Parallel algorithms for expression evaluation

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Parallel algorithms for expression evaluation

• Part1. Simultaneous substitution method (SimSub)
• Part2. A parallel pebble game

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Example 1 expression evaluation

• Summing the numbers 2, 1, 3, 2, 1, 3, 2, 1 we can
  write a sequential program
• x1 2 x2 x11 x3 x23 x4 x32
• x5 x41 x6 x53 x7 x62 x8x71

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Simultaneous Substitutions Method (SimSub)

• x Expression(y) and y Expression1
• a variable y can be substituted by its
  expression or by its value if it is known (if
  Expression1 is a constant)
• assume we have
• a sequence of expressions defining variables,
• each expression is of a constant size
• variables are numbered and
• i-th expression depends only on variables with
  numbers smaller than i e.g. x14 x2x15
  x3x1x27 x4x2x3

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• One parallel step for each expression substitute
  its variables by their expression if it is safe
• safe means that corresponding expression has at
  most one variable
• Example summing the numbers 2, 1, 3, 2, 1, 3, 2,
  1 we can write a sequential program
• x1 2 x2 x11 x3 x23 x4 x32 x5
  x41 x6 x53 x7 x62
• after parallel step 1
• x1 2 x2 3 x3 x14 x4 x25 x5 x33
  x6 x44 x7 x55
• after parallel step 2
• x1 2 x2 3 x3 6 x4 8 x5 x17 x6
  x29 x7 x38
• after parallel step 2
• x1 2 x2 3 x3 6 x4 8 x5 9 x6 12
  x7 14
• output sum x7
• In this way we can sum 1024 numbers in 10 rounds

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Tree-contraction

• Parallel algorithms related to expression
  evaluation correspond to parallel algorithmic
  technique called tree-contraction
• Tree-contraction is used in parallel expression
  evaluation
• Since the structure of a expression is a tree
  there are different tree-contraction techniques
• Basic operations are
• - redirecting edges of the tree
• - removing nodes marking (pebbling) nodes
• - creating additional edges
• the final aim is to guarantee that logarithmic
  number of contractions is sufficient

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Tree-contraction related to SimSub algorithm
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Example Substitutions method works in O(log n)
time. In each iteration the active tree is
reduced by 1/3 at least.
The active tree consists of nodes relevant to the
output computation, The root is the output
variable (the last one).
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Homework is it good object for SimSub? Why?
Fn is the smallest tree which needs n iterations
in tree contraction related to SimSub algorithm
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How the SimSub algorithm works for general graphs
?

• sometimes very poor
• (e.g. when computing Fibonacci numbers)

• generally with n processors we need
• n-1 iterations
• only one processor really works
• each time, so the same can be done with a single
  processor

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• SimSub will terminate after
• O(log (tree-size(graph))
  iterations
• for the dependency graph of Fibonacci numbers
  tree-size (number of paths) is exponential.

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Other tree contraction method parallel pebble
game (PPG)

• Parallel pebble game (PPG) works for full binary
  trees, full means each father has exactly 2 sons

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Parallel pebble game on binary tree

• Within the game each node v of the tree has
  associated with it similar node denoted by
  cond(v).
• At the outset of the game cond(v)v, for all v
• During the game the pairs (v,cond(v)) can be
  thought of as additional edges
• Node v is active if and only if cond(v)?v

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Pebbling

• Pebbling a node denotes the fact that in the
  current state of the game the processor
  associated with that node has sufficient
  information to evaluate the subtree rooted here

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Three operationsactive, square and pebble

• Activate
• for all non-leaf nodes v in parallel do
• if v is not active and precisely one of its
  sons is pebbled then
• cond(v) becomes the other son
• if v is not active and both sons are pebbled
  then
• cond(v) becomes one of the sons arbitrarily
• Square
• for all nodes v in parallel do cond(v)?
  cond(cond(v))
• Pebble
• for all nodes v in parallel do
• if cond(v) is pebbled then pebble v

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Key result

• At the outset of the game only the leaves of the
  tree are pebbled.
• One composite move of the pebbling game is the
  sequence of individual operations
• (activate, square, square, pebble)
• Theorem
• Let T be a binary tree with n leaves. If
  initially only the leaves are pebbled then after
  log2n moves of the pebbling game the root of T
  becomes pebbled.

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Example
square
activate
square
pebbling
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The application of the pebbling game

• Consider the arithmetic expression
  ((3(22))35)
• We assign a processor to each non-leaf node of
  the tree.

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X5
X3
X3
X2
3x5
3(2x3)5
3x9
3(2x)9
2x3
2x3
2x
2x
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Evaluation of arithmetic expressions

• Arithmetic expressions can be evaluated on a
  PRAM in O(log n) time using O(n) processors.