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Chapter 7 Sorting Part II

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7.3 QUICK SORT

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Example 4151264 left right pivot i j 5 > pivot and should go to the other side. 2 < pivot and should go to the other side. Interchange a[i] and a[j] 25 Stop, because i == j. a[j] will eventually stop at a position where a[j] < pivot. Interchange a[j] and pivot. 144 Stop; because i > j.

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Algorithm void QuickSort(int a[], int left, int right) { if (left < right) { pivot = a[left]; i = left; j = right+1; while (i < j) { for (i++; i<j and a[i] < pivot; i++) ; for (j--; i = pivot; j--) ; if (i < j) interchages a[i] and [i]; } interchange a[j] and pivot; QuickSort(a, left, j-1); QuickSort(a, j+1 right;); } void QuickSort(int a[], int left, int right) { if (left < right) { pivot = a[left]; i = left; j = right+1; while (i < j) { for (i++; i<j and a[i] < pivot; i++) ; for (j--; i = pivot; j--) ; if (i < j) interchages a[i] and [i]; } interchange a[j] and pivot; QuickSort(a, left, j-1); QuickSort(a, j+1 right;); }

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Analysis of QuickSort() Worst case: ◦ Consider a list is stored. The smallest one is always chosen as pivot. In 1st iteration, n-1 elements are examined. In second iterations, n-2 elements are examined... Totally, the execution steps are n-1 + n-2 + … + 1 = O(n 2 ) The time complexity is O(n 2 ); 16425

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Lemma 7.1 Let T avg (n) be the expect time for function QuickSort() to sort a list with n records. Then there exists a constant k such that T avg (n) ≦ knlog e n for n ≧ 2. ◦ In other words, T avg (n) = O(nlogn)

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Variations The position of the pivot decides the time complexity of QuickSort(). The best choice for the pivot is the median. ◦ Variations: Median-of-three: select the median among three records: the most left, the most right, and the middle one. Random: select the pivot randomly.

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7.4 HOW FAST CAN WE SORT? (DECISION TREE)

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Consideration What is the best computing time for sorting that we can hope for? ◦ Suppose the only operations permitted on keys are comparisons and exchanges. In this section, we shall prove O(n logn) is the best possible time. Using decision tree.

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Example 7.4 Decision tree for Insertion Sort working on [K 1, K 2, K 3 ] K1 ≦K2K1 ≦K2 [1, 2, 3] K2 ≦K3K2 ≦K3 Stop [1, 2, 3] K1 ≦K3K1 ≦K3 [1, 3, 2] Stop [3, 1, 2][1, 3, 2] K1 ≦K3K1 ≦K3 [2, 1, 3] Stop K2 ≦K3K2 ≦K3 [2, 3, 1] Stop [3, 2, 1][2, 3, 1] [2, 1, 3] Y Y Y Y Y N N N N N I IIIII IV VVI

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Observations The leaf nodes denote the states to terminate. The number of permutations is 3! = 6. ◦ n! possible permutation for n records to sort. A path from the root to some leaf node represents one of n! possibilities. The maximum depth of the tree is 3. ◦ The depth represents the number of comparisons.

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Theorem 7.1 Any decision tree that sorts n distinct elements has a height of at least log 2 (n!)+1. ◦ When sorting n elements, there are n! different possible results. Every decision tree for the sorting must at least have n! leaves. ◦ A decision tree is a binary tree; therefore 2 k-1 leaves if its height is k.

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Corollary Any algorithm that sorts only by comparisons must have a worst case computing time of Ω (n logn). Proof ◦ By theorem, there is a path of length log 2 n! So,

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7.5 MERGE SORT

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Merging Consider how to merge two ordered lists. initList mergeList l mm+1n sorted merge

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Example 45711236814 2 2 3 3 4 4 5 5 6 6 7 7 8 8 11 14 l mm+1n i1i2i1 iResult i1 i2 iResult iResult = l initList mergeList i1i2 iResult Copy the small one to mergeList Copy the rest to mergeList

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void Merge(int *initList, int *mergeList, int l, int m, int n) { int i1=l, iResult=l, i2=m+1; while (i1 <= m && i2 <= n) { if (initList[i1] <= initList[i2]) { mergeList[iResult++] = initList[i1]; i1++; } else { mergeList[iResult++] = initList[i2]; i2++; } for (i1; i1<=m; i1++) mergeList[iResult++] = initList[i1]; for (i2; i2<=n; i2++) mergeList[iResult++] = initList[i2]; }

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Analysis of Merge() Time complexity: ◦ The while-loop and two for-loops examine each element in initList exactly once. ◦ The time complexity is O(n- l +1). Space complexity: ◦ The additional array mergeList is required to store the merged result. ◦ Space complexity is O(n- l +1).

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7.5.2 Iterative Merge Sort L: the maximum number of records in a block. 265771611159154819 5 261 7711 6115 5919 48 1 5 26 7711 15 59 6119 48 1 5 11 15 26 56 61 7719 48 1 5 11 15 19 26 48 56 61 77 L = 1 L = 2 L = 4 L = 8 L = 16

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Merging blocks with length of L: 5 261 7711 6115 5919 48 L = 2 ii+L-1i+Li+2L-1i+2L 0123

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Adjacent pairs of blocks of size L are merged from initList to resultList. n is the number of records in initList. void MergePass(int *initList, int *resultList, int n, int L) { int i; for (i=0; i<=n-2*L; i+=2*L) Merge(initList, resultList, i i+L-1, i+2L-1); if (i + L - 1 < n-1) Merge(initList, resultList, i, i+L-1, n-1); else { for (i; i<n; i++) resultList[i] = resultList[i]; } void MergePass(int *initList, int *resultList, int n, int L) { int i; for (i=0; i<=n-2*L; i+=2*L) Merge(initList, resultList, i i+L-1, i+2L-1); if (i + L - 1 < n-1) Merge(initList, resultList, i, i+L-1, n-1); else { for (i; i<n; i++) resultList[i] = resultList[i]; } Merge remaining blocks of L < size < 2L

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01234567 i<=n-2L. Merge adjacent blacks. i<=n-2L. Merge adjacent blacks. 012346 i+L-1 < n-1. Merge from i to n-1. i+L-1 < n-1. Merge from i to n-1. 5 01234 i+L-1 >= n-1. Copy the rest. i+L-1 >= n-1. Copy the rest. 5

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Merge Sort L denotes the length of block currently being merged. void MergeSort(int *a, int n) { int *tempList = new int[n]; for (int L=1; L<n; L*=2) { MergePass(a, tempList, n, L); L *= 2; MergePass(tempList, a, n, L); } delete [] tempList; } The result is put into tempList Do the next pass directly. Merge records from tempList to a

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Analysis of MergeSort() Suppose there are n records. Space complexity: O(n). Time Complexity ◦ MergePass(): O(n). ◦ MergeSort(): A total passes are made over the data. Therefore, the time complexity O(nlogn). LL n

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7.5.2 Recursive Merge Sort We divide the list into two roughly equal parts and sort them recursively. sorted merge leftright

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265771611159154819 265771611159154819 265771611159154819 2651159 5 26 5 26 77 11 59 48 191 6111 15 59 1 5 26 61 7711 15 19 48 59 1 5 11 15 19 26 48 59 61 77 7715 26511594819161

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s void Merge(int *initList, int s, int m, int e) { int *temp = new int[e-s+1]; int i1=s, iResult=0, i2=m+1; while (i1 <= m && i2 <= e) { if (initList[i1] <= initList[i2]) { temp[iResult++] = initList[i1]; i1++; } else { temp[iResult++] = initList[i2]; i2++; } for (i1; i1<=m; i1++) temp[iResult++] = initList[i1]; for (i2; i2<=n; i2++) temp[iResult++] = initList[i2]; for (i=0; i<iResult; i++) initList[i] = temp[i]; delete [] temp; }

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MergeSort() start and end respectively denote the left end and right end to be sorted in the array a. void MergeSort(int *a, int start, int end) { if (end <= start) return; middle = (start + end) / 2; MergeSort(a, start, middle); MergeSort(a, middle+1, end); Merge(a, start, middle, end); }

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Analysis of Recursive Merge Sort Suppose there are n records to be sorted. Time complexity O(nlogn)

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Variation Natural Merge Sort ◦ Make an initial pass over the data To determine the sublists of records that are in order.

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7.6 HEAP SORT

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Discussion Merge Sort ◦ In worst case and average case, the time complexity is O(nlogn). ◦ However, additional storage is required. There is O(1) space merge algorithm, but it is much slower than the original one. Heap Sort ◦ Only a fixed amount of additional storage is required. ◦ The time complexity is also O(nlogn). ◦ Using max heap.

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Selection Sort Suppose there are n records in the list. How to sort the list? ◦ First, find the largest and put it at the position n-1. ◦ Second, find the largest from 0 to n-2 (the second largest), and put it at the position n-2. … ◦ In the i-th iteration, the i-th largest is selected and is put at the position of n-i. Consider how to sort [3, 4, 1, 5, 2].

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34152 Select 5. 34125 Select 4. 32145 34125 Select 3. 12345 32145 Select 2. 1234512345 Select 1. 1234512345

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Analysis of Selection In the i-th iteration, O(n-i+1) computing time is required to select the i-th largest. O(n) + O(n-1) + … + O(1) = O(n 2 ). ◦ How do we decrease the time complexity? Improve the approach of selecting the maximum. Using max heap The deletion of the maximum from a max heap is O(log n), when there are n elements in the heap. Note: when using heap sort, the data is stored in [1;n].

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Example 26 [1] 5 [2] 77 [3] 1 [4] 61 [5] 11 [6] 59 [7] 15 [8] 48 [9] 19 [10] 77 [1] 61 [2] 59 [3] 48 [4] 19 [5] 11 [6] 26 [7] 15 [8] 1 [9] 5 [10] Initial Array Initial Heap

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61 [1] 48 [2] 59 [3] 15 [4] 19 [5] 11 [6] 26 [7] 5 [8] 1 [9] 59 [1] 48 [2] 26 [3] 15 [4] 19 [5] 11 [6] 1 [7] 5 [8] Heap Size = 9 Sorted = [77] Heap Size = 8 Sorted = [61, 77]

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48 [1] 19 [2] 26 [3] 15 [4] 5 [5] 11 [6] 1 [7] 26 [1] 19 [2] 11 [3] 15 [4] 5 [5] 1 [6] Heap Size = 7 Sorted = [59, 61, 77] Heap Size = 69 Sorted = [48, 59, 61, 77]

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Adjust() Adjust binary tree with root to satisfy heap property. void Adjust(int *a, int root, int n) { int e = a[root]; for (int j=2*root; j<=n; j*=2) { if (j < n && a[j] < a[j+1]) //j is max child of its parent j++; if (e >= a[j]) break; a[j/2] = a[j]; } a[j/2] = e; }

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Heap Sort void HeapSort(int *a, int n) { for (int i=n/2; i>=1; i--) //heapify Adjust(a, i, n); for (int i=n-1; i>=1; i--) //sort { swap(a[1], a[i+1]); Adjust(a, 1, i); }

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Analysis of HeapSort() Space complexity: O(1). Time complexity: ◦ Suppose the tree has k levels. The number of nodes on level i is ≦ 2 i-1. ◦ In the first loop, Adjust() is called once for each node that has a child.

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Analysis of HeapSort() Space complexity: O(1). Time complexity: ◦ Suppose the tree has k levels. The number of nodes on level i is ≦ 2 i-1. ◦ In the first loop, Adjust() is called once for each node that has a child. Time complexity:

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Analysis of HeapSort() ◦ In the next loop, n-1 times of Adjust() are made with maximum depth k = and swap is invoked n-1 times. ◦ Consequently, time computing time for the loop is O(n logn). Overall, the time complexity for HeapSort() is O(n logn).

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