

• Embedding mesh into hypercube
• Show that the 3-by-3 mesh is a sub-graph of the
  4-cube.
• Show that 3-by-3 torus is not a sub-graph of the
  4-cube.
• Show that the 3-by-5 mesh is not a sub-graph of
  the 4-cube.
• Show an embedding of the 8-by-8-by-8 torus into
  the 8-cube and the 9-cube. What is the dilation
  and congestion of the embedding?

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Exercise 3

• Show that the -node complete binary tree
  is not a sub-graph of the q-cube but it is a
  sub-graph of (q1)-cube.
• (Disjoint-paths routing) In an n-cube Qn, given a
  source node and a destination node, show that
  there exist n disjoint paths of length at most
  n2 connecting the source node to the destination
  node.

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Exercise 3