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Quick Sort Roberto Hibbler Dept Quick Sort Roberto Hibbler Dept. of Computer Science Florida Institute of Technology Melbourne, FL 32901 [email protected] ABSTRACT 2. We show that theoretically Quick Sort performs no ͦ Given an array of elements, we want to arrange those elements worse than ͉ʚ͢ ʛ in the worst case, being equivalent into a sorted order. To sort those elements, we will need to make to current solutions. comparisons between the individual elements efficiently. Quick 3. We show that empirically Quick Sort is on average Sort uses a divide and conquer strategy to sort an array faster than Insertion Sort. efficiently while making the least number of comparisons between array elements. Our results show that for arrays with This paper will discuss in Section 2 comparison sort algorithms large numbers of array elements, Quick Sort is more efficient related to the problem, followed by the detailed approach of our than three other comparison sort algorithms, Bubble Sort, solution in Section 3, the evaluation of our results in Section 4, Insertion Sort, and Selection Sort. Our theoretical evaluation and our final conclusion in Section 5. shows that Quick Sort beats a quadratic time complexity, while our empirical evaluation shows that Quick Sort at times was 32 times faster than Insertion Sort, the current recognized most 2. RELATED WORK efficient comparison algorithm, with one real data set. The three algorithms that we will discuss are Bubble Sort, Selection Sort, and Insertion Sort. All three are comparison sort Keywords algorithms, just as Quick Sort. Quick Sort, sorting, comparisons, Selection Sort, time complexity The Bubble Sort algorithm works by continually swapping adjacent array elements if they are out of order until the array is 1. INTRODUCTION in sorted order. Every iteration through the array places at least The ability to arrange an array of elements into a defined order one element at its correct position. Although algorithmically is very important in Computer Science. Sorting is heavily used correct, Bubble Sort is inefficient for use with arrays with a with online stores, were the order that services or items were large number of array elements and has a ͉ʚͦ͢ʛ worst case time purchased determines what orders can be filled and who complexity. Knuth observed[1], also, that while Bubble Sort receives their order first. Sorting is also essential for the shares the worst-case time complexity with other prevalent database management systems used by banks and financial sorting algorithms, compared to them it makes far more element systems, such as the New York Stock Exchange, to track and swaps, resulting in poor interaction with modern CPU hardware. rank the billions of transactions that go on in one day. There are We intend to show that Quick Sort needs to make on average many algorithms which provide a solution to sorting arrays, fewer element swaps than Bubble Sort. including algorithms such as Bubble Sort[1], Insertion Sort[2], and Selection Sort[3]. While these algorithms are The Selection Sort algorithm arranges array elements in order by programmatically correct, they are not efficient for arrays with a first finding the minimum value in the array and swapping it large number of elements and exhibit quadratic time complexity. with the array element that is in its correct position depending on how the array is being arranged. The process is then repeated We are given an array of comparable values. We need to arrange with the second smallest value until the array is sorted. This these values into either an ascending or descending order. creates two distinctive regions within the array, the half that is sorted and the half that has not been sorted. Selection Sort shows We introduce the Quick Sort algorithm. The Quick Sort an improvement over Bubble Sort by not comparing all the algorithm is a divide-and-conquer algorithm. It takes an array elements in its unsorted half until it is time for that element to be and divides that array into sub arrays based on a pivot value. placed into its sorted position. This makes Selection Sort less Values less than the pivot value are shifted to within the array to affected by the input’s order. Though, it is still no less the left of the pivot while values greater are shifted to the right inefficient with arrays with a large number of array elements. of the pivot in a process called partitioning. These elements are Also, even with the improvements Selection Sort still shares the then divided into two sub arrays, one of the elements to the left same worst-case time complexity of ͉ʚͦ͢ʛ. We intend to show of the pivot and the other the right elements. The partitioning that Quick Sort will operate on average faster Selection Sort. process is done recursively until we have arrays of single elements. A single element is already sorted, and so the elements The Insertion Sort algorithm takes elements from the input array are combined back into sorted arrays two sub-arrays at a time, and places those elements in their correct place into a new array, until we are left with a final sorted array. We contribute the shifting existing array elements as needed. Insertion Sort following: improves over Selection Sort by only making as many comparisons as it needs to determine the correct position of the 1. We introduce the Quick Sort algorithm. current element, while Selection Sort makes comparisons against each element in the unsorted part of the array. In the As the pseudocode shows, after a new pivot element is randomly )v chosen (lines 3-4), the array is broken up into a left half and a average case, Insertion Sort’s time complexity is ͉ʚ ʛ, but its ͨ right half and sorted recursively (lines 5-6). In the partition ͦ worst case is ͉ʚ͢ ʛ, the same as Bubble Sort and Selection algorithm, the two sub arrays are compared to determine how Sort. The tradeoff of Insertion Sort is that on the average more they should be arranged (lines 12-15). In the following elements are swapped as array elements are shifted within the examples, using the given input, the partitioning of the array array with the addition of new elements. Even with its (Figure 3) and how the sub arrays are concatenated back into a limitations, Selection Sort is the current fastest comparison sorted array (Figure 4) are illustrated. based sorting algorithm since it equals Bubble Sort and Selection Sort in the worst case, but is exceptionally better in the Inputs – A: array of n elements average case. We intend to show that Quick Sort operates on (33 55 77 11 66 88 22 44) average faster than Insertion Sort. 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. Quick sort can solve this problem by using two key ideas. The first key idea of Quick sort is that a problem can be divided and conquered. The problem can be broken into smaller arrays, and those arrays can be solved. This is done by picking an element from the array, called the pivot, and reordering the array Figure 3: Shows the partitioning of the input array into so that all elements which are less than the pivot come before single element arrays. The red element is the pivot element the pivot and all elements greater than the pivot come after it. of the array. This process is called partitioning. Second, by dividing the array into two halves, elements less than the pivot and elements greater than the pivot, then partitioning those halves recursively into arrays of single elements, two sorted arrays are concatenated on either side of the pivot value into one array, as a single element array is already sorted. Refer to the following pseudocode: Input – A: array of n elements, left: left sub array from the pivot, right: right sub array from the pivot Output – array A sorted in ascending order 1. proc quicksort(A,left,right) 2. if right > left Figure 4: From the bottom up, shows the concatenating of 3. pivot = random(A) the single element arrays. The red element is the pivot 4. newPivot=partition(A,left,right,pivot) element. 5. quicksort(A, left, newPivot – 1) 6. quicksort(A,newPivot+1,right) 7. return A As the example shows, array A is broken in half based on the Figure 1: Pseudocode of the Quick Sort algorithm pivot until they are in arrays of only a single element, then those single elements are concatenated together until they form a single sorted array in ascending order. Input – A: array of n elements,left:array of m elements, right: array of k elements, pivot: int of pivot index 4. EVALUATION Output – array result sorted in ascending order 8. proc partition(A,left,right,pivot) 4.1 Theoretical Analysis 9. val = A[pivot] 4.1.1 Evaluation Criteria 10. swap(A[pivot],A[right]) 11. index = left All comparison based sorting algorithms count the comparisons 12. for I from left to right -1 of array elements as one of their key operations. The Quick Sort 13. if A[i] >= val algorithm can be evaluated by measuring the number of 14. swap(A[i],A[index]) comparisons between array elements. As the key operation, we 15. index=index+1 can measure the number of comparisons made to determine the 16. swap(A[i],A[right]) overall efficiency of the algorithm. We intend to show that 17. return index because the Quick Sort algorithm makes less comparisons over Figure 2: Pseudocode of the Partition step of Quick Sort the currently acknowledged most efficient algorithm, Insertion Sort[Sec 2], Quick Sort is the most efficient comparison sort algorithm.
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