Following are the steps involved in bubble sort(for sorting a given array in ascending order): 1. If the current element is smaller then the next element, we’ll move to the next element in the array. The non-optimized algorithm – which compares the elements until the end in each iteration – can be found in the class BubbleSort. Here is the result for the parallel algorithms, compared to the fastest sequential variant. But why is the runtime in the average case so much higher than in the worst case? We do not need to compare further since the 9 is already in the sorted area. The class in the repository implements the SortAlgorithm interface to be interchangeable within the test framework. The 6 and the 3, however, must be swapped to be in the correct order: The 6 and the 7 are in the right order and do not need to be swapped. The algorithm must perform n-1 comparisons; therefore: The best-case time complexity of Bubble Sort is: O(n). In the original variant, it is missing. Each and every element is compared with the other elements for array which takes n time; And the above steps continues for n iterations; Best Time Complexity: O(n^2) Average Time Complexity: O(n^2) Worst Time Complexity: O(n^2) Bubble Sort Space Complexity. C program to swap two numbers using third variable . Again we start at the beginning of the array. In the best case, when the given array is already sorted, the improved bubble sort achieves better time complexity compared to the standard version. If the branch prediction assumes that the result of a comparison is always the same as that of the previous comparison, then it is always right with this assumption – with one single exception: at the area boundary. The 2 and the 4 are positioned correctly to each other. The fourth and fifth element, the 9 and the 3, need to be swapped again: And finally, the fifth and sixth elements, the 9 and the 7, must be swapped. Hence, the best case time complexity of bubble sort is O(n). Isn't that wrong since best case can be defined as: Best case = fastest time to complete, with optimal inputs chosen. The “odd-even” variant is on my 6-core CPU (12 virtual cores with Hyper-threading) and with 20,000 unsorted elements thus 6.6 times faster than the sequential version. So bubble sort is slower than most of sorting algorithms. In the example above, n = 6. Therefore, in the best scenario, the time complexity of the standard bubble sort would be . In practice it is quadratic. Though there is an improvement in the efficiency and performance of the improved version in the average and the worst case. Bubble Sort Time Complexity. But as the array is already sorted, there will be no swaps. To see bubble sort in practice please refer to our article on implementing bubble sort in Java. * I explain the terms “time complexity” and “big O notation” in this article using examples and diagrams. Why Is Bubble Sort Faster for Elements Sorted in Descending Order Than for Unsorted Elements? My focus is on optimizing complex algorithms and on advanced topics such as concurrency, the Java memory model, and garbage collection. Given an array of size , the first iteration performs comparisons. To check this, I use the program CountOperations to display the number of different operations. Now let’s assume that the given input array is either nearly or already sorted. The main disadvantage of bubble sort is time complexity. The runtime is approximately quadrupled when doubling the input quantity for unsorted and descending sorted elements. There are so many alternative algorithms which take O(n*log(n)) time for sorting. If the given array is sorted, we traverse the array once. So the number of exchange operations is: It becomes even more complicated with the number of comparison operations, which amounts to (source: this German Wikipedia article; the English version doesn’t cover this): ½ (n² – n × ln(n) – ( + ln(2) – 1) × n) + O(√n). My focus is on optimizing complex algorithms and on advanced topics such as concurrency, the Java memory model, and garbage collection. Then please use the following form to subscribe to my newsletter. Now you perform one Bubble Sort iteration in all partitions in parallel. Set Flag: = True 2. 1. The pass through the list is repeated until the list is sorted. You repeat this process until there is no more swapping in one iteration. Bubble Sort requires no additional memory space apart from the loop variable max, and the auxiliary variables swapped, left, and right. Hence, the time complexity of the bubble sort in the worst case would be the same as the average case and best case: . Write a C program to plot and analyze the time complexity of Bubble sort, Insertion sort and Selection sort (using Gnuplot). Due to its simplicity, bubble sort is used to introduce sorting algorithms in computer science. Therefore, in the outer loop, we decrement the value max, starting at elements.length - 1, by one in every iteration. Wait until all threads are finished, and then compare the last element of one partition with the first of the next partition. Here on, I want to help you become a better Java programmer. In the second iteration, the second-largest moves to the second last position. Note that the goal is to take input array and sort its elements in ascending order. I assume it’s because saving and repeatedly (within one iteration) updating the last swapped element’s position is much more expensive than changing the swapped value only once (per iteration). This stands in contrast to the relatively high synchronization effort required by the phaser. Bubble Sort is an easy-to-implement, stable sorting algorithm with a time complexity of O(n²) in the average and worst cases – and O(n) in the best case. I compare the performance of the parallel variants with the CompareBubbleSorts test mentioned above. The following table … 1. However, the CompareBubbleSorts test shows that this variant is slower in practice: You can find the complete output of the test program in the file TestResults_BubbleSort_Algorithms.log. You will find more sorting algorithms in this overview of all sorting algorithms and their characteristics in the first part of the article series. Why is the second optimized version slower? The high level overview of all the articles on the site. In the first iteration, the largest element, the 6, moves from far left to far right. Time complexity of Bubble Sort during best case is explained as O(n) and not Theta(n)? The synchronization between the steps (the threads may not start with a step until all threads have finished the previous step) is realized with a Phaser. If the current element is less than the next element, move to the next element. C program to swap two number without using third variable. Can we improve on this? The performance of bubble sort in the modern CPU hardware is very poor. The runtime for elements sorted in ascending order increases linearly and is orders of magnitude smaller than for unsorted elements. When the input array contains a large number of elements, the efficiency of bubble sort decreases dramatically and the average time increases quadratically. With ascending presorted elements, Bubble Sort is so fast that the curve does not show any upward deflection. Let’s assume we want to sort the descending array [6, 5, 4, 3, 2, 1] with Bubble Sort. After the nth iteration, it is possible that not only the last n elements are sorted, but more than that – depending on how the elements were originally arranged. In fact, much of the code of both algorithms is the same, since the array is also divided into partitions for the odd-even approach. You find further information and options to switch off these cookies in our, overview of all sorting algorithms and their characteristics, explains how to derive its time complexity. Read more about me, This website uses cookies to analyze and improve the website. In the case of the standard version of the bubble sort, we need to do iterations. No auxiliary space is required in bubble sort implementation ; Hence space complexity is: O(1) Now we … After that, the first iteration is finished. After each iteration, let’s keep track of the elements which we swap. Unfortunately, the average time complexity of Bubble Sort cannot – in contrast to most other sorting algorithms – be explained in an illustrative way.