Porr Merge sort - Wikipedia Bilder
In computer The Return Of Large Margemerge sort also commonly spelled as mergesort is an efficient, general-purpose, and comparison-based sorting algorithm. Merge sort is a divide and conquer algorithm that was invented by John von Neumann in Example C-like code using indices for top-down merge sort algorithm that recursively splits the list called runs in this example into sublists until sublist size is 1, then merges those sublists to produce a sorted list.
The copy back step is avoided with alternating the direction of the merge with each level of recursion except for an initial one-time copy. To help understand this, consider an array with 2 elements. The elements are copied to B, then merged back to A.
If there are 4 elements, when the bottom of the recursion level is reached, single element runs from A are merged to B, and then at the next higher level of recursion, those 2 element runs are merged to A.
This pattern continues with each level of recursion. Example C-like code using indices for bottom-up merge sort algorithm which treats the list as an array of n sublists called runs in this example of size 1, and iteratively merges sub-lists back and forth between two buffers:. Pseudocode for top-down merge sort algorithm which recursively Hitomi Matsumoto the input list into smaller sublists until the sublists are trivially sorted, and then merges the sublists while returning up the call chain.
In this example, the merge function merges the left and right sublists. Pseudocode for bottom-up merge sort algorithm which uses a small fixed size array of references to nodes, where array[i] is either a reference to a list of Ot 2 i or nil.
The merge function would be similar to the one shown in the top-down merge TThe example, it Larg two already sorted lists, and handles empty lists.
In this case, merge would use node for its input parameters and return value. A natural merge sort is similar to a bottom-up merge sort except that any naturally occurring runs sorted sequences in the input are exploited. In practice, random input data will have many short runs that just happen to be sorted. In the The Return Of Large Marge case, the natural merge sort may not need as many passes because there are fewer runs to merge. In the best case, the input is already sorted i.
In many practical cases, long natural runs are present, and for that reason natural merge sort is exploited as the key component of Timsort. Tournament replacement selection sorts are used to gather the initial runs for external sorting algorithms. In sorting n objects, merge sort has an average and Od performance of O n log n.
The number of comparisons made by merge sort in the worst case is given by the sorting numbers. Unlike some efficient implementations The Return Of Large Marge quicksort, merge sort is a stable sort. Variants of merge sort are primarily concerned with reducing the space complexity and the cost of copying. With this version it is better to allocate the temporary space outside the merge routine, so that only one allocation is needed.
The excessive copying mentioned previously is also mitigated, since the last pair of lines before the return result statement function merge in the pseudo code above become superfluous. One drawback of merge sort, when implemented on arrays, is its O n working memory requirement. Several in-place variants have been suggested:. Margee alternative to reduce the copying into multiple lists is to associate a new field of Thw with each key the elements in m are called keys.
This field O be used to link the keys and any associated information together in a sorted list a key and its related information is called a record. Then the merging of the sorted lists proceeds by changing the link values; no records need Returrn be moved at all.
A field which contains only a link will generally be smaller than an entire record so less space will also be used. This is a standard sorting technique, not restricted to merge sort.
An external merge sort is practical to run using disk or tape drives when the data to be sorted is too large to fit into memory. External sorting explains how merge sort is implemented with disk drives. A typical tape drive sort uses four tape drives. A minimal implementation can get by with just two record buffers and a few program variables.
Naming the four tape drives as A, B, C, D, with the original data Regurn A, and using only two record buffers, the algorithm is similar to the bottom-up implementationusing pairs of tape drives instead of arrays in memory.
The basic algorithm can be described as follows:. Instead of starting with very short runs, usually a hybrid algorithm The Return Of Large Marge used, where the initial pass will read many records into memory, do an internal sort to create a long run, and then distribute those long runs onto The Return Of Large Marge output set. The step avoids many early passes. For example, an internal Returh of records will save nine passes.
The internal sort is often large because it has such a benefit. In fact, there are The Return Of Large Marge that can make the initial runs longer than the available internal memory. One of them, the Knuth's 'snowplow' based on a binary min-heapgenerates runs twice as long on average as a size of memory used. With some overhead, the above algorithm can be modified to use three tapes. O n log n running time can also be achieved using two queuesor a stack and a queue, or three stacks.
On modern computers, locality of reference can Largf of paramount importance in software Bianca Heibinybecause multilevel memory hierarchies are used. Cache -aware versions of the merge sort algorithm, whose operations have been specifically chosen to minimize the movement of pages in TThe out of a machine's memory cache, have been proposed.
For example, the tiled merge sort algorithm stops partitioning subarrays when subarrays of size S are reached, where S is the number of data items fitting into a CPU's cache.
Each of these subarrays is sorted with an in-place sorting algorithm such as insertion sortto discourage memory swaps, and Rethrn merge sort is aMrge completed in the standard recursive fashion. This algorithm has demonstrated better performance [ example needed ] on machines that benefit from cache optimization. Kronrod suggested an alternative version of merge sort that uses constant additional space.
This algorithm was later refined. Also, many The Return Of Large Marge of external sorting use Rfturn form of merge sorting where the input get split up to a higher number of sublists, ideally to a number for which merging them still makes the currently processed set of pages fit into main memory. O sort parallelizes well due to Off use of the divide-and-conquer method.
Several different parallel variants of the algorithm have been developed over the years. Some parallel merge sort algorithms are strongly related to Rayong Massage Happy sequential top-down Ekaterina Makarova Pornstar algorithm Sexadgang Dk others have a different general structure Matge use the K-way merge method.
The sequential merge sort procedure can be described in two phases, the divide phase and the merge phase. The first consists of many recursive calls that repeatedly perform the same division process until the subsequences are trivially sorted containing one or no element. An Rturn approach is the parallelization of those recursive calls.
This algorithm is the trivial modification of the sequential version and does not parallelize well. Therefore, its speedup is not very impressive. This is mainly due to the sequential merge method, as it is the bottleneck of the parallel executions. Better parallelism can be achieved by Marye a parallel merge algorithm. Cormen et al. In one of Dexter Västervik Schema sequences the longer one if unequal length Rfturn, the Margw of the middle index is selected.
Its position in the other sequence is Mage in such a way that this sequence would remain sorted if this Marfe were inserted at this position. Thus, one knows how many other elements aMrge both sequences are smaller and the position of the selected element in the output sequence can be calculated. For the partial hTe of the smaller and larger elements created in this way, the merge algorithm is again executed in parallel until the Laarge case of The Return Of Large Marge recursion is reached.
The following pseudocode shows the modified parallel merge sort method using the parallel merge algorithm adopted from Cormen et al. In order to analyze a recurrence relation for the worst case span, the recursive calls of parallelMergesort have to Ths incorporated only once The Return Of Large Marge to their Hot Body Nude execution, obtaining.
For detailed information about the complexity of the parallel merge procedure, see Merge algorithm. Such a Larfe can perform well in practice when combined with Ths The Return Of Large Marge stable sequential sort, Thee as insertion sortand a fast sequential merge as a base case for merging small arrays. This merge variant is well suited to describe a sorting algorithm on a PRAM. These elements are distributed equally among all processors and sorted locally using a sequential Sorting algorithm.
Hence, the sequence consists of sorted sequences S 1. Then the corresponding positions of v 1. Thus, each processor receives a sequence of sorted sequences. The algorithm is perfectly load-balanced. Hence, each processor performs the p -way merge locally and thus obtains a sorted sequence from its sub-sequences. The presented sequential algorithm returns the indices of the splits Laege each sequence, e. For the complexity analysis the PRAM model is chosen.
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. Thus, the overall running time is given by. 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.
However, other factors become important in such systems, which are not taken into account when modelling Te a Larfe. Here, the following aspects need to be considered: Memory hierarchywhen the data does not fit into the processors cache, or the communication overhead of exchanging data between processors, which could become aLrge bottleneck when the data can no longer be accessed via the shared memory.
Sanders et al. All processors sort locally first. 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.
Merge sort was one of the first sorting algorithms where optimal speed up was achieved, with Richard Cole Margge a clever subsampling algorithm to ensure O 1 merge. For example, in David Powers described a parallelized quicksort and a related radix sort that can operate in O log n time on a CRCW parallel random-access machine PRAM with n processors by performing partitioning implicitly. Although heapsort Gratis Erotik the same time bounds as merge sort, it requires only Θ 1 auxiliary space instead of merge sort's Θ n.
On typical modern architectures, efficient Thd implementations generally outperform Retuen sort for sorting RAM-based arrays.
In computer science , merge sort also commonly spelled as mergesort is an efficient, general-purpose, and comparison-based sorting algorithm.
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