Basic Idea

The parallel multiway mergesort process on four processors ${\displaystyle t_{0}}$ to ${\displaystyle t_{3}}$.

Given an unsorted sequence of ${\displaystyle n}$ elements, the goal is to sort the sequence with ${\displaystyle p}$ available processors. These elements are distributed equally among all processors and sorted locally using a sequential Sorting algorithm. Hence, the sequence consists of sorted sequences ${\displaystyle S_{1},...,S_{p}}$ of length ${\textstyle \lceil {\frac {n}{p}}\rceil }$. For simplification let ${\displaystyle n}$ be a multiple of ${\displaystyle p}$, so that ${\textstyle \left\vert S_{i}\right\vert ={\frac {n}{p}}}$ for ${\displaystyle i=1,...,p}$.

These sequences will be used to perform a multisequence selection/splitter selection. For ${\displaystyle j=1,...,p}$, the algorithm determines splitter elements ${\displaystyle v_{j}}$ with global rank ${\textstyle k=j{\frac {n}{p}}}$. Then the corresponding positions of ${\displaystyle v_{1},...,v_{p}}$ in each sequence ${\displaystyle S_{i}}$ are determined with binary search and thus the ${\displaystyle S_{i}}$ are further partitioned into ${\displaystyle p}$ subsequences ${\displaystyle S_{i,1},...,S_{i,p}}$ with ${\textstyle S_{i,j}:=\{x\in S_{i}|rank(v_{j-1})<rank(x)\leq rank(v_{j})\}}$.

Furthermore, the elements of ${\displaystyle S_{1,i},...,S_{p,i}}$ are assigned to processor ${\displaystyle i}$, means all elements between rank ${\textstyle (i-1){\frac {n}{p}}}$ and rank ${\textstyle i{\frac {n}{p}}}$, which are distributed over all ${\displaystyle S_{i}}$. Thus, each processor receives a sequence of sorted sequences. The fact that the rank ${\displaystyle k}$ of the splitter elements ${\displaystyle v_{i}}$ was chosen globally, provides two important properties: On the one hand, ${\displaystyle k}$ was chosen so that each processor can still operate on ${\textstyle n/p}$ elements after assignment. The algorithm is perfectly load-balanced. On the other hand, all elements on processor ${\displaystyle i}$ are less than or equal to all elements on processor ${\displaystyle i+1}$. Hence, each processor performs the p-way merge locally and thus obtains a sorted sequence from its sub-sequences. Because of the second property, no further p-way-merge has to be performed, the results only have to be put together in the order of the processor number.

Multi-sequence selection

In its simplest form, given ${\displaystyle p}$ sorted sequences ${\displaystyle S_{1},...,S_{p}}$ distributed evenly on ${\displaystyle p}$ processors and a rank ${\displaystyle k}$, the task is to find an element ${\displaystyle x}$ with a global rank ${\displaystyle k}$ in the union of the sequences. Hence, this can be used to divide each ${\displaystyle S_{i}}$ in two parts at a splitter index ${\displaystyle l_{i}}$, where the lower part contains only elements which are smaller than ${\displaystyle x}$, while the elements bigger than ${\displaystyle x}$ are located in the upper part.

The presented sequential algorithm returns the indices of the splits in each sequence, e.g. the indices ${\displaystyle l_{i}}$ in sequences ${\displaystyle S_{i}}$ such that ${\displaystyle S_{i}[l_{i}]}$ has a global rank less than ${\displaystyle k}$ and ${\displaystyle \mathrm {rank} \left(S_{i}[l_{i}+1]\right)\geq k}$.^[22]

algorithm msSelect(S : Array of sorted Sequences [S_1,..,S_p], k : int) is
    for i = 1 to p do 
	(l_i, r_i) = (0, |S_i|-1)
	
    while there exists i: l_i < r_i do
	// pick Pivot Element in S_j[l_j], .., S_j[r_j], chose random j uniformly
	v := pickPivot(S, l, r)
	for i = 1 to p do 
	    m_i = binarySearch(v, S_i[l_i, r_i]) // sequentially
	if m_1 + ... + m_p >= k then // m_1+ ... + m_p is the global rank of v
	    r := m  // vector assignment
	else
	    l := m 
	
    return l

For the complexity analysis the PRAM model is chosen. If the data is evenly distributed over all ${\displaystyle p}$, the p-fold execution of the binarySearch method has a running time of ${\displaystyle {\mathcal {O}}\left(p\log \left(n/p\right)\right)}$. The expected recursion depth is ${\displaystyle {\mathcal {O}}\left(\log \left(\textstyle \sum _{i}|S_{i}|\right)\right)={\mathcal {O}}(\log(n))}$ as in the ordinary Quickselect. Thus the overall expected running time is ${\displaystyle {\mathcal {O}}\left(p\log(n/p)\log(n)\right)}$.

Applied on the parallel multiway merge sort, this algorithm has to be invoked in parallel such that all splitter elements of rank ${\textstyle i{\frac {n}{p}}}$ for ${\displaystyle i=1,..,p}$ are found simultaneously. These splitter elements can then be used to partition each sequence in ${\displaystyle p}$ parts, with the same total running time of ${\displaystyle {\mathcal {O}}\left(p\,\log(n/p)\log(n)\right)}$.

Pseudocode

Below, the complete pseudocode of the parallel multiway merge sort algorithm is given. We assume that there is a barrier synchronization before and after the multisequence selection such that every processor can determine the splitting elements and the sequence partition properly.

/**
 * d: Unsorted Array of Elements
 * n: Number of Elements
 * p: Number of Processors
 * return Sorted Array
 */
algorithm parallelMultiwayMergesort(d : Array, n : int, p : int) is
    o := new Array[0, n]                         // the output array
    for i = 1 to p do in parallel                // each processor in parallel
        S_i := d[(i-1) * n/p, i * n/p] 	         // Sequence of length n/p
	sort(S_i)                                // sort locally
        synch
	v_i := msSelect([S_1,...,S_p], i * n/p)          // element with global rank i * n/p
        synch
	(S_i,1, ..., S_i,p) := sequence_partitioning(si, v_1, ..., v_p) // split s_i into subsequences
	    
	o[(i-1) * n/p, i * n/p] := kWayMerge(s_1,i, ..., s_p,i)  // merge and assign to output array
	
    return o

Analysis

Firstly, each processor sorts the assigned ${\displaystyle n/p}$ elements locally using a sorting algorithm with complexity ${\displaystyle {\mathcal {O}}\left(n/p\;\log(n/p)\right)}$. After that, the splitter elements have to be calculated in time ${\displaystyle {\mathcal {O}}\left(p\,\log(n/p)\log(n)\right)}$. Finally, each group of ${\displaystyle p}$ splits have to be merged in parallel by each processor with a running time of ${\displaystyle {\mathcal {O}}(\log(p)\;n/p)}$ using a sequential p-way merge algorithm. Thus, the overall running time is given by

${\displaystyle {\mathcal {O}}\left({\frac {n}{p}}\log \left({\frac {n}{p}}\right)+p\log \left({\frac {n}{p}}\right)\log(n)+{\frac {n}{p}}\log(p)\right)}$.

Practical adaption and application

The multiway merge sort algorithm is very scalable through its high parallelization capability, which allows the use of many processors. This makes the algorithm a viable candidate for sorting large amounts of data, such as those processed in computer clusters. Also, since in such systems memory is usually not a limiting resource, the disadvantage of space complexity of merge sort is negligible. However, other factors become important in such systems, which are not taken into account when modelling on a PRAM. Here, the following aspects need to be considered: Memory hierarchy, when the data does not fit into the processors cache, or the communication overhead of exchanging data between processors, which could become a bottleneck when the data can no longer be accessed via the shared memory.

Sanders et al. have presented in their paper a bulk synchronous parallel algorithm for multilevel multiway mergesort, which divides ${\displaystyle p}$ processors into ${\displaystyle r}$ groups of size ${\displaystyle p'}$. All processors sort locally first. Unlike single level multiway mergesort, these sequences are then partitioned into ${\displaystyle r}$ parts and assigned to the appropriate processor groups. These steps are repeated recursively in those groups. This reduces communication and especially avoids problems with many small messages. The hierarchical structure of the underlying real network can be used to define the processor groups (e.g. racks, clusters,...).^[21]