Parallel external memory (model)

In computer science, a parallel external memory (PEM) model is a cache-aware, external-memory abstract machine.^[1] It is the parallel-computing analogy to the single-processor external memory (EM) model. In a similar way, it is the cache-aware analogy to the parallel random-access machine (PRAM). The PEM model consists of a number of processors, together with their respective private caches and a shared main memory.

Model

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Read / Write conflicts

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Complexity measure

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Examples

Prefixsum

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Multiway partitioning

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Selection

The selection problem is about finding the k-th smallest item in an unordered list $A$ of size $N$ . The following code^[1] makes use of PRAMSORT which is a PRAM optimal sorting algorithm which runs in $O(\log N)$ , and SELECT, which is a cache optimal single-processor selection algorithm.

if  $N\leq P$  then 
   ${\texttt {PRAMSORT}}(A,P)$ 
  return  $A[k]$ 
end if 

for each processor  $i$  in parallel do 
  //Find median of each  $S_{i}$ 
   $m_{i}={\texttt {SELECT}}(S_{i},{\frac {N}{2P}})$ 
end for 

// Sort medians
 ${\texttt {PRAMSORT}}(\lbrace m_{1},\dots ,m_{2}\rbrace ,P)$ 

// Partition around median of medians
 $t={\texttt {PEMPARTITION}}(A,m_{P/2},P)$ 

if  $k\leq t$  then 
  return  ${\texttt {PEMSELECT}}(A[1:t],P,k)$ 
else 
  return  ${\texttt {PEMSELECT}}(A[t+1:N],P,k-t)$ 
end if

Under the assumption that the input is stored in contiguous memory, PEMSELECT has an I/O complexity of:

$O({\frac {N}{PB}}+\log PB\cdot \log({\frac {N}{P}}))$

Distribution sort

Distribution sort partitions an input list $A$ of size $N$ into $d$ disjoint buckets of similar size. Every bucket is then sorted recursively and the results are combined into a fully sorted list.

If $P=1$ the task is delegated to a cache-optimal single-processor sorting algorithm.

Otherwise the following algorithm^[1] is used:

// Sample  ${\tfrac {4N}{\sqrt {d}}}$  elements from  $A$ 
for each processor  $i$  in parallel do
  if  $M<|S_{i}|$  then
     $d=M/B$ 
    Load  $S_{i}$  in  $M$ -sized pages and sort pages individually
  else
     $d=|S_{i}|$ 
    Load and sort  $S_{i}$  as single page
  end if
  Pick every  ${\sqrt {d}}/4$ 'th element from each sorted memory page into contiguous vector  $R^{i}$  of samples
end for 

in parallel do
  Combine vectors  $R^{1}\dots R^{P}$  into a single contiguous vector  ${\mathcal {R}}$ 
  Make  ${\sqrt {d}}$  copies of  ${\mathcal {R}}$ :  ${\mathcal {R}}_{1}\dots {\mathcal {R}}_{\sqrt {d}}$ 
end do

// Find  ${\sqrt {d}}$  pivots  ${\mathcal {M}}[j]$ 
for  $j=1$  to  ${\sqrt {d}}$  in parallel do
   ${\mathcal {M}}[j]={\texttt {PEMSELECT}}({\mathcal {R}}_{i},{\tfrac {P}{\sqrt {d}}},{\tfrac {j\cdot 4N}{d}})$ 
end for

Pack pivots in contiguous array  ${\mathcal {M}}$ 

// Partition  $A$ around pivots into buckets  ${\mathcal {B}}$ 
 ${\mathcal {B}}={\texttt {PEMMULTIPARTITION}}(A[1:N],{\mathcal {M}},{\sqrt {d}},P)$ 

// Recursively sort buckets
for  $j=1$  to  ${\sqrt {d}}+1$  in parallel do
  recursively call  ${\texttt {PEMDISTSORT}}$  on bucket  $j$ of size  ${\mathcal {B}}[j]$ 
  using  $O\left(\left\lceil {\tfrac {{\mathcal {B}}[j]}{N/P}}\right\rceil \right)$  processors responsible for elements in bucket  $j$ 
end for

The I/O complexity of PEMDISTSORT is:

$O\left(\left\lceil {\frac {N}{PB}}\right\rceil \left(\log _{d}P+\log _{M/B}{\frac {N}{PB}}\right)+f(N,P,d)\cdot \log _{d}P\right)$

where

$f(N,P,d)=O\left(\log {\frac {PB}{\sqrt {d}}}\log {\frac {N}{P}}+\left\lceil {\frac {\sqrt {d}}{B}}\log P+{\sqrt {d}}\log B\right\rceil \right)$

If the number of processors is chosen that $f(N,P,d)=O\left(\left\lceil {\tfrac {N}{PB}}\right\rceil \right)$ and $M<B^{O(1)}$ the I/O complexity is then:

$O\left({\frac {N}{PB}}\log _{M/B}{\frac {N}{B}}\right)$

Other PEM algorithms


PEM Algorithm	I/O complexity	Constraints
Mergesort^[1]	$O\left({\frac {N}{PB}}\log _{\frac {M}{B}}{\frac {N}{B}}\right)={\textrm {sort}}_{P}(N)$	$P\leq {\frac {N}{B^{2}}},M=B^{O(1)}$
List ranking^[2]	$O\left({\textrm {sort}}_{P}(N)\right)$	$P\leq {\frac {N/B^{2}}{\log B\cdot \log ^{O(1)}N}},M=B^{O(1)}$
Euler tour^[2]	$O\left({\textrm {sort}}_{P}(N)\right)$	$P\leq {\frac {N}{B^{2}}},M=B^{O(1)}$
Expression tree evaluation^[2]	$O\left({\textrm {sort}}_{P}(N)\right)$	$P\leq {\frac {N}{B^{2}\log B\cdot \log ^{O(1)}N}},M=B^{O(1)}$
Finding a MST^[2]	$O\left({\textrm {sort}}_{P}(\|V\|)+{\textrm {sort}}_{P}(\|E\|)\log {\tfrac {\|V\|}{pB}}\right)$	$p\leq {\frac {\|V\|+\|E\|}{B^{2}\log B\cdot \log ^{O(1)}N}},M=B^{O(1)}$

Where ${\textrm {sort}}_{P}(N)$ is the time it takes to sort $N$ items with $P$ processors in the PEM model.

References

^ ^a ^b ^c ^d Arge, Lars; Goodrich, Michael T.; Nelson, Michael; Sitchinava, Nodari (2008). "Fundamental parallel algorithms for private-cache chip multiprocessors". Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures - SPAA '08. New York, New York, USA: ACM Press. doi:10.1145/1378533.1378573. ISBN 9781595939739.
^ ^a ^b ^c ^d Arge, Lars; Goodrich, Michael T.; Sitchinava, Nodari (2010). "Parallel external memory graph algorithms". 2010 IEEE International Symposium on Parallel & Distributed Processing (IPDPS). IEEE. doi:10.1109/ipdps.2010.5470440. ISBN 9781424464425.

[:0-1] Arge, Lars; Goodrich, Michael T.; Nelson, Michael; Sitchinava, Nodari (2008). "Fundamental parallel algorithms for private-cache chip multiprocessors". Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures - SPAA '08. New York, New York, USA: ACM Press. doi:10.1145/1378533.1378573. ISBN 9781595939739.

[:1-2] Arge, Lars; Goodrich, Michael T.; Sitchinava, Nodari (2010). "Parallel external memory graph algorithms". 2010 IEEE International Symposium on Parallel & Distributed Processing (IPDPS). IEEE. doi:10.1109/ipdps.2010.5470440. ISBN 9781424464425.

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