Mer4rrrrrrrrrrrrrrrrrrrrrrrrrrrrgeSort.ppt

Mer4rrrrrrrrrrrrrrrrrrrrrrrrrrrrgeSort.ppt

• 1.
  Merge Sort Merge Sort https://www.cs.uiowa.edu/~hzhang/c31/notes/ https://www.cs.uiowa.edu/~hzhang/c31/notes/mergesort mergesort. .ppt ppt
• 2.
  Merging Merging The key toMerge Sort is merging two sorted lists into one, such that if you have two lists X (x1x2… xm) and Y(y1y2… yn) the resulting list is Z(z1z2… zm+n) Example: L1 = { 3 8 9 } L2 = { 1 5 7 } merge(L1, L2) = { 1 3 5 7 8 9 }
• 3.
  Merging Merging (cont.) (cont.) 3 1023 54 1 5 25 75 X: Y: Result:
• 4.
  Merging Merging (cont.) (cont.) 3 1023 54 5 25 75 1 X: Y: Result:
• 5.
  Merging Merging (cont.) (cont.) 10 2354 5 25 75 1 3 X: Y: Result:
• 6.
  Merging Merging (cont.) (cont.) 10 2354 25 75 1 3 5 X: Y: Result:
• 7.
  Merging Merging (cont.) (cont.) 23 5425 75 1 3 5 10 X: Y: Result:
• 8.
  Merging Merging (cont.) (cont.) 54 2575 1 3 5 10 23 X: Y: Result:
• 9.
  Merging Merging (cont.) (cont.) 54 75 13 5 10 23 25 X: Y: Result:
• 10.
  Merging Merging (cont.) (cont.) 75 1 35 10 23 25 54 X: Y: Result:
• 11.
  Merging Merging (cont.) (cont.) 1 35 10 23 25 54 75 X: Y: Result:
• 12.
  Divide And Conquer DivideAnd Conquer  Merging a two lists of one element each is the same as sorting them.  Merge sort divides up an unsorted list until the above condition is met and then sorts the divided parts back together in pairs.  Specifically this can be done by recursively dividing the unsorted list in half, merge sorting the right side then the left side and then merging the right and left back together.
• 13.
  Merge Sort Algorithm MergeSort Algorithm Given a list L with a length k: If k == 1  the list is sorted Else: – Merge Sort the left side (1 thru k/2) – Merge Sort the right side (k/2+1 thru k) – Merge the right side with the left side
• 14.
  Merge Sort Example MergeSort Example 99 6 86 15 58 35 86 4 0
• 15.
  Merge Sort Example MergeSort Example 99 6 86 15 58 35 86 4 0 99 6 86 15 58 35 86 4 0
• 16.
  Merge Sort Example MergeSort Example 99 6 86 15 58 35 86 4 0 99 6 86 15 58 35 86 4 0 86 15 99 6 58 35 86 4 0
• 17.
  Merge Sort Example MergeSort Example 99 6 86 15 58 35 86 4 0 99 6 86 15 58 35 86 4 0 86 15 99 6 58 35 86 4 0 99 6 86 15 58 35 86 4 0
• 18.
  Merge Sort Example MergeSort Example 99 6 86 15 58 35 86 4 0 99 6 86 15 58 35 86 4 0 86 15 99 6 58 35 86 4 0 99 6 86 15 58 35 86 4 0 4 0
• 19.
  Merge Sort Example MergeSort Example 99 6 86 15 58 35 86 0 4 4 0 Merge
• 20.
  Merge Sort Example MergeSort Example 15 86 6 99 58 35 0 4 86 99 6 86 15 58 35 86 0 4 Merge
• 21.
  Merge Sort Example MergeSort Example 6 15 86 99 0 4 35 58 86 15 86 6 99 58 35 0 4 86 Merge
• 22.
  Merge Sort Example MergeSort Example 0 4 6 15 35 58 86 86 99 6 15 86 99 0 4 35 58 86 Merge
• 23.
  Merge Sort Example MergeSort Example 0 4 6 15 35 58 86 86 99
• 24.
  Implementing Merge Sort ImplementingMerge Sort  There are two basic ways to implement merge sort: – In Place: Merging is done with only the input array  Pro: Requires only the space needed to hold the array  Con: Takes longer to merge because if the next element is in the right side then all of the elements must be moved down. – Double Storage: Merging is done with a temporary array of the same size as the input array.  Pro: Faster than In Place since the temp array holds the resulting array until both left and right sides are merged into the temp array, then the temp array is copied over the input array.  Con: The memory requirement is doubled.
• 25.
  mergeSort(arr, int left,int right) mergeSort(arr, int left, int right) { { if (left >= right) if (left >= right) return; return; int mid = (left + right) / 2; int mid = (left + right) / 2; mergeSort(arr, left, mid); mergeSort(arr, left, mid); mergeSort(arr, mid + 1, right); mergeSort(arr, mid + 1, right); merge(arr, left, mid, right); merge(arr, left, mid, right); } } Merge(arr, beg, mid, end) { int start1=beg, start2=mid+1,pos=0; while(start1<=mid && start2<=end) { if(arr[start1]<=arr[start2]) {arr_temp[pos++]=arr[start1]; start1++;} else {arr_temp[pos++]=arr[start2]; start2++;} } if (start1>mid) while(start2<=end) {arr_temp[pos++]=arr[start2++];} if(start2>end) while(start1<=mid) {arr_temp[pos++]=arr[start1++];} copy arr_temp to arr(beg, end) } HW: Check if logic works
• 26.
  Merge Sort Analysis MergeSort Analysis The Double Memory Merge Sort runs O (N log N) for all cases, because of its Divide and Conquer approach. T(N) = 2T(N/2) + N = O(N logN)
• 27.
  Merge sort Merge sort Recurrenceequation: Recurrence equation: c c1 1 if n=1 if n=1 T(n) = 2T(n/2) + c.n if n>1 T(n) = 2T(n/2) + c.n if n>1 MergeSort(arr[], l, r) If r > l 1. Find the middle point to divide the array into two halves: middle m = (l+r)/2 2. Call mergeSort for first half: Call mergeSort(arr, l, m) 3. Call mergeSort for second half: Call mergeSort(arr, m+1, r) 4. Merge the two halves sorted in step 2 and 3: Call merge(arr, l, m, r)
• 28.
  L1.28 Recursion tree Recursion tree SolveT(n) = 2T(n/2) + cn, where c > 0 is constant. cn cn/4 cn/4 cn/4 cn/4 cn/2 cn/2 (1) … h = lg n cn cn cn #leaves = n (n) Total(n lg n) …
• 29.
  Solution Solution By Substitution: T(n) =2T(n/2) + c2n T(n/2) = 2T(n/4) + c2n/2 T(n) = 4T(n/4) + 2 c2n T(n) = 8T(n/8) + 3 c2n T(n) = 2i T(n/2i ) + ic2n Assuming n = 2k , expansion halts when we get T(1) on right side; this happens when i=k T(n) = 2k T(1) + kc2n Since 2k =n, we know k=logn; since T(1) = c1, we get T(n) = c1n + c2nlogn; thus an upper bound for TmergeSort(n) is O(nlogn)
• 30.
  Assume that amerge sort algorithm in the worst case takes 30 seconds for an input of size 64. Which of the following most closely approximates the maximum input size of a problem that can be solved in 6 minutes? (GATE 2015) 1. 256 2. 512 3. 1024 4. 2048 
• 32.
  Finally… Finally… There are othervariants of Merge Sorts including k- way merge sorting, but the common variant is the Double Memory Merge Sort. Though the running time is O(N logN) and runs much faster than insertion sort and bubble sort, merge sort’s large memory demands makes it not very practical for main memory sorting.
• 33.
  Questions Questions  H.W  If wepartition into three(or k) parts instead of two, what would be the complexity of the algorithm ? What if we do a n-way merge sort ?  Would you prefer to use a three part merge sort –why or why not ?  What about a ternary search instead of binary search  Would you prefer 2-way merge sort to merge sort. Why or why not 33