Samplesort
Samplesort is a sorting algorithm that is a divide and conquer algorithm often used in parallel processing systems. Conventional divide and conquer sorting algorithms partitions the array into sub-intervals or buckets. The buckets are then sorted individually and then concatenated together. However, if the array is non-uniformly distributed, the performance of these sorting algorithms can be significantly throttled. Samplesort addresses this issue by selecting a sample of size from the -element sequence, and determining the range of the buckets by sorting the sample and choosing elements from the result. These elements (called splitters) then divide the array  into approximately equal-sized buckets. Samplesort is described in the 1970 paper, "Samplesort: A Sampling Approach to Minimal Storage Tree Sorting", by W. D. Frazer and A. C. McKellar. Algorithm. Samplesort is a generalization of quicksort. Where quicksort partitions its input into two parts at each step, based on a single value called the pivot, samplesort instead takes a larger sample from its input and divides its data into buckets accordingly. Like quicksort, it then recursively sorts the buckets. To devise a samplesort implementation, one needs to decide on the number of buckets . When this is done, the actual algorithm  operates in three phases: The full sorted output is the concatenation of the buckets. A common strategy is to set equal to the number of processors available. The data is then distributed among the processors, which perform the sorting of buckets using some other, sequential, sorting algorithm. Pseudocode. The following listing shows the above mentioned three step algorithm as pseudocode and shows how the algorithm works in principle. In the following, is the unsorted data, is the oversampling factor, discussed later, and is the number of splitters. The pseudo code is different from the original Frazer and McKellar algorithm. In  the pseudo code, samplesort is called recursively. Frazer and McKellar called samplesort just once and used quicksort in all following iterations. Complexity. The complexity, given in Big O notation, for a parallelized implementation with formula_8 processors: Find the splitters. Send to buckets. Sort buckets. The number of comparisons, performed by this algorithm, approaches the information theoretical optimum formula_17 for big input sequences. In experiments, conducted by Frazer and McKellar, the algorithm needed 15% fewer comparisons than quicksort. Sampling the data. The data may be sampled through different methods. Some methods include: Oversampling. The oversampling ratio determines how many times more  data elements to pull as samples, before determining the splitters. The goal is to get a good representation of the distribution of the data. If the data values are widely distributed, in that there are not many duplicate values, then a small sampling ratio is sufficient. In other cases where there are many duplicates in the distribution, a larger oversampling ratio will be necessary. In the ideal case, after step 2, each bucket contains formula_18 elements. In this case, no bucket takes longer to sort than the others, because all buckets are of equal size. After pulling formula_19 times more  samples than necessary, the samples are sorted. Thereafter, the splitters used as bucket boundaries are the samples at position formula_20 of the sample sequence (together with formula_21 and formula_22 as left and right boundaries for the left most and right most buckets respectively). This provides a better heuristic for good splitters than just selecting formula_8 splitters randomly. Bucket size estimate. With the resulting sample size, the expected bucket size and especially the probability of a bucket exceeding a certain size can be estimated. The following will show that for an oversampling factor of formula_24 the probability that no bucket has  more than formula_25 elements is larger than formula_26. To show this let formula_27 be the input as a sorted sequence. For a processor to get more than formula_28 elements, there has to exist a subsequence of the input of length formula_28, of which a maximum of samples are picked. These cases constitute the probability formula_30. This can be represented as the random variable: For the expected value of formula_32 holds: This will be used to estimate formula_34: Using the chernoff bound now, it can be shown: Many identical keys. In case of many identical keys, the algorithm goes through many  recursion levels where sequences are sorted, because the whole sequence consists of identical keys. This can be counteracted by introducing equality buckets. Elements equal to a pivot are sorted into their respective equality bucket, which can be implemented with only one additional conditional branch. Equality buckets are not further sorted. This works, since keys occurring more than formula_37 times are likely to become pivots. Uses in parallel systems. Samplesort is often used in parallel systems, including distributed systems such as bulk synchronous parallel machines. Due to the variable amount of splitters (in contrast to only one pivot in Quicksort), Samplesort  is very well suited and intuitive for parallelization and scaling. Furthermore Samplesort is also more cache-efficient than implementations of e.g. quicksort. Parallelization is implemented by splitting the sorting for each processor or node, where the number of buckets is equal to the number of processors formula_8. Samplesort is efficient in parallel systems because each processor receives approximately the same bucket size formula_18. Since the buckets are sorted concurrently, the processors will complete the sorting at approximately the same time, thus not having a processor wait for others. On distributed systems, the splitters are chosen by taking formula_19 elements on  each processor, sorting the resulting formula_41 elements with a distributed sorting algorithm, taking every formula_19-th element and broadcasting the result to all processors. This costs formula_43 for sorting the formula_41 elements on formula_8 processors, as well as formula_46 for distributing the formula_8 chosen splitters to formula_8 processors. With the resulting splitters, each processor places its own input data into local buckets. This takes formula_49 with binary search. Thereafter, the local buckets are redistributed to the processors. Processor formula_50 gets the local buckets formula_51 of all other processors and sorts these locally. The distribution takes formula_52 time, where formula_53 is  the size of the biggest bucket. The local sorting takes formula_54. Experiments performed in the early 1990s on Connection Machine supercomputers showed samplesort to be particularly good at sorting large datasets on these machines, because its incurs little interprocessor communication overhead. On latter-day GPUs, the algorithm may be less effective than its alternatives. Efficient Implementation of Samplesort. As described above, the samplesort algorithm splits the elements according to the selected splitters. An efficient implementation strategy is proposed in the paper "Super Scalar Sample Sort". The implementation proposed in the paper uses two arrays of size formula_55 (the original array  containing the input data and a temporary one) for an efficient implementation. Hence, this version of the implementation is not an in-place algorithm. In each recursion step, the data gets copied to the other array in a partitioned fashion. If the data is in the temporary array in the last recursion step, then the data is copied back to the original array. Determining buckets. In a comparison based sorting algorithm the comparison operation is the most performance critical part. In Samplesort this corresponds to determining the bucket for each element. This needs formula_56 time for each element. Super Scalar  Sample Sort uses a balanced search tree which is implicitly stored in an array . The root is stored at 0, the left successor of formula_57 is stored at formula_58 and the right successor is stored at formula_59. Given the search tree , the algorithm calculates the bucket number of element formula_60 as follows (assuming formula_61 evaluates to 1 if it is "true" and 0 otherwise): Since the number of buckets is known at compile time, this loop can be unrolled by the compiler. The comparison operation is implemented with predicated instructions. Thus, there occur no branch mispredictions, which would slow down  the comparison operation significantly. Partitioning. For an efficient partitioning of the elements, the algorithm needs to know the sizes of the buckets in advance. To partition the elements of the sequence and put them into the array, we need to know the size of the buckets in advance. A naive algorithm could count the number of elements of each bucket. Then the elements could be inserted to the other array at the right place. Using this, one has to determine the bucket for each elements twice (one time for counting the number of elements in a bucket, and one time  for inserting them). To avoid this doubling of comparisons, Super Scalar Sample Sort uses an additional array formula_66 (called oracle) which assigns each index of the elements to a bucket. First, the algorithm determines the contents of formula_66 by determining the bucket for each element and the bucket sizes, and then placing the elements into the bucket determined by formula_66. The array formula_66 also incurs cost in storage space, but as it only needs to store formula_70 bits, these cost are small compared to the space of the input array. In-place samplesort. A key disadvantage of the efficient Samplesort  implementation shown above is that it is not in-place and requires a second temporary array of the same size as the input sequence during sorting. Efficient implementations of e.g. quicksort are in-place and thus more space efficient. However, Samplesort can be implemented in-place as well. The in-place algorithm is separated into four phases: One obvious disadvantage of this algorithm is that it reads and writes every element twice, once in the classification phase and once in the block permutation phase. However, the algorithm performs up to three times faster than other state of the art in -place competitors and up to 1.5 times faster than other state of the art sequential competitors. As sampling was already discussed above, the three later stages will be further detailed in the following. Local classification. In a first step, the input array is split up into formula_8 stripes of blocks of equal size, one for each processor. Each processor additionally allocates formula_19 buffers that are of equal size to the blocks, one for each bucket. Thereafter, each processor scans its stripe and moves the elements into the buffer of the according bucket. If a buffer is full, the buffer is  written into the processors stripe, beginning at the front. There is always at least one buffer size of empty memory, because for a buffer to be written (i.e. buffer is full), at least a whole buffer size of elements more than elements written back had to be scanned. Thus, every full block contains elements of the same bucket. While scanning, the size of each bucket is kept track of. Block permutation. Firstly, a prefix sum operation is performed that calculates the boundaries of the buckets. However, since only full blocks are moved in this phase, the boundaries are rounded up  to a multiple of the block size and a single overflow buffer is allocated. Before starting the block permutation, some empty blocks might have to be moved to the end of its bucket. Thereafter, a write pointer formula_73 is set to the start of the bucket formula_51 subarray for each bucket and a read pointer formula_75 is set to the last non empty block in the bucket formula_51 subarray for each bucket. To limit work contention, each processor is assigned a different primary bucket formula_77 and two swap buffers that can each hold a block. In each step, if both  swap buffers are empty, the processor decrements the read pointer formula_78 of its primary bucket and reads the block at formula_79 and places it in one of its swap buffers. After determining the destination bucket formula_80 of the block by classifying the first element of the block, it increases the write pointer formula_81, reads the block at formula_82 into the other swap buffer and writes the block into its destination bucket. If formula_83, the swap buffers are empty again. Otherwise the block remaining in the swap buffers has to be inserted into its destination bucket. If all blocks in the  subarray of the primary bucket of a processor are in the correct bucket, the next bucket is chosen as the primary bucket. If a processor chose all buckets as primary bucket once, the processor is finished. Cleanup. Since only whole blocks were moved in the block permutation phase, some elements might still be incorrectly placed around the bucket boundaries. Since there has to be enough space in the array for each element, those incorrectly placed elements can be moved to empty spaces from left to right, lastly considering the overflow buffer. External links. Frazer and McKellar's samplesort and derivatives : Adapted for use on parallel computers: