In the following steps, these are merged: The following source code shows a simple implementation where only areas sorted in ascending order are identified and merged: The signature of the merge() method differs from the example above as follows: The actual merge algorithm remains the same. Merge Sort has the advantage over Quicksort that, even in the worst case, the time complexity O(n log n) is not exceeded. The smaller of the two (1 in the example) is appended to a new array, and the pointer to that element is moved one field to the right: Now the elements above the pointers are compared again. to a maximum of 536,870,912 (= 2. Merge sort is a sorting technique based on divide and conquer technique. It is easy to understand time complexity with the help of recursion tree. You could also return the sorted array directly, but that would be incompatible with the testing framework. In the first step, the 4 and the 6 are merged to the subarray [4, 6]: Next, the 3 and the 7 are merged to the subarray [3, 7], 1 and 8 to the subarray [1, 8], the 2 and the 5 become [2, 5]. The following diagram shows all merge steps summarized in an overview: The following source code is the most basic implementation of Merge Sort. Since L > R, so we perform A = R. To gain better understanding about Merge Sort Algorithm. My focus is on optimizing complex algorithms and on advanced topics such as concurrency, the Java memory model, and garbage collection. In the merge phase, elements from two subarrays are copied into a newly created target array. The space complexity of merge sort algorithm is Θ (n). If you liked the article, feel free to share it using one of the share buttons at the end. Merge sort uses additional memory for left and right sub arrays. Efficiency of an algorithm depends on two parameters: 1. It then combines the results of sub problems to get the solution of the original problem. On the other hand, with Quicksort, only those elements in the wrong partition are moved. Finally, we merge these two sub arrays using merge procedure which takes Θ(n) time as explained above. With descending sorted elements, all elements of the right subarray are copied first, so that rightPos < rightLen results in false first. I'm a freelance software developer with more than two decades of experience in scalable Java enterprise applications. The merging itself is simple: For both arrays, we define a merge index, which first points to the first element of the respective array. This prevents the unnecessary further dividing and merging of presorted subsequences. Merge sort's most common implementation does not sort in place; therefore, the memory size of the input must be allocated for the sorted output to be stored in (see below for versions that need only n/2 extra spaces). Why do a third fewer operations lead to three times faster processing? Merge sort uses a divide and conquer paradigm for sorting. Merge sort first divides the array into equal halves and then combines them in a sorted manner. This can be circumvented by in-place merging, which is either very complicated or severely degrades the algorithm’s time complexity. The reason is simply that all elements are always copied when merging. The total effort is, therefore, the same at all merge levels. The first step identifies the “runs”. Insertion sort, selection sort, shellsort. Get more notes and other study material of Design and Analysis of Algorithms. In merge sort, we divide the array into two (nearly) equal halves and solve them recursively using merge sort only. Both algorithms process elements presorted in descending order slightly slower than those presorted in ascending order, so I did not add them to the diagram for clarity. Merge Sort is about three times faster for pre-sorted elements than for unsorted elements. Clearly, all the elements from right sub array have been added to the sorted output array. Enough theory! Therefore: The space complexity of Merge Sort is: O(n), (As a reminder: With linear effort, constant space requirements for helper and loop variables can be neglected.). This is because we are just filling an array of size n from left & right sub arrays by incrementing i and j at most Θ(n) times. you will find the source code of Merge Sort. Here is an example of the overall algorithm. Thus the order of identical elements to each other always remains unchanged. The 3 is smaller and is appended to the target array: And in the final step, the 6 is appended to the new array: The two sorted subarrays were merged to the sorted final array. These advantages are bought by poor performance and an additional space requirement in the order of O(n). Assume that a merge sort algorithm in the worst case takes 30 seconds for an input of size 64. 2. The following illustration shows Natural Merge Sort using our sequence [3, 7, 1, 8, 2, 5, 9, 4, 6] as an example. Instead of subarrays, the entire original array and the positions of the areas to be merged are passed to the method. So-called in-place algorithms can circumvent this additional memory requirement; these are discussed in the section “In-Place Merge Sort”. Would you like to be informed by e-mail when I publish a new article? Worst-case time complexity = O(NlogN) 3. Variants. The merge procedure of merge sort algorithm is used to merge two sorted arrays into a third array in sorted order. Natural Merge Sort is an optimization of Merge Sort: It identifies pre-sorted areas (“runs”) in the input data and merges them. Merge sort is a recursive sorting algorithm. Here is the result for Merge Sort after 50 iterations (this is only an excerpt for the sake of clarity; the complete result can be found here): Using the program CountOperations, we can measure the number of operations for the different cases. If both values are equal, first, the left one is copied and then the right one. The easiest way to show this is to use an example (the arrows represent the merge indexes): The elements over the merge pointers are compared. The cause lies in the branch prediction: If the elements are sorted, the results of the comparisons in the loop and branch statements, while (leftPos < leftLen && rightPos < rightLen). Mergesort uses extra space proportional to N. Pf. You can find the source code here in the GitHub repository. merge sort we focus on line 9,10,11 ( the before lines just O(1) time complexity ). The following diagram shows the runtimes for unsorted and ascending sorted input data. And that is regardless of whether the input elements are presorted or not. The difference between ascending and descending sorted elements corresponds approximately to the measured time difference. Recurrence relation for MergeSort is T(n) = T(n/2) + O(n) here n is the number of elements to be sorted. If T(n) is the time required by merge sort for sorting an array of size n, then the recurrence relation for time complexity of merge sort is-. Quicksort is about 50% faster than Merge Sort for a quarter of a billion unsorted elements. In the merge phase, we use if (leftValue <= rightValue) to decide whether the next element is copied from the left or right subarray to the target array. are always the same until the end of a merge operation. Since we repeatedly divide the (sub)arrays into two equally sized parts, if we double the number of elements n, we only need one additional step of divisions d. The following diagram demonstrates that for four elements, two division steps are needed, and for eight elements, only one more: Thus the number of division stages is log2 n. On each merge stage, we have to merge a total of n elements (on the first stage n × 1, on the second stage n/2 × 2, on the third stage n/4 × 4, etc. Since L < R, so we perform A = L. This time the 2 is smaller than the 4, so we append the 2 to the new array: Now the pointers are on the 3 and the 4. Otherwise, the array is split, and mergeSort() is called recursively for both parts. The above mentioned merge procedure takes Θ(n) time. It is more efficient as it is in worst case also the runtime is O (nlogn) The space complexity of Merge sort is O (n). Merge Sort is, therefore, a stable sorting process. Merge Sort is a famous sorting algorithm that uses divide and conquer paradigm.