Merge Sort Roberto Hibbler Dept. of Computer Science Florida Institute of Technology Melbourne, FL

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1 Merge Sort Roberto Hibbler Dept. of Computer Science Florida Institute of Technology Melbourne, FL ABSTRACT Given an array of elements, we want to arrange those elements into a sorted order. To sort those elements, we will need to make comparisons between the individual elements efficiently. Merge Sort uses a divide and conquer strategy to sort an array efficiently while making the least number of comparisons between array elements. Our results show that for arrays with large numbers of array elements, Merge Sort is more efficient than three other comparison sort algorithms, Bubble Sort[1], Insertion Sort[3], and Selection Sort[2]. Our theoretical evaluation shows that Merge Sort beats a quadratic time complexity, while our empirical evaluation shows that on average Merge Sort is 32 times faster than Insertion Sort[3], the current recognized most efficient comparison algorithm, with ten different data sets. Keywords Merge Sort, sorting, comparisons, Selection Sort[2], arrange 1. INTRODUCTION The ability to arrange an array of elements into a defined order is very important in Computer Science. Sorting is heavily used with online stores, were the order that services or items were purchased determines what orders can be filled and who receives their order first. Sorting is also essential for the database management systems used by banks and financial systems, such as the New Stock Exchange, to track and rank the billions of transactions that go on in one day. There are many algorithms, which provide a solution to sorting arrays, including algorithms such as Bubble Sort[1], Insertion Sort[3], and Selection Sort[2]. While these algorithms are programmatically correct, they are not efficient for arrays with a large number of elements and exhibit quadratic time complexity. We are given an array of comparable values. We need to arrange these values into either an ascending or descending order. We introduce the Merge Sort algorithm. The Merge Sort algorithm is a divide-and-conquer algorithm. It takes input of an array and divides that array into sub arrays of single elements. A single element is already sorted, and so the elements are sorted back into sorted arrays two sub-arrays at a time, until we are left with a final sorted array. We contribute the following: 1. We introduce the Merge Sort algorithm. 2. We show that theoretically Merge Sort has a worst-case time complexity better than. 3. We show that empirically Merge Sort is faster than Selection Sort[2] over ten data sets. This paper will discuss in Section 2 comparison sort algorithms related to the problem, followed by the detailed approach of our solution in Section 3, the evaluation of our results in Section 4, and our final conclusion in Section RELATED WORK The three algorithms that we will discuss are Bubble Sort[1], Selection Sort[2], and Insertion Sort[3]. All three are comparison sort algorithms, just as Merge Sort. The Bubble Sort[1] algorithm works by continually swapping adjacent array elements if they are out of order until the array is in sorted order. Every iteration through the array places at least one element at its correct position. Although algorithmically correct, Bubble Sort[1] is inefficient for use with arrays with a large number of array elements and has a time complexity. Knuth observed, also, that while Bubble Sort[1] shares the worstcase time complexity with other prevalent sorting algorithms, compared to them it makes far more element swaps, resulting in poor interaction with modern CPU hardware. We intend to show that Merge Sort needs to make on average fewer element swaps than Bubble Sort[1]. The Selection Sort[2]algorithm arranges array elements in order by first finding the minimum value in the array and swapping it with the array element that is in its correct position depending on how the array is being arranged. The process is then repeated with the second smallest value until the array is sorted. This creates two distinctive regions within the array, the half that is sorted and the half that has not been sorted. Selection Sort[2]shows an improvement over Bubble Sort[1] by not comparing all the elements in its unsorted half until it is time for that element to be placed into its sorted position. This makes Selection Sort[2]less affected by the input s order. Though, it is still no less inefficient with arrays with a large number of array elements. Also, even with the improvements Selection Sort[2]still shares the same worst-case time complexity of. We intend to show that Merge Sort will operate at a worst-case time complexity faster than. The Insertion Sort[3]algorithm takes elements from the input array and places those elements in their correct place into a new array, shifting existing array elements as needed. Insertion Sort[3]improves over Selection Sort[2]by only making as many comparisons as it needs to determine the correct position of the current element, while Selection Sort[2]makes comparisons against each element in the unsorted part of the array. In the average case, Insertion Sort[3] s time complexity is, but its worst case is, the same as Bubble Sort[1] and Selection Sort[2]. The tradeoff of Insertion Sort[3]is that on the average more elements are swapped as array elements are shifted within the array with the addition of new elements. We intend to show

2 that Merge Sort operates at an average case time complexity faster than. ( ) Output array A in ascending order 3. APPROACH A large array with an arbitrary order needs to be arranged in an ascending or descending order, either lexicographically or numerically. Merge sort can solve this problem by using two key ideas. The first key idea of merge sort is that a problem can be divided and conquered. The problem can be broken into smaller arrays, and those arrays can be solved. Second, by dividing the array into halves, then dividing those halves by recursively halving them into arrays of single elements, two sorted arrays are merged into one array, as a single element array is already sorted. Refer to the following pseudocode: Input A: array of n elements Output array A sorted in ascending order 1. proc mergesort(a: array) 2. var array left, right, result 3. if length(a)<=1 4. return(a) 5. var middle=length(a)/2 6. for each x in A up to middle 7. add x to left 8. for each x in A after middle 9. add x to right 10. left=mergesort(left) 11. right=mergesort(right) 12. result=merge(left,right) 13. return result Input left:array of m elements, right: array of k elements Output array result sorted in ascending order 14. proc merge(left: array, right: array) 15. var array result 16. which length(left) > 0 and length(right) > if first(left) <= first(right) 18. append first(left) to result 19. left=rest(left) 20. else 21. append first(right) to result 22. right=rest(right) 23. end while 24. if length(left) > append left to result 26. if length(right) > append right to result 28. return result As the pseudocode shows, after the array is broken up into a left half and a right half (lines 5-9), the two halves are divided recursively (lines 10 11) until they are all within a single element array. Then, the two halves elements are compared to determine how the two arrays should be arranged (lines 16-22). Should any one half contain elements not added to the sorted array after the comparisons are made, the remainder is added so no elements are lost (lines 24 27). In the following examples, using the given input, the division of the array (Figure 1) and how the array is merged back into a sorted array (Figure 2) are illustrated. Inputs A: array of n elements Figure 1: Shows the splitting of the input array into single element arrays. Figure 2: Shows the merging of the single element arrays during the Merge Step. As the example shows, array A is broken in half continuously until they are in arrays of only a single element, then those single elements are merged together until they form a single sorted array in ascending order. 4. EVALUATION 4.1 Theoretical Analysis Evaluation Criteria All comparison based sorting algorithms count the comparisons of array elements as one of their key operations. The Merge Sort algorithm can be evaluated by measuring the number of comparisons between array elements. As the key operation, we can measure the number of comparisons made to determine the overall efficiency of the algorithm. We intend to show that because the Merge Sort algorithm makes less comparisons over the currently acknowledged most efficient algorithm, Insertion Sort[3], Merge Sort is the most efficient comparison sort algorithm Merge Sort Case Scenarios Worst Case Merge Sort makes the element comparisons we want to measure during the merge step, where pairs of arrays are recursively

3 merged into a single array. Merge Sort s worst case, depicted in Figure 3, is the scenario where during each recursive call of the merge step, the two largest elements are located in different arrays. This forces the maximum number of comparisons to occur. In this case, the Merge Sort algorithm s efficiency can be represented by the number of comparisons made during each recursive call of the merge step, which is described in the following recurrence equation where variable n is denoted as the array size and T(n) refers to the total comparisons in the merge step: =2 + 1 (1) 1=0 (2) Equation (1) gives the total number of comparisons that occur in Merge Sort dependent on the number of array elements. The 2T(n/2) refers to the comparisons made to the two halves of the array before the arrays are merged. The n-1 refers to the total comparisons in the merge step. Equation (2) states the base case, which is no comparisons are needed for a single element array. With these two equations, we can determine the total number of comparisons by looking at each recursive call of the merge step. We will next solve equation (1) by expanding it to isolating n and perform substitution. = (3) = (4) = (5) = (6) By expanding equation (1) to get equations (3), (4), and (5) we can discern a pattern. By using the variable k to indicate the depth of the recursion, we get the following equation: = (7) We can solve equation (7) by using the base case in equation (2) and determine the value of k, which refers to the depth of recursion. 2 = (8) =log (9) = 0+log +1 (10) By making the statement in equation (8), equation (7) and equation (2) are equal. We can then solve equation (8) for k, and get equation (9). Equation (9) can then be used to reduce equation (7) to equation (10), which represents the total number of comparisons of array elements in the merge step This results in a Big O time complexity of log overall. total number of comparisons for the worst case, we get the following equations: =2 + (11) =2 + (12) = 0+ log (13) Similarly to earlier, equation (11) can be expanded to find a pattern; equation (12) can then be created by substituting k, and by solving for k get equation (13), which is the total number of comparisons for the best case of Merge sort This also results in a Big O time complexity of log, just like the worst case Insertion Sort[3] Case Scenarios Worst Case Insertion Sort[3] builds the sorted array one element at a time, removing one element from the input data and placing it in its correct location each iteration until the array is in sorted order. The worst case for Insertion Sort[3] is if the input array is in reverse from sorted order. In this case every iteration of the inner loop will scan and shift the entire sorted region of the array before inserting the next element. Denoting n as the number of array elements, the number of comparisons can be described by the following equation: = (14) This results in a Big O time complexity of in the worst case Best Case The best case for Insertion Sort[3] is when the input array is already sorted. In this scenario, one element is removed from the input array and placed into the sorted array without the need of shifting elements. This results in a Big O time complexity of in the best case Analysis Comparing the time complexities of Merge Sort and Insertion Sort[3], Insertion Sort[3] beats Merge Sort in the best case, as Merge Sort has a Big-O time complexity of log while Insertion Sort[3] is. However, taking the worst cases, Merge Sort is faster than Insertion Sort[3] with a time complexity of still log over Insertion Sort[3] s which is equivalent to. This adds to the theory that Merge Sort will overall be a better algorithm over Insertion Sort[3] with less than optimum input Best Case The best case of Merge Sort, depicted in Figure 3, occurs when the largest element of one array is smaller than any element in the other. In this scenario, only 2 comparisons of array elements are made. Using the same process that we used to determine the

4 Figure 4: Depicts the slower speed of Insertion Sort vs Merge Sort with array sizes greater than Figure 3: Depicts how single element arrays are merged together during the Merge step in the best and worst case 4.2 Empirical Analysis Evaluation Criteria The goal of this evaluation is to demonstrate the improved efficiency of the Merge Sort algorithm based on the execution s CPU runtime Evaluation Procedures The efficiency of Merge Sort can be measured by determining the CPU runtime of an implementation of the algorithm to sort a number of elements versus the CPU runtime it takes an implementation of the Insertion Sort[3] algorithm. The dataset used is a server log of connecting IP Addresses over the course of one day. All the IP Addresses are first extracted from the log, in a separate process, and placed within a file. Subsets of the IP Addresses are then sorted using both the Merge Sort algorithm and the Insertion Sort[3] algorithm with the following array sizes: 5, 10, 15, 20, 30, 50, 100, 1000, 5000, 10000, 15000, and Each array size was run ten times and the average of the CPU runtimes was taken. Both algorithms take as its parameter an array of IP Addresses. For reproducibility, the dataset used for the evaluation can be found at The original dataset has over thirty thousand records. For the purposes of this experiment, only twenty thousand were used. The tests were run on a PC running Microsoft Windows XP with the following specifications: Intel Core 2 Duo CPU E8400 at 3.00 GHz with 3 GB of RAM. The algorithms were implemented in Java using the Java 6 API Results and Analysis Compared to the Insertion Sort[3] algorithm,, the Merge Sort algorithm shows faster CPU runtimes for array sizes over A summary of the results is contained in Figure 4 and Figure 5. Figure 5: Depicts the faster speed of Insertion Sort until around array sizes of With Merge Sort being red and Insertion Sort being blue, Figure 5 shows the relative execution speed curves of the two algorithms. It can be seen that for an array sizess under 1000, the Insertion Sort[3] algorithm has much faster execution times, showing a clear advantage. This is probably due to the overhead incurred during the creation and deleting of new arrays during splitting. Around array sizes of 1000, the algorithm s curves intersect, and Merge Sort begins to show an advantage over Insertion Sort[3], which can be seen in Figure 4. As the size of the array grows larger, Insertion Sort[3] s execution runtime increases at a faster pace than Merge Sort. This shows that Insertion Sort[3] will progressively get worse than Merge Sort as you increase the array size, while Merge Sort s efficiency will decrease at a slower rate. It can be concluded from the results that for array sizes over 1000 Insertion Sort[3] will be unsuitable compared to the efficiency presented when using the Merge Sort algorithm.

5 5. CONCLUSION The purpose of this paper was to introduce the Merge Sort algorithm and show its improvements over its predecessors. Our theoretical analysis shows that compared to the Bubble Sort[1], Insertion Sort[3], and Selection Sort[2] algorithms, Merge Sort has a faster Big O worst-case time complexity of log. Or empirical analysis shows that compared to Insertion Sort[3], Merge Sort is 32 times faster for arrays larger than 1000 elements. This makes Merge Sort far more efficient than Insertion Sort for array sizes larger than Although Merge Sort has been shown to be better in the worst case for array sizes larger than 1000, it was still slower than Insertion Sort with array sizes less than This can be explained by the overhead required for the creation and merging of all the arrays during the merge step. A future improvement to the Merge Sort algorithm would be to determine a way to reduce this overhead. 6. REFERENCES [1] Knuth, D. Sorting by Exchanging. The Art of Computer Edition Pp [2] Knuth, D. Sorting by Selection. The Art of Computer Edition Pp [3] Knuth, D. Sorting by Insertion. The Art of Computer Edition Pp