Adding numbers

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Adding numbers
n data items, p processors
ts O(n)
tp O(n/p) if data on each proc gt Sts/tpO(p)
tp O(n n/p) if data needs broadcasting gt
Sts/tpo(1)
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Sequential Recursion
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Parallel Recursion
tcomm O(n/2 n/4 .. n/p) O(n)
So(1)
tcomp O(n/2 n/4 .. n/p) O(n)
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tcomm O(1 1 .. 1) O(log p)
SO(n / log p)
tcomp O(1 1 .. 1) O(log p)
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Parallel Bucket Sort
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Sequential
m buckets , n numbers
ts O(n m((n/m) log (n/m))) O(n log(n/m))
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m buckets , n numbers, pm processors
tp O(n (n/p) log (n/p))
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tp O(n/p (n/p) log (n/p)) O( (n/p) log
(n/p)) gt SO(p)
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Det. Sample Sort
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Det. Sample Sort

• sort locally and create p-sample

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Det. Sample Sort

• send all p-samples to processor 1

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Det. Sample Sort

• proc.1 sort all received samples and compute
  global p-sample

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Det. Sample Sort

• broadcast global p-sample
• bucket locally according to global p-sample
• send bucket i to proc.i
• resort locally

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Det. Sample Sort

• Lemma Each proc. receives at most 2 n/p data
  items

n/p2
n/p2
global sample
global sample
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Det. Sample Sort
n/p
n/p
n/p
n/p
n/p
n/p
n/p
n/p
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• Post-Processing Array Balancing

• 2 Rounds
• Each proc. sends rec. data size to all other
  proc.
• Move data to right location via one h-relation

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Det. Sample Sort

• 5 MPI_AlltoAllv
• for n/p gt p2
• O(n/p log n) local comp.
• Goodrich (FOCS'98)
• O(1) rounds
• for n/p gt pe

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Performance Det. Sample Sort
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Numerical Integration
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static assignment of processors to segments of
a,b
area d (f(p)f(q))/2
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Problem precision depends on curves shape
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Adaptive Quadrature
Terminate when C is sufficiently small Problem
different parts of the curve need different
resolution
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segment 1
segment 3
segment 4
segment 2
segment 5
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Gravitational N-Body Problem
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ts O(n2) per time step
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for each time step for each object traverse
tree to determine its forces Problemtraversal
s have different lengths
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object 1
object 3
object 5
object 2
object 4