Associative Nets: A GraphBased Parallel Computing Model

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Associative NetsA Graph-Based Parallel
Computing Model

• Alain Mérigot
• IEEE Transactions on Computer
• VOL 46, NO. 5, MAY 1997
• ??????

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Abstract

• Associative Net Model (ANM)
• a new model for parallel computing
• relies on basic primitives associations(apply
  an associative operator over connected components
  of a sub-graph of the physicalinter-processor
  connection graph.
• Aims at
• efficiently implemented in terms of hardware
  cost and processing time
• algorithm complexity roughly equals to that of
  the expensive PRAM method

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• a parallel computer
• consists of a set of processors processing
  elements (PE)
• is modeled by physical interconnection graph
• FTM(Fixed Topology Machine)
• simple, limited applications
• PRAM(Parallel Random Access Machine)
• fully interconnected PEs
• concurrent access to the same memory location
• expensive
• Re-configurable Machines
• asynchronous data propagation along the edges
  between PEs

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• ANM
• Like Re-configurable Machines
• relies on asynchronous data transfer, providing
  high speed basic communication primitives
• Contrary to Re-configurable Machines
• mixes communications and computations asso
  ciations
• applies operators on a set of interconnected PEs

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• ANM operations are applied on mgraph or
  connex set
• a set of connected components, that is,
• a subset of a graphwhose nodes are connected
  together,directly or indirectly,by edges.
• Dynamically defined in every PE

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• P number of processorsD degree of a
  processor
• the set of processors in the machine P Pi
  i ? 0, P-1
• the set of edges joining processors E eij
  0 ? i ? P-1 and 0 ? j ? P-1or E eik 0 ?
  i ? P-1 and 0 ? k ? D-1
• physical interconnection graph G (P, E)
• may be any physical graph
• assume 2-D mesh, symmetric directed, constant
  degree

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• Two Basic Objects - 1st
• pvar (parallel variable)
• represents an information distributed over the
  set of processors
• the value of the instance of x on processor
  Pi
• the jth bit of

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• Two Basic Objects - 2nd
• mgraph g
• represents the connex set of processors to
  communicate with
• g (P,E) where P P and E ? E
• g may be represented by the edges set E only
• Ex D-bit pvar
• gi incoming edges of processors Pi
• gik the kth incoming edge of processors Pi
• Ex g eik 0 ? k ? P-1 and 0 ? k ? D-1

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• Ascendants of Pi in mgraph g (neighbors)
• Ancestors of Pi in mgraph g
• Strict Ancestors of Pi in mgraph g

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• Basic Operations
• inversion
• g-1 eij eji ? g
• ?-association(g,p) ai 0 ? i ? P-1, ai ?
  pj
• ?-association-step(g,p) ai 0 ? i ? P-1, ai
  ? pj
• Note ? may be OR, AND, max, min, addition
  (which are associative and cumulative)

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• Implementation of Basic Primitives

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• Basic Primitive Algorithm 1spanning-tree(mgraph
  g, pvar r) mgraph1 mgraph t ? 2 while r
  has been modified in any processors3 where not
  r and or-association-step(g,r)4 let l be a
  random link among those incoming from a
  neighbor where r is set5 t ?l 6 r ?
  true7 end where8 end while9 return tend

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• Basic Primitive Algorithm 1spanning-tree-step(m
  graph g, pvar r) mgraph1 mgraph t ? 2
  where not r and or-association-step(g,r)3 let
  l be a random link among those incoming from a
  neighbor where r is set4 t ?l 5 r ?
  true6 end where7 return tend

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• Basic Primitive Algorithm 2max-association
  (mgraph g, pvar p) pvar1 pvar M 2 int i3
  for i ? b-1 downto 0 do4 Mi ?
  or-association(g,pi) 5 where (Mi1 and pi0)
  6 p ? 07 end where8 end for9 return
  Mend

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• Basic Primitive Algorithm 3prefix-max-associatio
  n (mgraph m, pvar p) pvar1 pvar M 2 int
  i3 for i ? b-1 downto 0 do4 Mi ?
  prefix-or-association(m, pi) 5 where Mi16
  m ? lop-off-zeroes(m, pi) 7 end where8
  end for9 return Mend

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• Basic Primitive Algorithm 4plus-association
  (mgraph g, pvar p) pvar1 pvar root ? true if
  (max-association(g,self-address())
  self-address()) 2 mgraph t ?
  spanning-tree(g,root)3 pvar result ?
  prefix-plus-association(t-1,p)4 result ?
  or-association(g,result and root) 5 return
  resultend

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• Basic Primitive Algorithm 5selection(mgraph g,
  pvar x, integer k) pvar1 pvar result, count,
  discarded ?0 2 pvar still-selected ? true3
  for i ? b-1 downto 0 do4 count ?
  plus-association(t,still-selected and
  xi0)5 where (countdiscarded) ? k6 where
  xi07 still-selected ? 08 end
  where9 discarded ? discardedcount10 result
  i ?111 elsewhere 12 where
  xi113 still-selected ?0 14 end
  where15 resulti ? 016 end where17 end
  for18 return resultend

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• Other Primitives Algorithm 6minimum-spanning-tre
  e (mgraph mg) mgraph
• ?
• Other Primitives Algorithm 7single-source-shorte
  st-path (mgraph g,pvar root, pvar weightsD)
  mgraph
• ?

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• Conclusion
• a new parallel programming model Association
  Net Model
• basic operations associations
• efficiently realized by asynchronous computations
• complex but efficient algorithms built by
  primitives
• Future Works
• a number of algorithms needed to be studied
• real time computer vision
• any graph-based applications