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Assignment 1

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  1. THE SUPERIOR UNIVERSITY LAHORE Analysis of Algorithm Course Title: Submitted To: DR. Imran Khan Department of CS & IT Submitted By: Zanib Adrees Reg. No. MCSW-S20-003 Department of CS Jan 31, 2022 Date of Submission:

  2. Marge Sort if len(arr) > 1: # Finding the mid of the array mid = len(arr)//2 # Dividing the array elements L = arr[:mid] # into 2 halves R = arr[mid:] # Sorting the first half mergeSort(L) # Sorting the second half mergeSort(R) i = j = k = 0 # Copy data to temp arrays L[] and R[] while i < len(L) and j < len(R): if L[i] < R[j]: arr[k] = L[i] i += 1

  3. else: arr[k] = R[j] j += 1 k += 1 # Checking if any element was left while i < len(L): arr[k] = L[i] i += 1 k += 1 while j < len(R): arr[k] = R[j] j += 1 k += 1 # Code to print the list def printList(arr): for i in range(len(arr)): print(arr[i], end=" ") print() # Driver Code if _name_ == '_main_':

  4. arr = [12, 11, 13, 5, 6, 7] print("Given array is", end="\n") printList(arr) mergeSort(arr) print("Sorted array is: ", end="\n") printList(arr) Output: Given array is: 12 11 13 5 6 7 Sorted array is: 5 6 7 11 12 13 Heap Sort def heapify(arr, n, i): largest = i # Initialize largest as root l = 2 * i + 1 # left = 2*i + 1 r = 2 * i + 2 # right = 2*i + 2 # See if left child of root exists and is # greater than root if l < n and arr[largest] < arr[l]: largest = l # See if right child of root exists and is # greater than root

  5. if r < n and arr[largest] < arr[r]: largest = r # Change root, if needed if largest != i: arr[i], arr[largest] = arr[largest], arr[i] # swap # Heapify the root. heapify(arr, n, largest) # The main function to sort an array of given size def heapSort(arr): n = len(arr) # Build a maxheap. for i in range(n//2 - 1, -1, -1): heapify(arr, n, i) # One by one extract elements for i in range(n-1, 0, -1): arr[i], arr[0] = arr[0], arr[i] # swap heapify(arr, i, 0)

  6. # Driver code arr = [12, 11, 13, 5, 6, 7] heapSort(arr) n = len(arr) print("Sorted array is") for i in range(n): print("%d" % arr[i],end=" ") Output Sorted array is: 5 6 7 11 12 13 Merge-sort recurrence by Master Theorem Formula for recurrences of the form T(n) = aT(n/b) + nc (T(1) = 1). –Merge-sort ⇒T(n) = 2T(n/2) + n (a = 2,b = 2, and c = 1). Proof (by iteration ) Recall geometric sum •a < bc

  7. bc • a = bc a= bc ⇔bac = 1 ⇒ Pklog=0b n−1 back = Plogk=0b n−1 1 = Θ(logb n) a = bc ⇔ logb a = logb bc = c T(n) = = ncΘ(logb n) + nlogb a = Θ(nc logb n) log n−1 •a > bc a> bc ⇔bac > 1 ⇒ Pklog=0b n−1 back = Θbaclogb n = Θ( baclog)logbbnn = Θ alogncbn T(n) = = Θ(nlogb a) + nlogb a = Θ(nlogb a)

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