Parallel external memory

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 ${\displaystyle B}$.

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.

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.

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.

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 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:

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

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.