Comparative Analysis between Selection Sort and Merge Sort Algorithms

Research on sorting algorithms is a very core research field in computer science. Today, any advancement in sorting algorithms would lead to less time and space required to perform the sorting task. This paper discusses a newly proposed algorithm—‘ledger-based sorting algorithm’. It is basically an advancement to ‘counting sort’, but not exactly counting sort. The purpose of this paper is to reduce the time and space required for sorting when the numbers given to you belong to a particular range (0–n) and all of them are distinct. This paper also compares the newly proposed algorithm with insertion sort, bubble sort, selection sort, quick sort, merge sort and mainly counting sort through graphical time analysis of these algorithms. The paper especially compares counting sort algorithm with the proposed algorithm in terms of both time and space complexity.

An alternative taxonomy (to that of Knuth and others) of sorting algorithms is proposed. It emerges naturally out of a top-down approach to the derivation of sorting algorithms. Work done in automatic program synthesis has produced interesting results about sorting algorithms that suggest this approach. In particular, all sorts are divided into two categories: hardsplit/easyjoin and easysplit/hardjoin. Quicksort and merge sort, respectively, are the canonical examples in these categories. Insertion sort and selection sort are seen to be instances of merge sort and quicksort, respectively, and sinking sort and bubble sort are in-place versions of insertion sort and selection sort. Such an organization introduces new insights into the connections and symmetries among sorting algorithms, and is based on a higher level, more abstract, and conceptually simple basis. It is proposed as an alternative way of understanding, describing, and teaching sorting algorithms.

Merging and sorting algorithms are the backbone of many modern computer applications. As such, efficient implementations are desired. Recent architectural advancements in CPUs (Central Processing Units), such as wider and more powerful vector instructions, allow for algorithmic improvements. This paper presents a new approach to merge sort using vector instructions. Traditional approaches to vectorized sorting typically utilize a bitonic sorting network (Batcher's Algorithm) which adds significant overhead. Our approach eliminates the overhead from this approach. We start with a branch-avoiding merge algorithm and then use the Merge Path algorithm to split up merging between the different SIMD lanes. Testing demonstrates that the algorithm not only surpasses the SIMD based bitonic counterpart, but that it is over 2.94X faster than a traditional merge, merging over 300M keys per second in one thread and over 16B keys per second in parallel. Our new sort reaches is over 5X faster than quicksort and 2X faster than Intel's IPP library sort, sorting over 5.3M keys per second for a single processor and in parallel over 500M keys per second and a speedup of over 2x from a traditional merge sort.

This paper presents the study of implementation and performance in sorting process, using two different algorithms, namely Insertion Sort and Merge Sort. In the first stage, the two algorithms were implemented in C++ language to sort several numbers typed by a user. In the second stage, the source codes for those algorithms were modified to enable sorting randomly generated numbers with the amount as requested by the user. To find out how well they performed in sorting out the data, therefore in the last stage the two algorithms were tested to sort random numbers with predetermined amount ranges and the results were compared. From the experiments performed, merge sort algorithm had shown a better performance, particularly in a large number of data (> 10000). Insertion sort algorithm has the advantage in lower complexity algorithm, notably in the best case condition and since it does not use recursion routines in sorting process, hence it does not require as much storage space or memory as needed by merge sort algorithm.

Sorting is the most fundamental operation in database system. There are many classical sorting algorithms and among them the most commonly‐used sorting algorithm in modern database system is merge sort. Merge sort is an efficient, general‐purpose, comparison‐based sorting algorithm. As merge sort is based on a divide and conquer model, it has found wide use in parallel computing algorithms. In this study, the authors present an adaptive sorting algorithm based on merge sort which is called array sort. Array sort not only takes advantages of merge sort, but also simplify the merge step by using a tag array. The proposed implementation consists of three phases: tag array creation phase, tag array split phase and list merge phase. Instead of just evaluating array sort in a single thread environment, the authors also run their experiments on multi‐thread to test how array sort performs.

Enhanced Merge Sort- A New Approach to the Merging Process

Quicksort is an important algorithm that uses the divide and conquer concept, and it can be run to solve any problem. The performance of the algorithm can be improved by implementing this algorithm in parallel. In this paper, the parallel sorting algorithm named the Multi-Deque Partition Dual-Deque Merge Sorting algorithm (MPDMSort) is proposed and run on a shared memory system. This algorithm contains the Multi-Deque Partitioning phase, which is a block-based parallel partitioning algorithm, and the Dual-Deque Merging phase, which is a merging algorithm without compare-and-swap operations and sorts the small data with the sorting function of the standard template library. The OpenMP library, which is an application programming interface used to develop the parallel implementation of this algorithm, is implemented in MPDMSort. Two computers (one with an Intel Xeon Gold 6142 CPU and the other with an Intel Core i7-11700 CPU) running Ubuntu Linux are used in this experiment. The results show that MPDMSort is faster than parallel balanced quicksort and multiway merge sort on the large random distribution data. A speedup of 13.81times and speedup per thread of 0.86 can be obtained. Thus, developers can use these parallel partitioning and merging algorithms to improve the performance of related algorithms.

We have a lot of sorting algorithms in practice in this modern informatica world. Most of them are operating in quadratic time, some have managed to get them down to logarithmic time combined with linear, but there is no sorting algorithm that is efficient enough to sort data in linear time alone. In this paper, we will try to devise some mathematical support to the failure of minimalizing the sorting algorithms in linear time by comparing three well-known approximations of ! with detailed mathematical proofs associated with it. Further, this paper deals with detailed mathematical analysis to support the fact the minimum time complexity that can be attained by a sorting algorithm is of order O( ) without taking into account any extrapolated system advancement or computer architecture modification. There are various sorting algorithms like Merge sort, Heap sort, Interpolation sort, etc. which attains this minimum possible time complexity (worst case). But if we consider some biased cases then better time complexity can be obtained. If the elements of the array are in arithmetic progression, then interpolation search takes lesser time and there are many such biased cases where a particular algorithm gives better output but this paper deals with the general case i.e., no conditions are imposed on the elements of the array.

Sorting is one of the most frequent concerns in Computer Science, various sorting algorithms were invented for specific requirements. As these requirements and capabilities grow, sequential processing becomes inefficient. Therefore, algorithms are being enhanced to run in parallel to achieve better performance. Performing algorithms in parallel differ depending on the degree of multi-threading. This study determines the optimal number of threads to use in parallel merge sort. Furthermore, it provides a comparative analysis of various degrees of multithreading. The implementation in this empirical experiment takes a group of devices with various specifications. For each device, it takes fixed-sized data set and executes merge sort for sequential and parallel algorithms. For each device, the lowest average runtime is used to measure the efficiency of the experiment. In all experiments, single-threaded is more efficient when the data size is less than 105 since it claimed 53% of the lowest runtime than the multithreaded executions. The overall average of the experiments shows either four or eight threads, with 72% and 28%, respectively, are most efficient when data sizes exceed 105.

Sorting algorithms play a crucial role in computing, but most are designed with rigid structure that are only efficient under certain conditions. Although some sorting algorithms perform well in some circumstances, they do not perform well on some resistant platforms. This study introduces Wall-L Merge Sort, which combines quadratic sorting with a modifiable multi-layer merging approach. By setting a single parameter, L, which determines the number of merge layers, Wall-L Sort shows a transition in the time complexity from O(n2) to without any modification in the unique idea. This degree of freedom enables a broad variety of input sizes to be encompassed and expands to several constraint platforms. The results show that Wall-L Sort and K-way Merge Sort have the built-in ability to handle different situations where other algorithms fail without assistance functions. Wall-L Merge Sort is the only sorting algorithm that combines complexity tuning, cache efficiency, recursion depth control, parallelism, and broad adaptability into one framework. It may not be the best choice for every situation, but its flexibility makes it a good fit for many different platforms, from small embedded systems to big computing systems. The theoretical and empirical evidence in this paper substantiates these advantages.

With the popularity of multi-core and many-core platform, multi-core-oriented parallel programming and optimization has become a research hotspot in computer field. However, the vast majority of programmers are still continuing the traditional serial programming habits. Meanwhile, the mainstream algorithms are still in serial. Therefore, how to parallelize the serial programs effectively and write the multi-core programs efficiently is becoming an urgent problem in the field of multi-core programming. In this paper, the optimization of the heap sort algorithm has been achieved adopting the parallel technology based on openMP. Firstly, based on the hypercube model, the heap data to be sorted are divided into n (n is the computer thread amount) blocks called sub-heap which will be sorted respectively. Thus, the scale of the problem is decomposed into 1/n, secondly, the sorted sub-heaps will be merged using merge sort algorithm, finally, the parallel optimized heap sort algorithm is compared to the traditional heap sort algorithm. When the data quantity comes up to ten million bytes, the execution efficiency of the optimized parallel heap sort algorithm of this paper is improved significantly compared with the traditional serial heap sort algorithm on the experimental computer. Furthermore, the efficiency gap between the two algorithms will be intensified prominently with the increasing data quantity.

One of the basic problem in computer science is to arrange the items in lexicographic order. Sorting is one of the major research topic. There are number of sorting algorithms. This paper presents the implementation and detailed analysis of library sort. Library sort is also called gapped insertion sort. It is a sorting algorithm that uses insertion sort with gaps. Time taken by insertion sort is O (n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) because each insertion takes O (n) time; and library sort has insertion time O (log n) with high probability. Total running time of library sort is O (n log n) time with high probability. Library sort has better run time than insertion sort, but the library sort also has some issues. The first issue is the value of gap which is denoted by `ε', the range of gap is given, but it is yet to be implemented to check that given range is satisfying the concept of library sort algorithm. The second issue is re-balancing which accounts the cost and time of library sort algorithm. The third issue is that, only a theoretical concept of library sort is given, but the concept is not implemented. So, to overcome these issues of library sort, in this paper, we have implemented the concept of library sort and done the detailed experimental analysis of library sort algorithm, and measure the performance of library sort algorithm on a dataset.

We present LCP Merge, a novel merging algorithm for merging two ordered sequences of strings. LCP Merge substitutes string comparisons with integer comparisons whenever possible to reduce the number of character-wise comparisons as well as the number of key accesses by utilizing the longest common prefixes (LCP) between the strings. As one of the applications of LCP Merge, we built a string merge sort based on recursive merge sort by replacing the merging algorithm with LCP Merge and we call it LCP Merge sort. In case of sorting strings, the computational complexity of recursive merge sort tends to be greater than O(n lg n) because string comparisons are generally not constant time and depend on the properties of the strings. However, LCP Merge sort improves recursive merge sort to the extent that its computational complexity remains O(n lg n) on average. We performed a number of experiments to compare LCP Merge sort with other string sorting algorithms to evaluate its practical performance and the experimental results showed that LCP Merge sort is efficient even in the real-world.

The Shell sort algorithm is one of the most practically effective in-place sorting algorithms. However, it is difficult to execute this algorithm with its intended running time complexity on data encrypted using fully homomorphic encryption (FHE), because the insertion sort in Shell sort has to be performed by considering the worst-case input data. In this paper, in order for the sorting algorithm to be used on the FHE data, we modify the Shell sort with an additional parameter α, allowing exponentially small sorting failure probability. For a gap sequence of powers of two, the modified Shell sort with input array length n is found to have the trade-off between the running time complexity of O(n3/2√α+loglogn) and the sorting failure probability of 2-α. Its running time complexity is close to the intended running time complexity of O(n3/2) and the sorting failure probability can be made very low with slightly increased running time. Further, the near-optimal window length of the modified Shell sort is also derived via convex optimization. The proposed analysis of the modified Shell sort is numerically confirmed by using randomly generated arrays. For the practical aspect, our modification can be applied to any gap sequence, and we show that Ciura’s gap sequence, which is known to have good practical performance, is also practically effective when our modified Shell sort is applied. We compare our modified Shell sort with other sorting algorithms with the FHE over the torus (TFHE) library, and it is shown that this modified Shell sort has the best performance in running time among in-place sorting algorithms on homomorphic encryption scheme.

Sorting is the procedure of arranging the elements in ascending or descending order. Sorting algorithms are not only used in computer science but also in our day-to-day life to reduce complexity. For optimizing the use of other algorithms, efficient sorting is required. This paper makes a comparison between merge sort, quick sort, selection sort and insertion sort by their time complexities.