• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Smart Bubble Sort:A Novel and Dynamic Variant of Bubble Sort Algorithm

    2022-08-23 02:17:38MohammadKhalidImamRahmani
    Computers Materials&Continua 2022年6期

    Mohammad Khalid Imam Rahmani

    College of Computing and Informatics,Saudi Electronic University,Riyadh,Saudi Arabia

    Abstract: In the present era, a very huge volume of data is being stored in online and offline databases.Enterprise houses, research, medical as well as healthcare organizations, and academic institutions store data in databases and their subsequent retrievals are performed for further processing.Finding the required data from a given database within the minimum possible time is one of the key factors in achieving the best possible performance of any computer-based application.If the data is already sorted,finding or searching is comparatively faster.In real-life scenarios,the data collected from different sources may not be in sorted order.Sorting algorithms are required to arrange the data in some order in the least possible time.In this paper, I propose an intelligent approach towards designing a smart variant of the bubble sort algorithm.I call it Smart Bubble sort that exhibits dynamic footprint: The capability of adapting itself from the average-case to the best-case scenario.It is an in-place sorting algorithm and its best-case time complexity is Ω(n).It is linear and better than bubble sort,selection sort,and merge sort.In averagecase and worst-case analyses,the complexity estimates are based on its static footprint analyses.Its complexity in worst-case is O(n2)and in average-case is Θ(n2).Smart Bubble sort is capable of adapting itself to the best-case scenario from the average-case scenario at any subsequent stages due to its dynamic and intelligent nature.The Smart Bubble sort outperforms bubble sort,selection sort,and merge sort in the best-case scenario whereas it outperforms bubble sort in the average-case scenario.

    Keywords: Sorting algorithms; smart bubble sort; footprint; dynamic footprint;time complexity;asymptotic analysis

    1 Introduction

    The digital world is producing a huge amount of data from almost every field of life and databases are storing those data in computer-based information systems.As said by Knuth in his book [1],“virtually every important aspect of programming arises somewhere in the context of sorting or searching”, many improvements have been conducted in sorting and searching algorithms since the advent of programming[2,3].Sorting various records in different application domains consumes 25%to 50% of the computational resources worldwide as mentioned in [2,4].It is argued that a single sorting algorithm cannot perform well in every situation that is why for a given scenario,an appropriate sorting algorithm must be applied[5].For example,bubble sort,insertion sort,and selection sort are best suited to small input data whereas quick sort and merge sort perform better when applied to a large volume of data [2,5–9].Researchers have been working to design more efficient algorithms for decades.Currently, new sorting and searching algorithms, as well as efficient modifications,have been proposed by the computer science community.In this connection, Mohammed et al.[10]developed a Bidirectional Conditional Insertion Sort algorithm, which demonstrated O(n1.5) time complexity in average case scenarios for normal and uniform distributed data.Appiah et al.[11]have designed a new algorithm:Magnetic Bubble sort with enhancements of bubble sort in the case where redundancies occur in the list.Alotaibi et al.[12] have proposed a new in-place sorting algorithm,which performs better than bubble sort and selection sort but similar to Insertion sort.Ranaa et al.[13]have developed a new approach called MinFinder to overcome some of the weaknesses of some conventional algorithms in terms of stability,computational time,and complexity analysis.Cheema et al.[14] have employed a hybrid approach that takes a minimum number of comparisons with less time and space complexity to sort example data with merge sort and bi-directional bubble sort approaches.Faujdar et al.[15] have evaluated the existing sorting algorithms using repeated data considering the average-case, worst-case, and best-case scenarios.Sodhi et al.[16] have designed Enhanced Insertion sort that achieved O(n) worst-case time complexity and O(n1.585) average time complexity showing better performance compared to insertion sort.

    In this article,a novel and dynamic variant of bubble sort is being proposed,which I call Smart Bubble sort.In the best-case scenario,our proposed algorithm will iterate only once and will terminate when it detects that the data is already in the sorted order.It has the capability of transforming itself from an average-case scenario to the best-case scenario when it detects that the remaining elements are now sorted after bubbling up some of the elements to their sorted positions during the sorting procedure.I have devised the term dynamic footprint to refer to this concept.The remaining of the paper is structured as follows.In Section 2,preliminary concepts have been discussed.In Section 3,the proposed algorithm is introduced and elaborated along with an example illustration of the working procedure followed by a detailed asymptotic analysis.In Section 4,a description of the new dataset is provided,the experimental setup is explained,and empirical results have been highlighted.In Section 5,conclusions are drawn at the end of the article.

    2 Preliminaries

    Searching and sorting of database entries are the two direct applications of sorting algorithms.Broader applications of sorting techniques are for the solution of many complex problems in the fields of information retrieval [5,17–19], image retrieval [20–25], image processing [26,27], database systems, networking [28], management information systems, decision support systems, operations research and optimization problems [29–31].Similarly, sorting algorithms play an important role in educational subjects like design&analysis of algorithms and data structures&programming where the problems require the use of syntax/semantics of every important programming construct related to any programming language.Bubble sort,selection sort,and exchange sort are best suited to data samples of small to medium sizes.These algorithms are comparison-based sorting algorithms;therefore,their lower bound performance is proven to be O(nlogn).A small number of algorithms exist,which claim to have linear bound,however,they are designed for some special cases of input data.Sorting algorithms with O(n)time complexity are considered best performing algorithms.On the other hand,algorithms with O(n2)time complexity are thought to be bad performers whereas algorithms with O(nlogn)time complexity are identified as good algorithms.

    2.1 Performance Measurement and Analysis of Sorting Algorithms

    The performance of an algorithm is estimated in terms of computational complexity that is measured as its time complexity and space complexity.The time complexity of an algorithm is more important because the availability of sufficiently large memory space in the current era of technology is not a big deal.The efficiency of an algorithm is always expressed as a function of its input size as T(n)or S(n);where n is the input size,T(n)and S(n)are time and space complexities,respectively.

    The relative efficacy of sorting algorithms can be characterized by the order of growth of their running times.While calculating the exact running times of a very large volume of input data, the input itself dominates the coefficients and other lower-order terms.For measuring the efficiency of algorithms and comparing the performance of two or more sorting algorithms,the order of growth of their running times is significantly used.However,this technique is not useful when the order of growth of two or more algorithms is the same.On such occasions,the determination of the exact running time of the algorithms is essential which I have done in this article.Asymptotic efficiency of an algorithm refers to the estimation of the order of growth of the running time for very large inputs[7].That is,I am concerned with how the running time of an algorithm increases with the size of the input in the limit, as the size of the input increases without bound.Usually, an algorithm that is asymptotically more efficient will be the best choice for all but very small inputs.To compare different algorithms for the problem of sorting,I break the analysis of algorithms into three different cases so that I can formulate a way that can provide a better estimate of the resources required.In the first case known as the best-case,the minimum number of computational steps are taken by the algorithm because the input data is sorted in the same order which would be the expected outcome of the algorithm.In the second case known as the average case,the average number of computational steps are taken for the algorithm.This is the most complex kind of analysis that is often based on the probability theory because the input data is randomly collected without any explicit ordering.In the third case known as the worst-case,the maximum number of computational steps are taken by the algorithm.This case is analyzed most often because the maximum time complexity of the algorithm matters the most where the input data is sorted in reverse order.

    2.2 Internal Sorting Algorithms

    Internal sorting algorithms maintain the entire sequence of data into the main memory of the computer.These algorithms are based on a comparison of items and are further grouped into simple or complex categories.The simple ones are mainly iterative,simple to understand,and easy to implement but their time complexity is more i.e., O(n2).Bubble sort, Selection sort, and Insertion sort are in this category.The complex ones are mainly divide-and-conquer based and difficult to implement but their time complexity is less i.e.,O(nlogn).Quicksort,Merge sort,and Heapsort algorithms are in this category.

    3 The Proposed Algorithm

    In this section, I have first identified the problem of the bubble sort algorithm with its detailed asymptotic analysis.Next,I described our idea about the Smart Bubble sort algorithm and provided its detailed asymptotic analysis with a proper derivation of best case, worst case, and average case notations.A sorting algorithm consists of instructions arranged in a sequence that puts the input data elements in an ordered sequence.Designing an efficient sorting algorithm is necessary for the optimal performance of different computer applications that require sorted data elements to smoothly perform their operations.Sorting algorithms are usually selected based on their computational requirements and ease of implementation.The efficiency of a comparison-based sorting algorithm can be enhanced by reducing either the number of comparison operations or swapping operations or both.

    3.1 Problem with Bubble Sort

    I start with the formal description of the Bubble sort algorithm in terms of its steps as below:

    BUBBLE-SORT(A,n)

    A is an array of size n;each element of the array will be sorted after the algorithm gets terminated.

    Step 1.for i=1 to n-1 do

    Step 2.for j=1 to n–i-1 do

    Step 3.if A[j]>A[j+1]then

    Step 4.val=A[j]

    Step 5.A[j]=A[j+1]

    Step 6.A[j+1]=val

    It is well known that bubble sort is considered the slowest sorting algorithm because it performs the maximum number of comparisons without looking at the instance of input data.It fails to exploit the instance of input data in the best case and the average case.I have highlighted these phenomena in Figs.1 and 2 through simple and effective illustrations where the steps of bubble sort are shown on the left side and that of Smart Bubble sort are shown on the right side of the figures.

    Figure 1:Illustration of the footprint of smart-bubble-sort in case of sorted data

    In Fig.1, I can see that the data is already sorted.However, bubble sort will go through all the required passes and steps even if the data is already sorted.On the other hand, Smart Bubble sort will perform only one iteration(Please see Iteration 1 of Pass 1 in Fig.1)and will not continue once it detects that the data is already sorted.Similarly,the data instance in Fig.2 is nearly sorted.Again,bubble sort will go through all the passes and steps to complete its operation blindly.After bubbling up the element 20 to the highest position of the array,the remaining elements,1,4,6,8,and 14 are now in their proper positions of the sorted array at the end of Iteration 2 of Pass 2.Bubble sort is not able to exploit this advantage.It will complete all the remaining iterations and steps blindly doing no further swap operation.However, Smart Bubble sort will exploit this situation to its advantage through its dynamic footprint detection capability and will terminate without performing the remaining iterations and passes.

    Figure 2:Illustration of the dynamic footprint of smart-bubble-sort in case of nearly sorted data

    3.2 Analysis of Bubble Sort

    Analysis of the algorithms is shown as the general expression of the cost function in terms of the input data size in the form of asymptotic notationsΩ(best case),Θ(average case), and O(worst case).The computational complexity for individual steps is provided in Tab.1 that corresponds to the algorithm provided in BUBBLE-SORT(A,n).

    Table 1: Analysis of bubble sort algorithm

    3.2.1 Best Case Analysis

    The best case of input instance for the sorting algorithms occurs when the elements are sorted in the same order that the algorithm will produce.Therefore,T(n)for the best case can be derived as below:

    Since Eq.(1)is a quadratic equation,the time complexity of Bubble sort in best-case is O(n2).

    3.2.2 Worst Case Analysis

    The worst case of input instance for the sorting algorithms occurs when the elements are sorted in the opposite order that the algorithm will produce.Therefore,T(n)for the worst case can be derived as below:

    3.2.3 Average Case Analysis

    The average case of input instance for the sorting algorithms occurs when the elements are arranged in random order.This is the most expected case.Therefore, T(n) for the average case can be derived as below:

    Since Eq.(3)is a quadratic equation,the algorithm’s time complexity in the average case is O(n2).

    3.3 The Concept of Smart Bubble Sort

    In this section,I have materialized the idea of our proposed Smart Bubble sort algorithm.I begin with the description of the algorithm followed by its asymptotic analysis.

    The bubble sort algorithm usesn-1 passes for sortingnnumbers.In each pass, the number of iterations to compare the key values are(n-pass number).For controlling the passes of the algorithm,the outerforloop is used.The innerforloop is used for making the actual comparisons for swapping the adjacent key values whenever they are found out of order as shown in BUBBLE-SORT(A, n).Bubble sort is considered the slowest algorithm in average-case and best-case scenarios.But its bestcase, as well as average-case running times, can be improved by doing some smart enhancements in the classical bubble sort algorithm.The bubble sort does unnecessary work even if the input data is already sorted as demonstrated in Figs.1 and 2.It keeps on comparing the items blindly even if the remaining items are already sorted thereby losing the opportunity of exploiting the scenario for terminating it quickly whereas there is no need of running the remaining passes.The Smart Bubble sort algorithm exploits the scenarios and terminates quickly giving much rise in its performance against not only bubble sort but other sorting algorithms as well.In case,the algorithm completes a pass for the remaining elements of the array without detecting any swapping of adjacent elements,it is concluded that the array is sorted and so the algorithm should stop immediately.Similarly,the Smart Bubble sort can transform the average-case into best-case whenever it detects that no swap has been performed at any stage of the algorithm execution as shown in Figs.1 and 2.

    To implement the Smart Bubble sort procedure,I have used a boolean variableflagto detect the swapping of elements.The variable will be set totrueat the beginning of every pass and if a swap is detected in that pass,it would be reset tofalse.If at the end of a pass,the variable is foundtrue,then it would be concluded that no swapping of elements has occurred and hence the algorithm is terminated immediately.

    A formal description of the Smart Bubble sort algorithm is given as follows where the order of sorting is considered as ascending order.

    Step 1.Forn-1number of passes,carry out each pass.

    Step 2.Bubble up the largest element to the top position.

    Step 3.If during the bubbling up no swap is encountered,Exit.

    Step 4.Else,carry out the next pass.

    The formal algorithm for Smart Bubble sort is provided in SMART-BUBBLE-SORT(A,n).

    SMART-BUBBLE-SORT(A,n)

    Ais an array of sizen;elements of the array will be sorted in increasing order after the algorithm gets terminated.

    Step 1.fori=1 ton–1 do

    Step 2.flag=true

    Step 3.forj=1 ton–i–1 do

    Step 4.ifA[j]>A[j+1]then

    Step 5.flag=false

    Step 6.val=A[j]

    Step 7.A[j]=A[j+1]

    Step 8.A[j+1]=val

    Step 9.ifflagthen

    Step 10.break

    The time complexity of Smart Bubble sort isΩ(n) in the best case, O(n2) in the worst case, andΘ(n2)in the average case.It is worth noting that the Smart Bubble sort is capable of converting itself from an average-case scenario to the best-case scenario as discussed earlier.This is because the Smart Bubble sort detects the dynamic footprint in its data instance during the execution of the algorithm well before the end of the execution.

    3.4 Analysis of Smart Bubble Sort

    Analysis of the algorithms is shown as the general expression of the cost function in terms of the input data size in the form of asymptotic notationsΩ(best case),Θ(average case), and O(worst case).The computational complexity for individual steps is provided in Tab.2 that corresponds to the algorithm provided in SMART-BUBBLE-SORT(A,n).

    Table 2: Analysis of smart bubble sort algorithm

    3.4.1 Best Case Analysis

    The best-case scenario for the algorithm arises when the input array is sorted in the same order in which the output is desired.The algorithm checks this scenario by using a boolean variableflag,which is initialized totrueat the beginning of the external for loop(for Passes).Inside the internal for loop(for comparison and swapping)at step 3 when any two adjacent elements are found out-of-order by the if-then structure(i.e.,ifA[j]>A[j+1])of step 4 then these elements are swapped and theflagvariable assumes afalsevalue.In case, the value of the variableflagis foundtrueby the if-then structure of step 9,it is concluded that the remaining elements of the array are already sorted.Therefore,the Smart Bubble sort algorithm should stop immediately.This is ensured through the break statement of step 10.As a result, the algorithm does the minimum possible work due to the ability to detect dynamic footprint smartly.

    Since the if-then structure of step 4 will never be satisfied in this case,steps 5 through 8 will not be contributing anything to the overall time complexity of the algorithm.Therefore,the total time taken by the algorithm is computed by taking the sum of the products of the corresponding time(cost)and repetition for the contributing steps only.Mathematically,the expression for the time complexity can be derived as follows:

    where,A=C3+C4,B=C1+C2+C3+C9+C10

    Since Eq.(4) is a linear equation, therefore, the coefficient of the highest order term and the constant term are neglected to get the asymptotic notation of the algorithm,which isΩ(n).Therefore,the time complexity of Smart Bubble sort in the best-case isΩ(n).

    3.4.2 Worst Case Analysis

    The worst case of an input instance for sorting algorithms occurs when the input array is sorted in descending order and the output is desired in ascending order or vice-versa.The algorithm will have to do the maximum work in this case.The for loop of step 1 is testedntimes withn-1 time for executing its body and one more time when the loop condition becomes false.Steps 2 and 9 will be executedn-1 time.Since each pair of adjacent elements will be out-of-order,the number of swapping will be equal to the number of comparisons.So,the if-then structure of step 9 will never be satisfied and as a result,the break statement of step 10 will never be executed.Steps 3 through 8 will be executed for the maximum possible number of times,which arefor the step 3 andfor the rest of the steps.Mathematically,the expression for the time complexity can be derived as follows:

    Since Eq.(5) is a quadratic equation, therefore, the coefficient of the highest order term and the lower order terms are neglected to get the asymptotic notation of the algorithm,which is O(n2).Therefore,the time complexity of Smart Bubble sort in the worst case is O(n2).

    3.4.3 Average Case Analysis

    The average case of input instance for the sorting algorithms occurs when the elements are arranged in random order.This is the most expected case.The algorithm might have to do as much work as in the worst case[7].However,I have safely assumed that only half of the adjacent elements are out-of-order.Therefore, the number of swapping will become almost half of the number of comparisons.Similar to the worst-case analysis, the for loop in step 1 is testedntimes and steps 2 and 9 will be executedn–1 time.Step 3 will be executed fortimes and steps 4 through 8 will be executed fortimes.Unlike the worst-case scenario, the if-then structure of step 9 may be satisfied at any successive pass after the first pass.When this happens,the footprint of the algorithm changes to the best-case scenario,and the break statement of step 10 is executed and as a result,the algorithm successfully terminates.This is the situation,which I call thedynamic footprintof the Smart Bubble sort.Thedynamic footprintis defined as the ability to convert the footprint from one case of input instance to another case of input instance.Therefore,the total time taken by the algorithm is computed by taking the sum of the products of the corresponding time(cost)and repetition for the contributing steps only.Mathematically,the expression for the time complexity can be derived as given below:

    Since Eq.(6) is a quadratic equation, therefore, the coefficient of the highest order term and the lower order terms are neglected to get the asymptotic notation of the algorithm,which isΘ(n2).Therefore,the time complexity of Smart Bubble sort in the average case isΘ(n2).

    4 Results and Discussion

    In this section, I have constructed a new dataset containing random, sorted, and reverse sorted data as input data to the algorithms.Then I have provided details about the experimental setup.Lastly,I have discussed the results.

    4.1 Dataset

    Using a perfect dataset is one of the important considerations for establishing an experimental setup for measuring the performance analysis of computer algorithms.In this paper, I have used datasets of randomly created positive integer values generated by a dedicated Java program, which are stored intextformat.The datasets are of 3 categories of sizes 500, 2500, 5000, 50000, 100000,625000,1250000,and 2500000.

    a.Random sets:Sets of the specified sizes were created in random order to analyze the average case scenario.

    b.Sorted sets:The Random sets were first sorted and then used for the analysis of the best-case scenario.

    c.Reverse Sorted sets:The Sorted sets were reversed and then used for the analysis of the worstcase scenario.

    I have designed a Java program that used the Random class along with its nextInt() method.This class was instantiated for calling the nextInt() method.The value of the argumentrangein the nextInt()method was given as 2147483647 to maintain as much uniqueness of elements in the data set as possible.

    Besides, I have created another similar dataset with approximately 50% sorted data items to provide evidence ofdynamic footprintdetection capability for which the results are provided in Section 4.2.4.

    4.2 Experimental Setup

    The proposed algorithm’s steps are written in a formal way suitable to be implemented in any programming language but its Java implementation is the most efficient one in terms of CPU time(measured in nanoseconds)and memory requirement[32,33].I have used IntelliJ IDEA version 11.0.9 with JDK 1.8.0_231 for implementing the algorithms and testing & validating the results.In our implementation, the variable,flagof type boolean is used to check if any swapping is made inside the inner for loop or not.A boolean variable may assume only two values;trueorfalse.So, its implementation requires just 1 bit of memory[34]whereas C/C++implementation[35]will use either short or int,which requires 2 to 4 bytes of memory.

    The experiments are performed on a 3.6 GHz Intel Core i9(9thgeneration)processor with 64 GB RAM,and Windows 10(64-bit OS)platform.

    4.2.1 Performance Analysis of Smart Bubble Sort on Sorted Data

    The experimental results for best-case analysis on sorted data are presented in Supplementary Tab.1 where the names of the sorting algorithms are given along the top row and the data sizes are arranged along the first column.The best-performing values are highlighted in blue.As evident from the obtained results,Smart Bubble sort is outperforming all the listed algorithms by a huge margin in terms of the number of comparisons,swapping,and CPU clock time.It can be noted down that from this point onward swapping will refer to merging operation in case of merge sort analysis.The growth of the Smart Bubble sort algorithm is linear to the size of the input data in the best-case scenario.Therefore,CPU clock time is the least.This is the edge of the Smart Bubble sort over bubble sort and the rest of the algorithms under consideration.This is because the number of swapping in Bubble sort is zero but the number of comparisons is higher than the Smart Bubble sort.In the case of merge sort and selection sort,both the number of comparisons and swapping are higher than those of the Smart Bubble sort.For a comprehensive understanding of better performing results and achievements of the Smart Bubble sort, I illustrated the number of comparisons, swapping, and CPU clock time for all data sizes including 500,2500,5000,50000,100000,625000,1250000,and 2500000 in Fig.3.

    Figure 3:Results for applying sorting algorithms on the sorted dataset:(a)data size 500,(b)data size 2500,(c)data size 5000,(d)data size 50000,(e)data size 1000000,(f)data size 625000,(g)data size 150000 and(h)2500000

    4.2.2 Performance Analysis of Smart Bubble Sort on Reverse Sorted Data

    The experimental results for worst-case analysis on reverse sorted data are presented in Supplementary Tab.2 where the names of the sorting algorithms are given along the top row and the data sizes are arranged along the first column.The best-performing values are highlighted in blue.It is evident from the results that in the worst-case scenario,the performance of Smart Bubble sort is much better than bubble sort and selection sort in terms of CPU clock time however, it is much slower than the merge sort algorithm.For data size 500,the CPU clock time is higher for Smart Bubble sort because the number of steps of SMART-BUBBLE-SORT(A, n) is more than those of BUBBLE-SORT(A,n).Therefore,the cost of the coefficient is higher for Smart Bubble sort as evident from Eqs.(2)and(5).However,for the data sizes greater than 500,this overhead is overtaken by the performance of the algorithm as evident from Supplementary Tab.2.The best values of CPU clock time for Smart Bubble sort are given in blue whereas the value where the CPU clock time is more for Smart Bubble sort while processing data size 500 is given in red.For a comprehensive understanding of the achieved results,I highlighted the number of comparisons,swapping,and CPU clock time for all sizes of reverse sorted data including 500,2500,5000,50000,100000,625000,1250000,and 2500000 in Fig.4.

    Figure 4: Continued

    Figure 4:Results for applying sorting algorithms on the reverse sorted dataset:(a)data size 500,(b)data size 2500,(c)data size 5000,(d)data size 50000,(e)data size 1000000,(f)data size 625000,(g)data size 150000 and(h)2500000

    4.2.3 Performance Analysis of Smart Bubble Sort on Random Data

    The experimental results for average-case analysis on random data are demonstrated in Supplementary Tab.3 where the names of the sorting algorithms are mentioned along the top row whereas the data sizes are presented along the first column.The best-performing values are highlighted in blue.The results show the superior performance of Smart Bubble sort in terms of the number of comparisons and CPU clock time over bubble sort.It can be observed from Supplementary Tab.3 that the Smart Bubble sort is performing a lesser number of comparisons than the Bubble sort as well as Selection sort for all data sizes.However,CPU clock time is lesser for data sizes 50000 and above as compared to the Bubble sort algorithm.In the case of Selection sort and Merge sort,both the number of comparisons and swapping(merging in case of merge sort)are less than those of the Smart Bubble sort.For data size 500,2500,and 5000,the CPU clock time is higher for Smart Bubble sort despite a lesser number of comparisons.This is because the number of steps of SMART-BUBBLE-SORT(A,n) is more than those of BUBBLE-SORT(A, n).Therefore, the cost of the coefficient is higher for Smart Bubble sort as evident from Eqs.(3) and (6).However, for the data sizes greater than 50000,this overhead is overtaken by the performance of the algorithm as evident from Supplementary Tab.3.The best values of comparisons and CPU clock time for Smart Bubble sort are given in blue whereas the values where the CPU clock time is more for Smart Bubble sort while processing data size of 500,2500,and 5000 are given in red.I highlighted the number of comparisons,swapping,and CPU clock time for all sizes of random data including 500, 2500, 5000, 50000, 100000, 625000, 1250000, and 2500000 in Fig.5.

    Figure 5: Results for applying sorting algorithms on the random dataset: (a) data size 500, (b) data size 2500,(c)data size 5000,(d)data size 50000,(e)data size 1000000,(f)data size 625000,(g)data size 150000 and(h)2500000

    4.2.4 Performance Analysis of Smart Bubble Sort for Validation of Dynamic Footprint

    The experimental results for dynamic footprint analysis on partially sorted data are presented in Supplementary Tab.4 where names of the sorting algorithms are mentioned along the top row and the data sizes are given along the first column.The best-performing values are highlighted in blue.The number of swapping performed by Smart Bubble sort have now become half of the total number of comparisons for a given data as claimed in the columnAverage Caseof Tab.2 and supported by the results in Supplementary Tab.4.It is because the input data is now partially sorted to establish a basis for detecting the dynamic footprint by Smart Bubble sort for transforming itself from the average case scenario to the best-case scenario.The number of comparisons performed by Smart Bubble sort is less than those of bubble sort and selection sort algorithms.I can observe that the number of swapping performed by Smart Bubble sort is equal to those of bubble sort.As I discussed previously the Smart Bubble sort can detect the dynamic footprint of the data instance and as a result, it can transform itself from an average case to the best-case scenario.The number of comparisons is less than Bubble sort is because Smart Bubble sort detected that the remaining data items are already sorted so it terminated earlier whereas bubble sort kept on comparing the already sorted elements blindly,which caused more number of comparisons.On the other hand, for data size 500, the CPU clock time is higher for Smart Bubble sort despite a lesser number of comparisons.This is because the number of steps of SMART-BUBBLE-SORT(A,n)is more than those of BUBBLE-SORT(A,n).Therefore,the cost of the coefficient is higher for the Smart Bubble sort.However,for the data sizes greater than 500,this overhead is overtaken by the performance of the algorithm as evident from Supplementary Tab.4.The best values of comparisons for Smart Bubble sort are given in blue whereas the value where the CPU clock time is more for Smart Bubble sort while processing data size 500 is given in red.It is now evident that the lesser number of comparisons by Smart Bubble sort is due to its dynamic footprint detection capability(Please,refer to Fig.6).

    Figure 6: Continued

    Figure 6:Results for applying sorting algorithms on the random dataset with 50%sorted data:(a)data size 500,(b)data size 2500,(c)data size 5000,(d)data size 50000,(e)data size 1000000,(f)data size 625000,(g)data size 150000 and(h)2500000

    5 Conclusion

    In this article, I have designed and verified a novel and dynamic variant of the bubble sort algorithm with performance improvements in the best case and average case scenarios while keeping the worst-case performance well within the upper bound.The proposed Smart Bubble sort algorithm demonstrated better performance compared to bubble sort,selection sort,and merge sort in the best case scenario with a time complexity ofΩ(n).

    To achieve the improvements for the average-case scenario,the concept of dynamic footprint has been elaborated and empirically verified.It is observed that the performance of an algorithm can be enhanced by identifying useful patterns present in the data and consequently exploiting the scenarios for improving the performance and efficiency of the algorithm.Smart Bubble sort can detect the useful patterns in average case data favoring its transformation to the best-case scenario and finally,it can adapt from average case to the best-case scenario.This enables it to terminate early in the sorting process while giving about a 25%performance raise over bubble sort.

    Data Availability Statement:The constructed datasets are available at https://drive.google.com/file/d/112pWWW_cDWgM7O3oa80Tn6O1ASR38mEh/view?usp=sharing.

    Funding Statement:The author received no specific funding for this study.

    Conflicts of Interest:The author declares that he has no conflicts of interest to report regarding the present study.

    狂野欧美激情性xxxx在线观看| 麻豆成人午夜福利视频| 午夜福利视频1000在线观看| 可以在线观看的亚洲视频| 欧美3d第一页| 欧美最黄视频在线播放免费| 免费av不卡在线播放| 看黄色毛片网站| 在线观看66精品国产| 久久鲁丝午夜福利片| 久久久久性生活片| 久99久视频精品免费| 毛片女人毛片| 久久久国产成人免费| 少妇裸体淫交视频免费看高清| 91麻豆精品激情在线观看国产| 免费观看a级毛片全部| 国产黄片美女视频| 日韩欧美在线乱码| 精品国内亚洲2022精品成人| 日本三级黄在线观看| 女人被狂操c到高潮| 欧美xxxx性猛交bbbb| 国产成人a∨麻豆精品| 日韩三级伦理在线观看| 亚洲真实伦在线观看| 三级男女做爰猛烈吃奶摸视频| 国产毛片a区久久久久| 久久久久久久久久久丰满| 人妻系列 视频| 在线免费十八禁| 午夜福利在线观看吧| 我要搜黄色片| 淫秽高清视频在线观看| 最新中文字幕久久久久| 久久久午夜欧美精品| 日本黄大片高清| 熟女电影av网| 九草在线视频观看| 免费av观看视频| 丝袜喷水一区| 中文字幕制服av| 日本黄色视频三级网站网址| 国产欧美日韩精品一区二区| 国产伦理片在线播放av一区 | 91精品国产九色| 精品日产1卡2卡| 有码 亚洲区| 黑人高潮一二区| 国产一区二区三区在线臀色熟女| 免费观看人在逋| 亚洲熟妇中文字幕五十中出| 亚洲精品自拍成人| 国产一区二区三区av在线 | 亚洲精品影视一区二区三区av| 久久久久久国产a免费观看| 熟妇人妻久久中文字幕3abv| 精品久久久久久久久久免费视频| 非洲黑人性xxxx精品又粗又长| av女优亚洲男人天堂| 男人的好看免费观看在线视频| 欧美zozozo另类| 国产精品三级大全| 久久精品夜夜夜夜夜久久蜜豆| 2021天堂中文幕一二区在线观| 在线观看av片永久免费下载| 亚洲中文字幕一区二区三区有码在线看| 美女脱内裤让男人舔精品视频 | 网址你懂的国产日韩在线| 特级一级黄色大片| 久久99精品国语久久久| 亚州av有码| 99热网站在线观看| 国产精品嫩草影院av在线观看| 国产老妇伦熟女老妇高清| 欧美成人精品欧美一级黄| 国内精品久久久久精免费| 亚洲一区二区三区色噜噜| 国产高清激情床上av| 我要搜黄色片| 91精品国产九色| 97超碰精品成人国产| 寂寞人妻少妇视频99o| 国内揄拍国产精品人妻在线| 国产黄色小视频在线观看| 久久久国产成人免费| 99久久成人亚洲精品观看| 97超碰精品成人国产| 亚洲一区二区三区色噜噜| 国产黄a三级三级三级人| 成人无遮挡网站| 亚洲欧美精品综合久久99| 久久久欧美国产精品| 1000部很黄的大片| 变态另类丝袜制服| 日本欧美国产在线视频| 在线播放无遮挡| 欧美一区二区精品小视频在线| 亚洲一级一片aⅴ在线观看| 成人午夜高清在线视频| 中文字幕精品亚洲无线码一区| 人人妻人人看人人澡| 一进一出抽搐gif免费好疼| 国产私拍福利视频在线观看| 亚洲欧美清纯卡通| 国产成人a区在线观看| 婷婷亚洲欧美| 91av网一区二区| 狂野欧美激情性xxxx在线观看| 久久久久网色| 国产欧美日韩精品一区二区| eeuss影院久久| 一级毛片久久久久久久久女| 久久久久性生活片| 国产免费男女视频| 成人三级黄色视频| 性插视频无遮挡在线免费观看| 干丝袜人妻中文字幕| 国产精品人妻久久久久久| 两个人的视频大全免费| 国产一区二区在线av高清观看| 国产黄色视频一区二区在线观看 | 成人一区二区视频在线观看| av.在线天堂| 精品国产三级普通话版| 色播亚洲综合网| 最近2019中文字幕mv第一页| 国产伦一二天堂av在线观看| av天堂在线播放| 成年女人永久免费观看视频| 麻豆精品久久久久久蜜桃| 麻豆一二三区av精品| 我的老师免费观看完整版| 国产精品av视频在线免费观看| 舔av片在线| 国产麻豆成人av免费视频| 久久久久久久亚洲中文字幕| 国产高潮美女av| av又黄又爽大尺度在线免费看 | av卡一久久| 国产v大片淫在线免费观看| 日韩欧美国产在线观看| 97超碰精品成人国产| 国产精品一区二区三区四区免费观看| 夫妻性生交免费视频一级片| 国语自产精品视频在线第100页| 婷婷亚洲欧美| 欧美区成人在线视频| 天堂中文最新版在线下载 | 久久精品国产99精品国产亚洲性色| 听说在线观看完整版免费高清| 超碰av人人做人人爽久久| 久久6这里有精品| 日日摸夜夜添夜夜爱| 亚洲aⅴ乱码一区二区在线播放| 日日摸夜夜添夜夜爱| 狠狠狠狠99中文字幕| 高清毛片免费观看视频网站| 给我免费播放毛片高清在线观看| 国产人妻一区二区三区在| 韩国av在线不卡| 国产单亲对白刺激| 久久精品国产鲁丝片午夜精品| 亚洲不卡免费看| 国产成人精品一,二区 | av.在线天堂| 永久网站在线| 国产高清视频在线观看网站| 中文字幕制服av| 国产精品国产高清国产av| av国产免费在线观看| 99九九线精品视频在线观看视频| 美女内射精品一级片tv| 别揉我奶头 嗯啊视频| 老熟妇乱子伦视频在线观看| 亚洲欧洲日产国产| 麻豆精品久久久久久蜜桃| 国产成人一区二区在线| 日韩欧美一区二区三区在线观看| 亚洲最大成人手机在线| 精品久久久久久久久久久久久| 国产精品国产高清国产av| 26uuu在线亚洲综合色| 天天躁日日操中文字幕| 国产在线精品亚洲第一网站| 寂寞人妻少妇视频99o| 青青草视频在线视频观看| 成人二区视频| 国语自产精品视频在线第100页| 日韩欧美一区二区三区在线观看| av女优亚洲男人天堂| 日韩一区二区视频免费看| 日韩一本色道免费dvd| 亚洲欧美日韩卡通动漫| 天堂影院成人在线观看| 男的添女的下面高潮视频| 九色成人免费人妻av| 日韩亚洲欧美综合| 天堂av国产一区二区熟女人妻| 国内少妇人妻偷人精品xxx网站| 一级毛片电影观看 | 久久九九热精品免费| 99热这里只有是精品50| 悠悠久久av| 淫秽高清视频在线观看| 亚洲精品456在线播放app| 全区人妻精品视频| 国产黄a三级三级三级人| 韩国av在线不卡| 日韩,欧美,国产一区二区三区 | 亚洲熟妇中文字幕五十中出| 国产日本99.免费观看| 国产高清不卡午夜福利| 天堂√8在线中文| 51国产日韩欧美| 尾随美女入室| 精品国产三级普通话版| 人妻少妇偷人精品九色| 婷婷亚洲欧美| 久久中文看片网| 欧美3d第一页| 亚洲人成网站在线播放欧美日韩| 97在线视频观看| 亚洲一区高清亚洲精品| 在线a可以看的网站| 丰满乱子伦码专区| 国产91av在线免费观看| 一进一出抽搐gif免费好疼| 中文字幕熟女人妻在线| 校园春色视频在线观看| 在线观看免费视频日本深夜| 深爱激情五月婷婷| 九九爱精品视频在线观看| 在线天堂最新版资源| 色哟哟·www| 欧美色欧美亚洲另类二区| 亚洲国产高清在线一区二区三| 国产av不卡久久| 少妇人妻精品综合一区二区 | 此物有八面人人有两片| 丝袜美腿在线中文| 国产一级毛片在线| 国产在线男女| 两个人的视频大全免费| 我的女老师完整版在线观看| 嫩草影院新地址| 欧美丝袜亚洲另类| 亚洲av中文字字幕乱码综合| 亚洲成人久久性| 又粗又硬又长又爽又黄的视频 | 一卡2卡三卡四卡精品乱码亚洲| 青春草视频在线免费观看| 九草在线视频观看| 99国产极品粉嫩在线观看| 免费av毛片视频| a级毛片a级免费在线| 夜夜爽天天搞| 国产麻豆成人av免费视频| 老师上课跳d突然被开到最大视频| 国产成人午夜福利电影在线观看| 天堂中文最新版在线下载 | 国产精品久久视频播放| 国产精品久久电影中文字幕| 91精品一卡2卡3卡4卡| 你懂的网址亚洲精品在线观看 | 国产精品久久久久久精品电影| 毛片一级片免费看久久久久| 久久草成人影院| 搞女人的毛片| 欧美日韩精品成人综合77777| 丝袜美腿在线中文| 亚洲国产精品国产精品| 欧美三级亚洲精品| 性插视频无遮挡在线免费观看| 国产伦理片在线播放av一区 | 日韩高清综合在线| 一卡2卡三卡四卡精品乱码亚洲| 美女脱内裤让男人舔精品视频 | 欧美日韩精品成人综合77777| 亚洲五月天丁香| 老女人水多毛片| 日韩一本色道免费dvd| а√天堂www在线а√下载| 内地一区二区视频在线| 精品熟女少妇av免费看| 国产一区二区在线av高清观看| 久久草成人影院| 亚洲电影在线观看av| 成人特级av手机在线观看| 蜜桃亚洲精品一区二区三区| 三级国产精品欧美在线观看| 69av精品久久久久久| 国产日本99.免费观看| 国产黄片美女视频| 亚洲欧洲日产国产| 亚洲第一电影网av| 日韩一本色道免费dvd| 欧美在线一区亚洲| 欧美变态另类bdsm刘玥| 久久99热这里只有精品18| 亚洲av免费高清在线观看| 91狼人影院| 免费电影在线观看免费观看| 啦啦啦啦在线视频资源| 欧美精品一区二区大全| 69av精品久久久久久| 国产在线精品亚洲第一网站| 久久亚洲精品不卡| 毛片一级片免费看久久久久| 亚洲av电影不卡..在线观看| 在线国产一区二区在线| 三级国产精品欧美在线观看| 九九久久精品国产亚洲av麻豆| 精品免费久久久久久久清纯| 久久精品夜色国产| 日韩一区二区三区影片| 蜜桃久久精品国产亚洲av| 国产在线精品亚洲第一网站| 婷婷精品国产亚洲av| 国产精品无大码| 国产av不卡久久| 成年女人永久免费观看视频| 免费观看在线日韩| 亚洲丝袜综合中文字幕| 国产黄片美女视频| 日产精品乱码卡一卡2卡三| 观看美女的网站| 好男人视频免费观看在线| 久久久久久久久久久丰满| 日韩成人伦理影院| 免费av不卡在线播放| 91av网一区二区| 99九九线精品视频在线观看视频| 人妻久久中文字幕网| 美女脱内裤让男人舔精品视频 | 久久精品人妻少妇| 大又大粗又爽又黄少妇毛片口| 九九爱精品视频在线观看| 九草在线视频观看| 亚洲内射少妇av| 午夜福利视频1000在线观看| 国产91av在线免费观看| 麻豆成人av视频| 亚洲激情五月婷婷啪啪| 91aial.com中文字幕在线观看| 男人舔女人下体高潮全视频| 69人妻影院| 可以在线观看的亚洲视频| 99热这里只有是精品50| 亚洲欧美成人精品一区二区| 麻豆久久精品国产亚洲av| 男的添女的下面高潮视频| 国产亚洲5aaaaa淫片| 人人妻人人澡欧美一区二区| 亚洲精品成人久久久久久| 久久这里只有精品中国| 亚洲18禁久久av| 一个人免费在线观看电影| 给我免费播放毛片高清在线观看| 日韩欧美国产在线观看| 在线a可以看的网站| 色综合亚洲欧美另类图片| 欧美激情久久久久久爽电影| 日韩欧美 国产精品| 少妇裸体淫交视频免费看高清| 久久精品国产亚洲av天美| 婷婷精品国产亚洲av| 亚洲国产欧洲综合997久久,| 最近的中文字幕免费完整| 精品久久久久久久久久免费视频| 午夜福利高清视频| 久久久久久久午夜电影| 国产单亲对白刺激| 欧美激情久久久久久爽电影| 国产精品一区二区三区四区久久| 精品无人区乱码1区二区| 伦理电影大哥的女人| 欧美一区二区国产精品久久精品| 人妻久久中文字幕网| 日本黄色视频三级网站网址| 青青草视频在线视频观看| 中文在线观看免费www的网站| 亚洲成人精品中文字幕电影| 毛片一级片免费看久久久久| 别揉我奶头 嗯啊视频| 欧美区成人在线视频| 九九热线精品视视频播放| 国产久久久一区二区三区| 人体艺术视频欧美日本| 国产精品国产三级国产av玫瑰| 成人性生交大片免费视频hd| 色综合色国产| 色吧在线观看| 好男人视频免费观看在线| 亚洲欧美成人精品一区二区| 亚洲欧美日韩无卡精品| 禁无遮挡网站| 欧美潮喷喷水| 日韩精品青青久久久久久| 亚洲高清免费不卡视频| 精品久久久久久久久久久久久| h日本视频在线播放| 国产精品无大码| 岛国在线免费视频观看| 亚洲图色成人| 人妻制服诱惑在线中文字幕| 高清日韩中文字幕在线| 熟女人妻精品中文字幕| 精品国内亚洲2022精品成人| 国产亚洲5aaaaa淫片| 国产精品女同一区二区软件| 久久精品国产99精品国产亚洲性色| 伦理电影大哥的女人| 国产亚洲91精品色在线| 麻豆乱淫一区二区| 亚洲经典国产精华液单| 深夜精品福利| 嘟嘟电影网在线观看| 高清午夜精品一区二区三区 | 国产单亲对白刺激| 全区人妻精品视频| 欧美一区二区亚洲| 亚洲第一电影网av| 亚洲自拍偷在线| 毛片女人毛片| 国产探花极品一区二区| 美女黄网站色视频| 99热这里只有是精品50| 男女做爰动态图高潮gif福利片| 国产真实乱freesex| 乱人视频在线观看| 久久精品夜色国产| 成人永久免费在线观看视频| 男女边吃奶边做爰视频| 国产色婷婷99| 免费电影在线观看免费观看| 99热这里只有精品一区| 国产一区二区在线观看日韩| 色视频www国产| 亚洲精品自拍成人| 能在线免费观看的黄片| 免费观看在线日韩| 日本一二三区视频观看| 在线观看免费视频日本深夜| 亚洲av熟女| 国产成人一区二区在线| 99热这里只有是精品在线观看| 国产乱人偷精品视频| 身体一侧抽搐| 狠狠狠狠99中文字幕| 成人综合一区亚洲| 欧美+亚洲+日韩+国产| 久久欧美精品欧美久久欧美| 亚洲av熟女| 欧美潮喷喷水| 91午夜精品亚洲一区二区三区| 国产精品久久久久久av不卡| 亚洲精华国产精华液的使用体验 | 九九爱精品视频在线观看| 一本—道久久a久久精品蜜桃钙片 精品乱码久久久久久99久播 | 亚洲真实伦在线观看| 99久国产av精品| 九九久久精品国产亚洲av麻豆| 99热这里只有是精品50| 国产一区二区三区在线臀色熟女| 老熟妇乱子伦视频在线观看| 免费观看的影片在线观看| 边亲边吃奶的免费视频| 国产69精品久久久久777片| 男女下面进入的视频免费午夜| 精品久久久久久久久久久久久| 在线免费观看的www视频| 国产激情偷乱视频一区二区| 国产黄色小视频在线观看| 黄色日韩在线| 人妻少妇偷人精品九色| 黄片无遮挡物在线观看| 天堂影院成人在线观看| 九九热线精品视视频播放| 国产精品乱码一区二三区的特点| 最近手机中文字幕大全| 亚洲av中文av极速乱| 午夜精品国产一区二区电影 | 国产精品久久久久久久久免| 国产精品,欧美在线| 看片在线看免费视频| 亚洲av第一区精品v没综合| 美女脱内裤让男人舔精品视频 | 日本黄色视频三级网站网址| 中文字幕精品亚洲无线码一区| 精品久久久久久久久亚洲| av在线老鸭窝| 激情 狠狠 欧美| 亚洲经典国产精华液单| 亚洲欧美日韩高清专用| 深夜a级毛片| 国产综合懂色| a级毛片a级免费在线| 99久久九九国产精品国产免费| 久久午夜亚洲精品久久| 亚洲自拍偷在线| 尤物成人国产欧美一区二区三区| 亚洲国产欧美人成| www.av在线官网国产| 男人的好看免费观看在线视频| 亚洲精品自拍成人| 国产久久久一区二区三区| 九色成人免费人妻av| 欧美激情国产日韩精品一区| 一本久久精品| 日韩精品青青久久久久久| 欧美高清性xxxxhd video| 一边亲一边摸免费视频| 看免费成人av毛片| av女优亚洲男人天堂| 18禁黄网站禁片免费观看直播| av在线老鸭窝| 观看美女的网站| 日本黄色视频三级网站网址| 桃色一区二区三区在线观看| 久久婷婷人人爽人人干人人爱| 欧美潮喷喷水| 免费搜索国产男女视频| 两个人的视频大全免费| 免费看av在线观看网站| 色吧在线观看| 麻豆成人av视频| 高清毛片免费观看视频网站| 自拍偷自拍亚洲精品老妇| 中文字幕人妻熟人妻熟丝袜美| 欧美一区二区国产精品久久精品| 色综合色国产| 麻豆成人午夜福利视频| 久久精品国产自在天天线| 99精品在免费线老司机午夜| 国产黄片视频在线免费观看| 国内精品美女久久久久久| 国产精品久久久久久久久免| 久久人人爽人人爽人人片va| 人妻制服诱惑在线中文字幕| 在线免费十八禁| 亚洲精品日韩av片在线观看| 亚洲人成网站高清观看| 观看美女的网站| 天堂中文最新版在线下载 | 日韩av不卡免费在线播放| 中文字幕制服av| 精品久久国产蜜桃| h日本视频在线播放| 狂野欧美白嫩少妇大欣赏| 九九久久精品国产亚洲av麻豆| 人人妻人人看人人澡| 午夜福利在线观看吧| 亚洲精品久久久久久婷婷小说 | 大又大粗又爽又黄少妇毛片口| 日日干狠狠操夜夜爽| 国产高潮美女av| 99热这里只有精品一区| 国产视频首页在线观看| 在线a可以看的网站| 欧美色欧美亚洲另类二区| 我的女老师完整版在线观看| 亚洲,欧美,日韩| 一本久久精品| 深夜a级毛片| 免费人成在线观看视频色| 男女视频在线观看网站免费| 亚洲精品日韩在线中文字幕 | 淫秽高清视频在线观看| 欧美区成人在线视频| 国产69精品久久久久777片| 亚洲国产欧美在线一区| 国产毛片a区久久久久| 51国产日韩欧美| 久久精品国产清高在天天线| 麻豆成人av视频| 久久久久网色| 亚洲欧美成人综合另类久久久 | 只有这里有精品99| 亚洲av成人精品一区久久| 亚洲欧美日韩卡通动漫| 男人舔奶头视频| 最近最新中文字幕大全电影3| 一个人免费在线观看电影| 一进一出抽搐gif免费好疼| 高清毛片免费看| 一级二级三级毛片免费看| 寂寞人妻少妇视频99o| 欧美成人精品欧美一级黄| 欧美一区二区国产精品久久精品| 九九在线视频观看精品| 秋霞在线观看毛片| 国产精品,欧美在线| 看免费成人av毛片| 美女 人体艺术 gogo| 99热只有精品国产| 九草在线视频观看| 三级毛片av免费| 亚州av有码| 免费无遮挡裸体视频| 桃色一区二区三区在线观看| 亚洲av成人av| 国语自产精品视频在线第100页| 亚洲成a人片在线一区二区| 日韩三级伦理在线观看| 白带黄色成豆腐渣| 亚洲成a人片在线一区二区| 亚州av有码| 网址你懂的国产日韩在线| 久久99热6这里只有精品| 亚洲精品色激情综合| 欧美不卡视频在线免费观看| 99热这里只有是精品在线观看| АⅤ资源中文在线天堂| 精品不卡国产一区二区三区|