Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Question
Chapter 2, Problem 2P
(a)
Program Plan Intro
To prove that BUBBLESORT terminates and
(b)
Program Plan Intro
To state loop invariant for the for loop in line 2-4 and also prove that it holds.
(c)
Program Plan Intro
To state loop invariant for the for loop in line 1-4 and also prove that will allow to prove the inequality 2.3.
(d)
Program Plan Intro
To find the worst case running time bubble sort and compare it with the running time of the insertion sort.
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Q2: a. Write an algorithm that searches a sorted list of n items by dividing it into three sublists of almost n/3 items.
This algorithm finds the sublist that might contain the given item and divides it into three smaller sublists of
almost equal size. The algorithm repeats this process until it finds the item or concludes that the item is not in
the list. Dry run the above algorithm to find the value 240.
A[] = {10,15,20,60,65,110,150,220,240,245,260,290,300,460,470,501}
Language: Python 3
Autocomplete Ready O
1 v import ast
3. Hybrid Sort
input()
lst
%3D
3
lst = ast.literal_eval(lst)
4
Insertion sort is a simple sorting algorithm that builds the final sorted array one
item at a time. In each iteration, insertion sort inserts an element into an already
sorted list (on left). The position where the item will be inserted is found through
linear search. You decided to improve insertion sort by using binary search to
find the position p where
the new insertion should take place.
6
print(BinaryInsertionSort(lst))
Algorithm BinarylnsertionSort
Input/output: takes an integer array a = {a[0], ..., a[n – 1]} of size n
begin BinarylnsertionSort
for i =1 to n
val = a[i]
p = BinarySearch(a, val, 0, i – 1)
for j = i-1 to p
a[j + 1]= a[i]
j= j-1
end for
a[p] = val
i i+1
end for
end BinarylnsertionSort
Here, val = a[i] is the current value to be inserted at each step i into the already
sorted part a[0], ..., ați – 1] of the array a. The binary search along that part…
Select the appropriate code that performs selection sort.
a)
int min;
for(int j=0; j<arr.length-1; j++)
{
min = j;
for(int k=j+1; k<=arr.length-1; k++)
{
if(arr[k] < arr[min])
min = k;
}
int temp = arr[min];
arr[min] = arr[j];
arr[j] = temp;
}
b)
int min;
for(int j=0; j<arr.length-1; j++)
{
min = j;
for(int k=j+1; k<=arr.length; k++)
{
if(arr[k] < arr[min])
min = k;
}
int temp = arr[min];
arr[min] = arr[j];
arr[j] = temp;
}
c)
int min;
for(int j=0; j<arr.length-1; j++)
{
min = j;
for(int k=j+1; k<=arr.length-1; k++)
{
if(arr[k] > arr[min])
min = k;
}
int temp = arr[min];
arr[min] = arr[j];
arr[j] = temp;
}
d)
int min;
for(int j=0; j<arr.length-1; j++)
{
min = j;
for(int k=j+1; k<=arr.length; k++)
{
if(arr[k] > arr[min])
min = k;
}
int temp = arr[min];
arr[min] = arr[j];
arr[j] = temp;
}
Chapter 2 Solutions
Introduction to Algorithms
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Similar questions
- Language: Python 3 • Autocomplete Ready O 1 v import ast lst = input(O lst = ast.literal_eval(lst) def binarysearch(lst,x,low,high): if low - high x: 10 11 Algorithm BinarylnsertionSort 12 Input/output: takes an integer array a = {a[0], ..., a[n – 1]} of size n 13 begin BinarylnsertionSort return binarysearch (lst, x, mid, high) 14 for i =1 to n val = a[i] p = BinarySearch(a, val, 0, i – 1) for j = i-1 to p alj + 1]= a[j] j= j-1 end for 15 else: 16 return mid 17 18 def BinaryInsertionSort(lst): 19 print (BinaryInsertionSort(lst)) 20 a[p] = val j=i+1 end for end BinarylnsertionSort Here, val = a[i] is the current value to be inserted at each step i into the already sorted part a[0], ..., ați – 1] of the array a. The binary search along that part returns the position p where the val will be inserted. After finding p, the data values in the subsequent positions j = i- 1, ..., p are sequentially moved one position up to i, ..., p+1 so that the value val can be inserted into the proper…arrow_forwardArray implementation of List ADT Display the array elements from Left to Right and Right to left. Input: Enter the size of array MAX Enter number of elements N Enter the element to be inserted. Output: Display the Array For example: Test Input 1 2 10 W3 123 10 5 11 12 13 14 15 Result Forward: 1 2 3 Backward:3 2 1 Forward: 11 12 13 14 15 Backward:15 14 13 12 11arrow_forwardT/F: Selection Sort has both a best case and worst case runtime of O(n^2).arrow_forward
- Q4- For each of the following situations, name the best sorting algorithm we studied. (For one or two questions, there may be more than one answer deserving full credit, but you only need to give one answer for each.) (a) The array is mostly sorted already (a few elements are in the wrong place). (b) You need an O(n log n) sort even in the worst case and you cannot use any extra space except for a few local variables.(c) The data to be sorted is too big to fit in memory, so most of it is on disk. (d) You have many data sets to sort separately, and each one has only around 10 elements. (e) You have a large data set, but all the data has only one of about 10 values for sorting purposes (e.g., the data is records of elementary-school students and the sort is by age in years).(f) Instead of sorting the entire data set, you only need the k smallest elements where k is an input to the algorithm but is likely to be much smaller than the size of the entire data set.arrow_forwardPseudo Code: Algorithm A2: (Selection Sort) SELECTION (A, N) This algorithm sorts the array A with N elements. Initialize a loop: For (i = 0; i <N; i++) Set MIN: = A[i] Initialize another loop For (j = i+1 ; j<N ; j++) If (MIN >A[j]) SWAP (MIN,A[j]) Exit Q: Implement the above pseudo-code Sorting algorithms of Selection sort in C++arrow_forwardCreate a program that can find the node of a sorted dynamic array and print it. If there is more than one node found print them in ascending order. Example of array: {20, 30, 30, 50, 70 , 80, 80, 90, 100}. no sorting necessaryarrow_forward
- Implement the following two sorting algorithms in a program called p3.py. Write two separate functions for these algorithms. Both functions must take a list of integers as the input parameter.1) Bogosort: first shuffle the list argument (i.e., randomize the positions of every element) and then check to see if the result is in sorted order. If it is, the algorithm terminates successfully and returns True, but if it is not then the process must be repeated.2) Bozosort: choose two elements in the list at random, swap them, and then check if the result is in sorted order. If it is, the algorithm terminates successfully and returns True, but if it is not then the process must be repeated.Write a main() function and call both sorting functions using the same list as their arguments. The list can be of any size (try a small list first). Does any of your algorithms terminate? If yes, count the number of iterations it uses to sort the list. Does it always use the same number of repetitions? If…arrow_forward1r. ""Implementation of the Misra-Gries algorithm.Given a list of items and a value k, it returns the every item in the listthat appears at least n/k times, where n is the length of the array By default, k is set to 2, solving the majority problem. For the majority problem, this algorithm only guarantees that if there isan element that appears more than n/2 times, it will be outputed. If thereis no such element, any arbitrary element is returned by the algorithm.Therefore, we need to iterate through again at the end. But since we have filtredout the suspects, the memory complexity is significantly lower thanit would be to create counter for every element in the list. For example:Input misras_gries([1,4,4,4,5,4,4])Output {'4':5}Input misras_gries([0,0,0,1,1,1,1])Output {'1':4}Input misras_gries([0,0,0,0,1,1,1,2,2],3)Output {'0':4,'1':3}Input misras_gries([0,0,0,1,1,1]Output None""".arrow_forwardQuick Sort is another sorting algorithm that follows a divide-and-conquer approach. The algorithm can be summarized in 3 steps: A pivot element is chosen, usually the first element. All elements smaller than the pivot are placed to the left of the pivot. This creates 2 partitions, elements greater than the pivot and elements less than the pivot. The 2 partitions are sorted using Quick Sort. Sample code in python3: def quick_sort(arr): def quick_sort_r(arr, start, end): if end - start < 2: # single element base case return # choose a pivot pivot = start # you may choose other elements store = pivot+1 # index to store less than elements # for all elements after the pivot for i in range(pivot+1, end): if arr[i] < arr[pivot]: # if element is less than pivot arr[i], arr[store] = arr[store], arr[i] # swap store += 1 # increment store index # swap pivot with last element in less than…arrow_forward
- 25. Minimum Difference Sum Given an array of n integers, rearrange them so that the sum of the absolute differences of all adjacent elements is minimized. Then, compute the sum of those absolute differences. Example n = 5 arr = [1, 3, 3, 2, 4] If the list is rearranged as arr' = [1, 2, 3, 3, 4], the absolute differences are /1-2/ = 1, 12-3|= 1, 13- 3|=0, 13-4/= 1. The sum of those differences is 1+1+0+1 = 3. Function Description Complete the function minDiff in the editor below. minDiff has the following parameter: arr: an integer array Returns: int: the sum of the absolute differences of adjacent elements Constraints 1 > #include ... 'PRENENANG 19 20 21 22 23 24 25 26 28 * Complete the 'minDiff' function below. ★ 27 int minDiff(int arr_count, int* arr) { * The function is expected to return an INTEGER. * The function accepts INTEGER_ARRAY arr as parameter. */ 29 } 30 31 > int main() ...arrow_forwardQ1: Suppose you are given an array A of n Your task is to sort n numbers stored in array A by reading the first element of A and placing it on its original position (position after sorting). Then read the second element of A, and place it on its original position. Continue in this manner for the first n-1 elements of A. What type of sorting is this? Write the algorithm and also mention the name of this sorting algorithm. What loop invariant does this algorithm maintain? Give the best-case and worst-case running times of this sorting algorithm.arrow_forwardJAVAsimulate the process of selection sort for array {3,1,2,4}arrow_forward
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