Outline. Quadratic-Time Sorting. Linearithmic-Time Sorting. Conclusion. Bubble/Shaker Sort Insertion Sort Odd-Even Sort
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1 Outline Quadratic-Time Sorting Bubble/Shaker Sort Insertion Sort Odd-Even Sort Linearithmic-Time Sorting Heap Sort Merge Sort Quick Sort Conclusion Check out this link for animation of various sorting algorithms: 1
2 Heap Sort Already saw this: (Refer back to Lec 8 notes) Insert all elements in the list into a min-heap Repeat until heap is empty - Remove smallest element from the heap - Put it in tail of new list public static int[] heapsort(int[] list) { Heap<Integer> heap = new Heap<>(); // build min-heap -- add all items from the list for (int i = 0; i < list.length; i++) { heap.add(list[i]); } } // re-build the list by continuously pulling the min from the heap for (int i = 0; i < list.length; i++) { list[i] = heap.remove(0); } return list; 2
3 Heap Sort Analysis Pros: It's fast: O(n log n) time in best/worst/average case Cons: Our version uses up n space (need to create a heap of size n) But there is a better version that does not create auxiliary space 3
4 Outline Quadratic-Time Sorting Bubble/Shaker Sort Insertion Sort Odd-Even Sort Linearithmic-Time Sorting Heap Sort Merge Sort Quick Sort Conclusion Check out this link for animation of various sorting algorithms: 4
5 Merging Two Sorted Arrays To merge two sorted arrays into a new sorted array: Repeatedly compare the two smallest items Copy the smaller of the two to the new array first second
6 Define merge() First public static int[] merge(int[] first, int[] second) { // merge the two pre-sorted lists int[] merged = new int[first.length + second.length]; int i = 0, j = 0, k = 0; while (j < first.length && k < second.length) { if (first[j] <= second[k]) { merged[i++] = first[j++]; } else { merged[i++] = second[k++]; } } // finish off any stragglers from first list while (j < first.length) { merged[i++] = first[j++]; } } // finish off any stragglers from second list while (k < second.length) { merged[i++] = second[k++]; } return merged; 6
7 Merging Two Sorted Arrays To merge two sorted arrays into a new sorted array: Repeatedly compare the two smallest items (tiebreak: take from first) Copy the smaller of the two to the new array first second j k i Complexity: O(m + n), where m, n refer to sizes of the 2 arrays 7
8 Merge Sort Observations: Just established merging 2 sorted lists into a sorted list takes linear time A list containing one element is sorted Merge Sort exploits the above two observations Input an unsorted list Recursively split the list in two unsorted halves - Until each of the halves contains one element - Merge the sorted halves together 8
9 Define mergesort() Now /** * Merge Sort list a list of ints a sorted list of ints */ public static int[] mergesort(int[] list) { // base case: list is sorted when empty or contains one item if (list.length <= 1) { return list; } int midpt = list.length/2; // first half of list int[] first = Arrays.copyOfRange(list, 0, midpt); first = mergesort(first); // sort it // second half of list int[] second = Arrays.copyOfRange(list, midpt, list.length); second = mergesort(second); // sort it } // merge the sorted lists return merge(first, second); 9
10 Merge Sort (Cont.) Recursive call to mergesort() Return from previous call to mergesort() 10
11 Merge Sort (Cont.) Recursive call to mergesort() Return from previous call to mergesort() 10
12 Merge Sort (Cont.) Recursive call to mergesort() Return from previous call to mergesort() 10
13 Merge Sort (Cont.) Recursive call to mergesort() Return from previous call to mergesort() 10
14 Merge Sort (Cont.) Recursive call to mergesort() Return from previous call to mergesort() 10
15 Merge Sort (Cont.) merge() Recursive call to mergesort() Return from previous call to mergesort() 10
16 Merge Sort (Cont.) merge() Recursive call to mergesort() Return from previous call to mergesort() 10
17 Merge Sort (Cont.) merge() Recursive call to mergesort() Return from previous call to mergesort() 10
18 Merge Sort (Cont.) merge() Recursive call to mergesort() Return from previous call to mergesort() 10
19 Merge Sort (Cont.) merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
20 Merge Sort (Cont.) merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
21 Merge Sort (Cont.) merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
22 Merge Sort (Cont.) merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
23 Merge Sort (Cont.) merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
24 Merge Sort (Cont.) merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
25 Merge Sort (Cont.) merge() merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
26 Merge Sort (Cont.) merge() merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
27 Merge Sort (Cont.) merge() merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
28 Merge Sort (Cont.) merge() merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
29 Merge Sort (Cont.) merge() merge() merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
30 Merge Sort (Cont.) merge() merge() merge() merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
31 Merge Sort (Cont.) merge() merge() merge() merge() merge() merge() merge() Recursive call to mergesort() Return from previous call to mergesort() 10
32 Class Activity: MergeSort Given: A list of numbers someone asked me to sort If list is empty or has one item, - Hand it back to the student who gave you the list Else: - Divide list in two halves (if it doesn't divide evenly, give extra item to the left list) - Turn around (and you should be facing 2 students behind you) Hand off the left list to a student» Wait for them to hand it back to you sorted Hand off the right list to a student» Wait for them to hand it back to you sorted Once you receive both halves, merge() them, and hand it forward! 11
33 Complexity of Merge First, determine the time complexity of merge() Just need to examine each item once As we know from before, T merge (m, n) =m + n But we're always merging two roughly-equal halves: So, in our case, T merge (n/2,n/2) = n 12
34 Complexity of MergeSort Intuitively... Problem Size: n Steps, T(n), per level 2n+1 (why?) n/ [2(n/2)+1]+1 = 2n+2 n/ [2(n/4)+1]+2+1 = 2n MergeSort's time complexity: T (n) = ( 1, if n apple 1 2T (n/2) + 2n +1, otherwise 13
35 Complexity of MergeSort Intuitively... Problem Size: n Steps, T(n), per level 2n+1 (why?) n/ [2(n/2)+1]+1 = 2n+2 n/ [2(n/4)+1]+2+1 = 2n MergeSort's time complexity: T (n) = ( 1, if n apple 1 2T (n/2) + 2n +1, otherwise T (n) =O(n log 2 n) 13
36 Analysis of MergeSort Pros: Linearithmic time complexity Time complexity is consistent (best/avg/worst) and predictable Cons: Space complexity can be a problem - Requires 2 additional copies of the list each recursive call One created when splitting, another created when merging Semi-sorted lists don't help, unlike the linear-time sorts 14
37 Outline Quadratic-Time Sorting Bubble/Shaker Sort Insertion Sort Odd-Even Sort Linearithmic-Time Sorting Heap Sort Merge Sort Quick Sort Conclusion Check out this link for animation of various sorting algorithms: 15
38 Quick Sort Input: two indices, start and end, of current list If the list isn't empty (i.e., start < end): Use last item in the list as the pivot - Rearrange (partition) all items in the list so that: Items greater than it appear on its right Items less than pivot appear on its left Then recurse on both unsorted sublists start pivot end 16
39 Pivot and Partitioning Trace for: and for /** * Rearranges items in the list and returns the pivot's position */ private static int partition(int[] list, int start, int end) { int pivot = list[end]; // use last item in the sublist as pivot int i = start; for (int j = start; j < end; j++) { if (list[j] < pivot) { swap(list, i, j); i++; } } // swap pivot into proper position swap(list, i, end); return i; } private static void swap(int[] list, int i, int j) { int tmp = list[i]; list[i] = list[j]; list[j] = tmp; } 17
40 Complexity of Partition Determine the time complexity of partition() Select a pivot item Examine each item, comparing with pivot - Swapping, if needed One last swap to get pivot in place Return list Complexity: T part (n) 4n +4 18
41 QuickSort /** * QuickSort list A list of ints a sorted list of ints */ public static int[] quicksort(int[] list) { return quicksort(list, 0, list.length-1); } private static int[] quicksort(int[] list, int start, int end) { if (start < end) { // partitions the list, and gets the position of the pivot int pivot_idx = partition(list, start, end); } // sort the two other halves quicksort(list, start, pivot_idx-1); quicksort(list, pivot_idx+1, end); } return list; 19
42 Quick Sort Trace List
43 Quick Sort Trace: First partition(list,0,7) List i j pivot quicksort(list, 0, 7)
44 Quick Sort Trace: First partition(list,0,7) List i j pivot quicksort(list, 0, 7)
45 Quick Sort Trace: First partition(list,0,7) List i j swap pivot quicksort(list, 0, 7)
46 Quick Sort Trace: First partition(list,0,7) List i j swap pivot quicksort(list, 0, 7)
47 Quick Sort Trace: First partition(list,0,7) List i j swap pivot quicksort(list, 0, 7)
48 Quick Sort Trace: First partition(list,0,7) List i j swap pivot quicksort(list, 0, 7)
49 Quick Sort Trace: First partition(list,0,7) List i j swap pivot quicksort(list, 0, 7)
50 Quick Sort Trace: First partition(list,0,7) List i j swap pivot quicksort(list, 0, 7)
51 Quick Sort Trace: First partition(list,0,7) List i j swap pivot quicksort(list, 0, 7)
52 Quick Sort Trace: First partition(list,0,7) List i j swap pivot quicksort(list, 0, 7)
53 Quick Sort Trace: First partition(list,0,7) List i j swap pivot quicksort(list, 0, 7)
54 Quick Sort Trace: First partition(list,0,7) List i j (Stop!) swap pivot quicksort(list, 0, 7)
55 Quick Sort Trace: First partition(list,0,7) List i j swap pivot into place! quicksort(list, 0, 7)
56 Quick Sort Trace: First partition(list,0,7) List i j swap pivot into place! quicksort(list, 0, 7)
57 Quick Sort Trace: First partition(list,0,7) List i j swap pivot into place! quicksort(list, 0, 7)
58 Quick Sort Trace: First partition(list,0,7) List i j (Stop!) swap pivot into place! quicksort(list, 0, 7)
59 Quick Sort Trace (Omitting partition() Trace) List quicksort(list, 0, 7) quicksort(list, 0, 2)
60 Quick Sort Trace (Omitting partition() Trace) List quicksort(list, 0, 7) quicksort(list, 0, 2) quicksort(list, 0, 0)
61 Quick Sort Trace (Omitting partition() Trace) List quicksort(list, 0, 7) quicksort(list, 0, 2) quicksort(list, 0, 0) quicksort(list, 2, 2)
62 Quick Sort Trace (Omitting partition() Trace) List quicksort(list, 0, 7) quicksort(list, 0, 2) quicksort(list, 4, 7)
63 Quick Sort Trace (Omitting partition() Trace) List quicksort(list, 0, 7) quicksort(list, 0, 2) quicksort(list, 4, 7) quicksort(list, 4, 3) (base case: do nothing) 30
64 Quick Sort Trace (Omitting partition() Trace) List quicksort(list, 0, 7) quicksort(list, 0, 2) quicksort(list, 4, 7) quicksort(list, 5, 7)
65 Quick Sort Trace (Omitting partition() Trace) List quicksort(list, 0, 7) quicksort(list, 0, 2) quicksort(list, 4, 7) quicksort(list, 5, 7) quicksort(list, 5, 5) 6 32
66 Quick Sort Trace (Omitting partition() Trace) List quicksort(list, 0, 7) quicksort(list, 0, 2) quicksort(list, 4, 7) quicksort(list, 5, 7) quicksort(list, 5, 5) quicksort(list, 7, 7)
67 Class Activity: QuickSort Given: A list of numbers If start >= end - Hand list back to the student who gave it to you Else: - Choose end element as pivot - Partition() the sublist indicated from start to end - Give left student the list, and: Tell them your start and end = pivot_index - 1» Wait to receive the list back - Give right student the list, and: Tell them start = pivot_index + 1 and your end» Wait to receive the list back 34
68 Another Trace! List quicksort(list, 0, 3) quicksort(list, 4, 3) base case: do nothing 35
69 Another Trace! List quicksort(list, 0, 3) quicksort(list, 0, 2) quicksort(list, 4, 3) base case: do nothing 36
70 Another Trace! List quicksort(list, 0, 3) quicksort(list, 0, 2) quicksort(list, 4, 3) base case: do nothing quicksort(list, 3, 2) base case: do nothing 37
71 Another Trace! List quicksort(list, 0, 3) quicksort(list, 0, 2) quicksort(list, 0, 1) quicksort(list, 4, 3) base case: do nothing quicksort(list, 3, 2) base case: do nothing 38
72 Another Trace! List quicksort(list, 0, 3) quicksort(list, 0, 2) quicksort(list, 0, 1) quicksort(list, 4, 3) base case: do nothing quicksort(list, 3, 2) base case: do nothing quicksort(list, 2, 1) base case: do nothing 39
73 Another Trace! List quicksort(list, 0, 3) quicksort(list, 0, 2) quicksort(list, 0, 1) quicksort(list, 4, 3) base case: do nothing quicksort(list, 3, 2) base case: do nothing quicksort(list, 0, 0) 5 quicksort(list, 2, 1) base case: do nothing 40
74 Food for Thought Notice how the performance of QuickSort is related to its "Recursion Tree" (results from the trace) The first trace, the tree looked more balanced Like MergeSort's recursion tree (which is always balanced) The second trace, the tree looked like a linked list Like a worst-case binary search tree... 41
75 Pivot: Connection to Performance Important: Performance of QuickSort depends on the pivot partition() selects pivot as the last item in the sublist How to get trees that look like the first trace? Want pivot to be the median (or near it!) item in the sublist Splits the list in half (or close to half!) partition(list, 0, 7) 42
76 Pivot: Connection to Performance Important: Performance of QuickSort depends on the pivot partition() selects pivot as the last item in the sublist How to get trees that look like the first trace? Want pivot to be the median (or near it!) item in the sublist Splits the list in half (or close to half!) partition(list, 0, 7)
77 Pivot: Connection to Performance Important: Performance of QuickSort depends on the pivot partition() selects pivot as the last item in the sublist How to get trees that look like the first trace? Want pivot to be the median (or near it!) item in the sublist Splits the list in half (or close to half!) n partition(list, 0, 7) n/2 n/2 42
78 Pivot: Connection to Performance How to get trees that look like the second trace? When pivot is sorted into one of the extreme ends Occurs when list is sorted in either ascending or descending order partition(list, 0, 7) partition(list, 0, 7) 43
79 Pivot: Connection to Performance How to get trees that look like the second trace? When pivot is sorted into one of the extreme ends Occurs when list is sorted in either ascending or descending order partition(list, 0, 7) partition(list, 0, 7)
80 Pivot: Connection to Performance How to get trees that look like the second trace? When pivot is sorted into one of the extreme ends Occurs when list is sorted in either ascending or descending order n partition(list, 0, 7) partition(list, 0, 7) n
81 Pivot: Connection to Performance How to get trees that look like the second trace? When pivot is sorted into one of the extreme ends Occurs when list is sorted in either ascending or descending order n partition(list, 0, 7) partition(list, 0, 7) n
82 Pivot: Connection to Performance How to get trees that look like the second trace? When pivot is sorted into one of the extreme ends Occurs when list is sorted in either ascending or descending order n n partition(list, 0, 7) partition(list, 0, 7) n n 1 43
83 Pivot: Connection to Performance The fix: What if we selected a pivot randomly? Instead of always being the last item RNG says: "Choose 5 as pivot" Swap pivot to the back partition(list, 0, 7) 44
84 Pivot: Connection to Performance The fix: What if we selected a pivot randomly? Instead of always being the last item RNG says: "Choose 5 as pivot" Swap pivot to the back partition(list, 0, 7)
85 Pivot: Connection to Performance The fix: What if we selected a pivot randomly? Instead of always being the last item RNG says: "Choose 5 as pivot" n Swap pivot to the back partition(list, 0, 7) n/2 n/2 44
86 Pivot: Connection to Performance The fix: What if we selected a pivot randomly? Instead of always being the last item RNG says: "Choose 5 as pivot" n Swap pivot to the back partition(list, 0, 7) n/2 n/2 This is awesome!!! 44
87 Pivot: Connection to Performance The fix: What if we selected a pivot randomly? Instead of always being the last item RNG says: "Choose 8 as pivot" Swap pivot to the back partition(list, 0, 7) 45
88 Pivot: Connection to Performance The fix: What if we selected a pivot randomly? Instead of always being the last item RNG says: "Choose 8 as pivot" Swap pivot to the back partition(list, 0, 7)
89 Pivot: Connection to Performance The fix: What if we selected a pivot randomly? Instead of always being the last item RNG says: "Choose 8 as pivot" n Swap pivot to the back partition(list, 0, 7) n
90 Pivot: Connection to Performance The fix: What if we selected a pivot randomly? Instead of always being the last item RNG says: "Choose 8 as pivot" n Swap pivot to the back partition(list, 0, 7) n
91 Pivot: Connection to Performance The fix: What if we selected a pivot randomly? Instead of always being the last item RNG says: "Choose 8 as pivot" n Swap pivot to the back partition(list, 0, 7) n 1 1 Or... you can s=ll get preby unlucky 45
92 Analysis of QuickSort Best case: Median value of the sublist is always picked to be pivot List split in two roughly equal halves Best Case Analysis: ( 1, if n apple 1 T (n) = 2T (n/2) + T part (n), otherwise T (n) =O(n log 2 n) 46
93 Analysis of QuickSort Worst case: List splits into lopsided "halves" Extreme case is when list size only reduces by 1 item Can be mitigated by using random pivot points Complexity: T (n) = ( 1, if n apple 1 T (n 1) + T part (n)+1, otherwise T (n) =O(n 2 ) 47
94 Outline Quadratic-Time Sorting Bubble/Shaker Sort Insertion Sort Odd-Even Sort Linearithmic-Time Sorting Heap Sort Merge Sort Quick Sort Conclusion 48
95 Summary of Sorting Algorithms Algorithm Best Average Worst Bubble O(n) O(n^2) O(n^2) Shaker O(n) O(n^2) O(n^2) OddEven O(n) O(n^2) O(n^2) Inser=on O(n) O(n^2) O(n^2) Merge O(n log n) O(n log n) O(n log n) Heap O(n log n) O(n log n) O(n log n) Quick O(n log n) O(n log n) O(n^2) 49
96 In Practice... The only quadratic-time algorithms in use is probably insertion sort Good for semi-sorted data Good when n is relatively small Doesn't require auxiliary space Otherwise it's a toss-up: QuickSort vs. MergeSort QuickSort doesn't require auxiliary space; merge sort does MergeSort doesn't have a quadratic-time worst case 50
97 Administrivia 4/20 Solutions posted Hwk 5: Expression Trees Lab 11: HashMap Next Tuesday is a work day Attendance still required 51
98 Administrivia 4/25 Review guide 3 posted Solution will be posted Monday Small Hwk 6 correction Previously... Quadratic-time sorts (sometimes can perform in linear time) Today: MergeSort (linearithmic time) 52
99 Administrivia 4/27 Sorting notes posted Review Monday Look over review guide 3 Solution will be posted Monday Previously... We learned about MergeSort and why it's fast - Eats up lots of auxiliary space though... (can we do better on space?) Today: Yes we can. QuickSort 53
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