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Algorithms

The Project for the Establishing the Korea ㅡ Vietnam College of Technology in Bac Giang. Algorithms. April-May 2013. Dr. Youn-Hee Han. Divide & Conquer. The most-well known algorithm design strategy: 1. Divide instance of problem into two or more smaller instances

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Algorithms

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  1. The Project for the Establishing the Korea ㅡ Vietnam College of Technology in BacGiang Algorithms April-May 2013 Dr. Youn-Hee Han

  2. Divide & Conquer • The most-well known algorithm design strategy: 1. Divide instance of problem into two or more smallerinstances 2. Solve smaller instances recursively 3. Obtain solution to original (larger) instance by combining these solutions - “Combining these solutions” is optional

  3. Divide & Conquer a problem of size n (instance) subproblem 1 of size n/2 subproblem 2 of size n/2 a solution to subproblem 1 a solution to subproblem 2 a solution to the original problem It general leads to a recursive algorithm!

  4. Divide & Conquer • Examples • Sorting: mergesort and quicksort • Binary tree traversals • Binary search • Multiplication of large integers • Matrix multiplication: Strassen’s algorithm

  5. MergeSort • Algorithm Summary – Part1 • Split array A[0..n-1] into about equal halves and make copies of each half in arrays B and C • MergeSortarrays B and C recursively • Merge the sorted arrays B and C into array A

  6. MergeSort • Algorithm Summary – Part1 void mergesort(index low, index high){ index mid; if (low < high) { mid = (low + high) / 2  ; mergesort(low, mid); mergesort(mid+1, high); merge(low, mid, high); } } ... // in main function type[] S = new type[0..n-1]; ... mergesort(0, n-1); ...

  7. MergeSort • Algorithm Summary – Part 2 • Merge sorted arrays B and C into array A as follows: • Repeat the following until no elements remain in one of the arrays B and C: • compare the first elements in the remaining unprocessed portions of the two arrays B and C • copy the smaller of the two into A, while incrementing the index indicating the unprocessed portion of that array • Once all elements in one of the arrays are processed, copy the remaining unprocessed elements from the other array into A.

  8. MergeSort • Algorithm Summary – Part 2 keytype[] U = new keytype[0..n-1]; // temporary array … void merge(index low, index mid, index high){ index i, j, k; i = low; j = mid + 1; k = low; while (i <= mid && j <= high) { if (S[i] < S[j]) { U[k] = S[i]; i++; } else { U[k] = S[j]; j++; } k++; } if(i > mid)copy S[j] through S[high] to U[k] through U[high]; else copy S[i] through S[mid] to U[k] through U[high]; copy U[low] through U[high] to S[low] through S[high]; } i=0 j=5 i=0 j=4 j=6 j=5 i=2 i=2 j=6 i=4

  9. MergeSort • Complexity of MergeSort • The number of unit operation: W(n) • Asymptotic Complexity • O(nlogn)by using Master Theorem! W(n) = 2W(n/2) + O(n), n>1 W(1) = 0

  10. Master Theorem T(n) = aT(n/b) + f (n)where f(n)(nd), d  0 Master Theorem: If a < bd, T(n) O(nd) If a = bd, T(n) O(ndlog n) If a > bd, T(n) O(nlogb a ) Examples: T(n) = 4T(n/2) + n  T(n)  ? T(n) = 4T(n/2) + n2  T(n)  ? T(n) = 4T(n/2) + n3  T(n)  ?

  11. p A[i]<p A[i]p Quick Sort • Algorithm Summary • Select a pivot (partitioning element) in the array • usually, the first element in the array • Rearrange the list so that… • all the elements in the first subarrayare smaller than the pivot • all the elements in the second subarray are larger than or equal to the pivot • Exchange the pivot with the last element in the first subarray • the pivot is now in its final position in the first subarray • Sort the two subarrays recursively

  12. Quick Sort • Example: 15, 22, 13, 27, 12, 10, 20, 25

  13. Quick Sort • Algorithm Summary void quicksort (index low, index high){ index pivotpoint; if (high > low) { pivotpoint = partition(low, high); quicksort(low, pivotpoint-1); quicksort(pivotpoint+1, high); } } ... // in main function type[] S = new type[0..n-1]; ... quicksort(0, n-1); ...

  14. Quick Sort • Algorithm Summary index partition (index low, index high){ index i, j, pivotpoint; keytypepivotitem; pivotitem = S[low]; //j’s role here: it points the index of elements less than the pivotitem j = low; for(i = low + 1; i <= high; i++) if (S[i] < pivotitem) { j++; exchange S[i] and S[j]; } //j’s role here: it points the index of the location for pivotitem //after the for-loop pivotpoint = j; exchange S[low] and S[pivotpoint]; return pivotpoint; }

  15. Quick Sort • Algorithm Summary low=1, high=8 pivotitem = s[low] pivotpoint

  16. Quick Sort • Algorithm Summary high i j low to be investigated < pivotitem >= pivotitem ifS[i] >= pivotitem, index “i” increases high i j low to be investigated < pivotitem >= pivotitem ifS[i] < pivotitem, 1) index “j” increase, 2) swap two values in the indexs “i” and “j”, 3) index “i” increases high j low i > pivotitem < pivotitem to be investigated

  17. Quick Sort • Complexity of Quick Sort • Worst case: sorted array! — O(n2) • Average case: random arrays — O(n log n) • Proof: skip • W(n) = W(n - 1) + n - 1, if n > 0 W(0) = 0 O(n2)

  18. [Programming Practice 3] • Sort: MergeSortvs. Quick Sort • Visit • http://link.koreatech.ac.kr/courses/2013_1/AP-KOICA/AP-KOICA20131.html • Download “SortMain.java” and run it • Analyze the source codes • Complete the source codes while insert right codes within the two functions • public static void mergeSort(int low, int high) • public static void merge(int low, int mid, int high) • public static void quickSort(int low, int high) • public static int partition(int low, int high) • Compare the execution times of the two sort algorithms

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