Parallel external memory

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.

PEM Model

Model

Definition

The PEM model^[1] is a combination of the EM model and the PRAM model. The PEM model is a computation model which consists of ${\displaystyle P}$  processors and a two-level memory hierarchy. This memory hierarchy consists of a large external memory (main memory) of size ${\displaystyle N}$  and ${\displaystyle P}$  small internal memories (caches). The processors share the main memory. Each cache is exclusive to a single processor. A processor can't access another’s cache. The caches have a size ${\displaystyle M}$  which is partitioned in blocks of size ${\displaystyle B}$ . The processors can only perform operations on data which are in their cache. The data can be transferred between the main memory and the cache in blocks of size ${\displaystyle B}$ .

I/O complexity

The complexity measure of the PEM model is the I/O complexity,^[1] which determines the number of parallel blocks transfers between the main memory and the cache. During a parallel block transfer each processor can transfer a block. So if ${\displaystyle P}$  processors load parallelly a data block of size ${\displaystyle B}$  form the main memory into their caches, it is considered as an I/O complexity of ${\displaystyle O(1)}$  not ${\displaystyle O(P)}$ . A program in the PEM model should minimize the data transfer between main memory and caches and operate as much as possible on the data in the caches.

Read/write conflicts

In the PEM model, there is no direct communication network between the P processors. The processors have to communicate indirectly over the main memory. If multiple processors try to access the same block in main memory concurrently read/write conflicts^[1] occur. Like in the PRAM model, three different variations of this problem are considered:

• Concurrent Read Concurrent Write (CRCW): The same block in main memory can be read and written by multiple processors concurrently.
• Concurrent Read Exclusive Write (CREW): The same block in main memory can be read by multiple processors concurrently. Only one processor can write to a block at a time.
• Exclusive Read Exclusive Write (EREW): The same block in main memory cannot be read or written by multiple processors concurrently. Only one processor can access a block at a time.

The following two algorithms^[1] solve the CREW and EREW problem if ${\displaystyle P\leq B}$  processors write to the same block simultaneously. A first approach is to serialize the write operations. Only one processor after the other writes to the block. This results in a total of ${\displaystyle P}$  parallel block transfers. A second approach needs ${\displaystyle O(\log(P))}$  parallel block transfers and an additional block for each processor. The main idea is to schedule the write operations in a binary tree fashion and gradually combine the data into a single block. In the first round ${\displaystyle P}$  processors combine their blocks into ${\displaystyle P/2}$  blocks. Then ${\displaystyle P/2}$  processors combine the ${\displaystyle P/2}$  blocks into ${\displaystyle P/4}$ . This procedure is continued until all the data is combined in one block.

Comparison to other models

Model	Multi-core	Cache-aware
Random-access machine (RAM)	No	No
Parallel random-access machine (PRAM)	Yes	No
External memory (EM)	No	Yes
Parallel external memory (PEM)	Yes	Yes

Examples

Multiway partitioning

Let ${\displaystyle M=\{m_{1},...,m_{d-1}\}}$  be a vector of d-1 pivots sorted in increasing order. Let A be an unordered set of N elements. A d-way partition^[1] of A is a set ${\displaystyle \Pi =\{A_{1},...,A_{d}\}}$  , where ${\displaystyle \cup _{i=1}^{d}A_{i}=A}$  and ${\displaystyle A_{i}\cap A_{j}=\emptyset }$  for ${\displaystyle 1\leq i<j\leq d}$ . ${\displaystyle A_{i}}$  is called the i-th bucket. The number of elements in ${\displaystyle A_{i}}$  is greater than ${\displaystyle m_{i-1}}$  and smaller than ${\displaystyle m_{i}^{2}}$ . In the following algorithm^[1] the input is partitioned into N/P-sized contiguous segments ${\displaystyle S_{1},...,S_{P}}$  in main memory. The processor i primarily works on the segment ${\displaystyle S_{i}}$ . The multiway partitioning algorithm (PEM_DIST_SORT^[1]) uses a PEM prefix sum algorithm^[1] to calculate the prefix sum with the optimal ${\displaystyle O\left({\frac {N}{PB}}+\log P\right)}$  I/O complexity. This algorithm simulates an optimal PRAM prefix sum algorithm.

// Compute parallelly a d-way partition on the data segments ${\displaystyle S_{i}}$  for each processor i in parallel do Read the vector of pivots M into the cache. Partition ${\displaystyle S_{i}}$  into d buckets and let vector ${\displaystyle M_{i}=\{j_{1}^{i},...,j_{d}^{i}\}}$  be the number of items in each bucket. end for Run PEM prefix sum on the set of vectors ${\displaystyle \{M_{1},...,M_{P}\}}$  simultaneously. // Use the prefix sum vector to compute the final partition for each processor i in parallel do Write elements ${\displaystyle S_{i}}$  into memory locations offset appropriately by ${\displaystyle M_{i-1}}$  and ${\displaystyle M_{i}}$ . end for Using the prefix sums stored in ${\displaystyle M_{P}}$  the last processor P calculates the vector B of bucket sizes and returns it. 

If the vector of ${\displaystyle d=O\left({\frac {M}{B}}\right)}$  pivots M and the input set A are located in contiguous memory, then the d-way partitioning problem can be solved in the PEM model with ${\displaystyle O\left({\frac {N}{PB}}+\left\lceil {\frac {d}{B}}\right\rceil >\log(P)+d\log(B)\right)}$  I/O complexity. The content of the final buckets have to be located in contiguous memory.

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 ${\displaystyle O(\log N)}$ , and SELECT, which is a cache optimal single-processor selection algorithm.

if ${\displaystyle N\leq P}$  then ${\displaystyle {\texttt {PRAMSORT}}(A,P)}$  return ${\displaystyle A[k]}$  end if //Find median of each ${\displaystyle S_{i}}$  for each processor i in parallel do ${\displaystyle m_{i}={\texttt {SELECT}}(S_{i},{\frac {N}{2P}})}$  end for // Sort medians ${\displaystyle {\texttt {PRAMSORT}}(\lbrace m_{1},\dots ,m_{2}\rbrace ,P)}$  // Partition around median of medians ${\displaystyle t={\texttt {PEMPARTITION}}(A,m_{P/2},P)}$  if ${\displaystyle k\leq t}$  then return ${\displaystyle {\texttt {PEMSELECT}}(A[1:t],P,k)}$  else return ${\displaystyle {\texttt {PEMSELECT}}(A[t+1:N],P,k-t)}$  end if 

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

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

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 ${\displaystyle P=1}$  the task is delegated to a cache-optimal single-processor sorting algorithm.

Otherwise the following algorithm^[1] is used:

// Sample ${\displaystyle {\tfrac {4N}{\sqrt {d}}}}$  elements from A for each processor i in parallel do if ${\displaystyle M<|S_{i}|}$  then ${\displaystyle d=M/B}$  Load ${\displaystyle S_{i}}$  in M-sized pages and sort pages individually else ${\displaystyle d=|S_{i}|}$  Load and sort ${\displaystyle S_{i}}$  as single page end if Pick every ${\displaystyle {\sqrt {d}}/4}$ 'th element from each sorted memory page into contiguous vector ${\displaystyle R^{i}}$  of samples end for in parallel do Combine vectors ${\displaystyle R^{1}\dots R^{P}}$  into a single contiguous vector ${\displaystyle {\mathcal {R}}}$  Make ${\displaystyle {\sqrt {d}}}$  copies of ${\displaystyle {\mathcal {R}}}$ : ${\displaystyle {\mathcal {R}}_{1}\dots {\mathcal {R}}_{\sqrt {d}}}$  end do // Find ${\displaystyle {\sqrt {d}}}$  pivots ${\displaystyle {\mathcal {M}}[j]}$  for ${\displaystyle j=1}$  to ${\displaystyle {\sqrt {d}}}$  in parallel do ${\displaystyle {\mathcal {M}}[j]={\texttt {PEMSELECT}}({\mathcal {R}}_{i},{\tfrac {P}{\sqrt {d}}},{\tfrac {j\cdot 4N}{d}})}$  end for Pack pivots in contiguous array ${\displaystyle {\mathcal {M}}}$  // Partition Aaround pivots into buckets ${\displaystyle {\mathcal {B}}}$  ${\displaystyle {\mathcal {B}}={\texttt {PEMMULTIPARTITION}}(A[1:N],{\mathcal {M}},{\sqrt {d}},P)}$  // Recursively sort buckets for ${\displaystyle j=1}$  to ${\displaystyle {\sqrt {d}}+1}$  in parallel do recursively call ${\displaystyle {\texttt {PEMDISTSORT}}}$  on bucket jof size ${\displaystyle {\mathcal {B}}[j]}$  using ${\displaystyle O\left(\left\lceil {\tfrac {{\mathcal {B}}[j]}{N/P}}\right\rceil \right)}$  processors responsible for elements in bucket j end for 

The I/O complexity of PEMDISTSORT is:

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

where

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

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

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

Other PEM algorithms

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

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

See also

• Parallel random-access machine (PRAM)
• Random-access machine (RAM)
• External memory (EM)

References

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. New York, New York, USA: ACM Press. pp. 197–206. doi:10.1145/1378533.1378573. ISBN 9781595939739. S2CID 11067041.
2. Arge, Lars; Goodrich, Michael T.; Sitchinava, Nodari (2010). "Parallel external memory graph algorithms". 2010 IEEE International Symposium on Parallel & Distributed Processing (IPDPS). IEEE. pp. 1–11. doi:10.1109/ipdps.2010.5470440. ISBN 9781424464425. S2CID 587572.