A taxonomy of parallel sorting

TR 84-601In this paper, we propose a taxonomy of parallel sorting that includes a broad range of array and file sorting algorithms. We analyze the evolution of research on parallel sorting, from the earliest sorting networks to the shared memory algorithms and the VLSI sorters. In the context of sorting networks, we describe two fundamental parallel merging schemes - the odd-even and the bitonic merge. Sorting algorithms have been derived from these merging algorithms for parallel computers where processors communicate through interconnection networks such as the perfect shuffle, the mesh and a number of other sparse networks. After describing the network sorting algorithms, we show that, with a shared memory model of parallel computation, faster algorithms have been derived from parallel enumeration sorting schemes, where keys are first ranked and then rearranged according to their rank

Fast Parallel Algorithms for Basic Problems

Parallel processing is one of the most active research areas these days. We are interested in one aspect of parallel processing, i.e. the design and analysis of parallel algorithms. Here, we focus on non-numerical parallel algorithms for basic combinatorial problems, such as data structures, selection, searching, merging and sorting. The purposes of studying these types of problems are to obtain basic building blocks which will be useful in solving complex problems, and to develop fundamental algorithmic techniques. In this thesis, we study the following problems: priority queues, multiple search and multiple selection, and reconstruction of a binary tree from its traversals. The research on priority queue was motivated by its various applications. The purpose of studying multiple search and multiple selection is to explore the relationships between four of the most fundamental problems in algorithm design, that is, selection, searching, merging and sorting; while our parallel solutions can be used as subroutines in algorithms for other problems. The research on the last problem, reconstruction of a binary tree from its traversals, was stimulated by a challenge proposed in a recent paper by Berkman et al. ( Highly Parallelizable Problems, STOC 89) to design doubly logarithmic time optimal parallel algorithms because a remarkably small number of such parallel algorithms exist

Algorithms for the NJIT turbonet parallel computer

Element selection for arrays, array merging, and sorting are very frequent operations in many of today\u27s important applications. These operations are of interest to scientific, as well as other applications where high-speed database search, merge, and sort operations are necessary and frequent. Therefore, their efficient implementation on parallel computers should be a worthwhile objective. Parallel algorithms are presented in this thesis for the implementation of these operations on the NET TurboNet system, an in-house built experimental parallel computer with TMS320C40 Digital Signal Processors interconnected in a 3-D hypercube structure. The first algorithm considered is selection. It involves finding the k-th smallest element in an unsorted sequence of n elements, where 1≤k≤n. The second algorithm involves the merging of two sequences sorted in nondecreasing order to form a third sequence, also sorted in nondecreasing order. The third parallel algorithm is sorting. For a given unsorted sequence S of size n, we want to sort the sequence such that st\u27≤i+1\u27 for all n elements. Performance results show that the robust structure of TurboNet results in significant speedups

An Efficient Multiway Mergesort for GPU Architectures

Sorting is a primitive operation that is a building block for countless algorithms. As such, it is important to design sorting algorithms that approach peak performance on a range of hardware architectures. Graphics Processing Units (GPUs) are particularly attractive architectures as they provides massive parallelism and computing power. However, the intricacies of their compute and memory hierarchies make designing GPU-efficient algorithms challenging. In this work we present GPU Multiway Mergesort (MMS), a new GPU-efficient multiway mergesort algorithm. MMS employs a new partitioning technique that exposes the parallelism needed by modern GPU architectures. To the best of our knowledge, MMS is the first sorting algorithm for the GPU that is asymptotically optimal in terms of global memory accesses and that is completely free of shared memory bank conflicts. We realize an initial implementation of MMS, evaluate its performance on three modern GPU architectures, and compare it to competitive implementations available in state-of-the-art GPU libraries. Despite these implementations being highly optimized, MMS compares favorably, achieving performance improvements for most random inputs. Furthermore, unlike MMS, state-of-the-art algorithms are susceptible to bank conflicts. We find that for certain inputs that cause these algorithms to incur large numbers of bank conflicts, MMS can achieve up to a 37.6% speedup over its fastest competitor. Overall, even though its current implementation is not fully optimized, due to its efficient use of the memory hierarchy, MMS outperforms the fastest comparison-based sorting implementations available to date

A faster all parallel Mergesort algorithm for multicore processors

The problem addressed in this paper is that we want to sort an integer array a[] of length n in parallel on a multi core machine with p cores using mergesort. Amdahl’s law tells us that the inherent sequential part of any algorithm will in the end dominate and limit the speedup we get from parallelisation. This paper introduces ParaMerge, an all parallel mergesort algorithm for use on an ordinary shared memory multi core machine that has just a few simple statements in its sequential part. The new algorithm is all parallel in the sense that by recursive decent it is two parallel in the top node, four parallel on the next level in the recursion, then eight parallel until we at least have started one thread for all the p cores. After parallelization, each thread then uses sequential recursion mergesort with a variant of insertion sort for sorting short subsections at the end. ParaMerge can be seen as an improvement over traditional parallelization of the mergesort algorithm where one follows the sequential algorithm and substitute recursive calls with the creation of parallel threads in the top of the recursion tree. This traditional parallel mergesort finally does a sequential merging of the two sorted halves of a[]. First at the next level it goes two-parallel, then four parallel on the next level, and so on. After parallelization my implementation of this traditional algorithm also use the same sequential mergesort and insertion sort algorithm as the ParaMerge algorithm in each thread. There are two main improvements in Paramerge: First the observation that merging can both be done from the start left to right picking the smallest elements of the two sections to be merged, and at the same time from the end of the same sections from right to left picking the largest elements. The second improvement is that the contract between a node and its two sub-nodes is changed. In a traditional parallelization a node is given a section of a[], sort this by merging two sorted halves it recursively receives from its own two sub nodes and returns its to its mother node. In Paramerge the two sub nodes each receive a full sorting from its two own sub nodes of the section itself got from its mother node (so this problem is already solved). Every node has a twin node. In parallel these two twin nodes then merge their two sorted sections, one from left and the other from right as described above. The two twin sub nodes have then sorted the whole section given to their common mother node. This goes also for the top node. We have thus raised the level of parallelization by a factor of two at each level of the top of the recursion tree. The ParaMerge algorithm also contains other improvements, such as a controlled sorting back and forth between a[] and a scratch area b[] of the same size such that the sorted result always ends up in a[] without any copy, and a special insertion sort that is central for achieving this copy-free feature. ParaMerge is compared with other published algorithms, and in only one case is one of the ‘new’ features in Paramerge found. This other algorithm is described and compared in some detail.Finally, ParaMerge is empirically compared with three other algorithms sorting arrays of length n =10,20,…,50 m, and ..1000m when p=32. demonstrating that it is significantly faster than two other merge algorithms, the sequential and the traditional parallel algorithm, and Arrays.sort(), a sequential Quicksort algorithm from the Java library