Sorting Algorithms Day 2 4/5/17

Transcription

1 Sorting Algorithms Day 2 4/5/17

2 Agenda HW Sorting Algorithms: Review Selection Sort, Insertion Sort Introduce MergeSort

3 Sorting Algorithms to Know Selection Sort Insertion Sort MergeSort Know their relative efficiencies in terms of time and space and how they work. *Knowing the algorithm is more important than the code.

4 Selection vs Insertion Both are O(n 2 ) algorithms but Insertion Sort performs better than this if the list is almost sorted. Selection Sort performs the same either way. Why is this? How would you describe to someone what it means for an algorithm to be an O(n 2 )?

5 Sorting Dances Selection: 8whw Insertion: 9l3U

6 MergeSort Algorithm 1. If the array has one element, do nothing. 2. If the array has two elements, swap if necessary 3. Split array into two equal halves. 4. Sort the first half and second half (recursion) 5. Merge both halves into sorted array.

7 MergeSort - recursive

8 MergeSort Analysis MergeSort is an O(nlog n ) algorithm- is better than an O(n 2 ) algorithm on average. A divide and conquer recursive algorithm. A downside to MergeSort is that it takes more space. It requires a new array to be allocated for each merge. If space is an issue, mergesort may not be the right choice.

9 Mergesort (cont d) public void mergesort (double[ ] arr, int from, int to) { if (to <= from) return; int middle = (from + to ) / 2; mergesort (arr, from, middle); mergesort (arr, middle + 1, to); Base case Optional shortcut: if not yet sorted... } if (arr [middle] > arr [middle + 1]) { copy (arr, from, to, temp) ; merge (temp, from, middle, to, arr); } double[ ] temp is initialized outside the mergesort method 14-9

10 Visualize Sort

11 Sorting Dances 8whw Insertion: 9l3U MergeSort NVoo

12 Visual Notice performance based on initial conditions of an array. When does InsertionSort out perform MergeSort? What other reasons might you choose Insertion Sort over MergeSort? Why might you use Selection Sort?

13 Ideal Sorting Algorithm Properties Stable: Equal keys aren't reordered. Operates in place, requiring no extra space. Worst-case O(n log(n)) key comparisons. Worst-case O(n) swaps. Adaptive: Speeds up to O(n) when data is nearly sorted or when there are few unique keys. No sorting algorithm is ideal so choice is based on application

14 Comparison of Sorting Algorithms SelectionSort InsertionSort MergeSort Average Case O(n 2 ) O(n 2 ) O(nlog n) Description Always performs n(n-1)/2 number of comparisons. Performs n swaps. Does not get any faster if working with an already sorted list always the same. Not Stable Non adaptive O(n) swaps Can approach an Order of n algorithm if list nearly sorted. Comparisons are minimized in this case because only need to compare last element of sorted section. Generally performs n 2 number of swaps (more than Selection Sort). Stable O(n 2 ) comparisons and swaps Adaptive: O(n) time when nearly sorted A divide and conquer algorithm. Uses recursion to split the list until 1 element and then each list of roughly equal size gets merged together. This minimizes comparisons because each sublist is sorted so only looking at first element of each. Requires O(n) extra space to store sublists. Stable Θ(n) extra space for arrays Θ(n lg(n)) time Not adaptive When to Use If space is an issue and swapping is expensive (Large data records) Good for roughly sorted data. The winner between the quadratic sorts except for the rare case noted to the left. Beats MergeSort when nearly sorted data In general fastest among these 3. Use as long as space is not an issue. Often the best choice of divide and conquer sorting algorithms. When Not to Use Generally not the best choice except for conditions noted above Not a good choice if list is sorted backwards. Not a good choice if space is an issue.

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19 Sorting/Searching Questions from packet