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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Chapter 2.2, Problem 2E
Program Plan Intro
To write the pseudo code of selection sort and also describe the loop invariants and the running time complexity in best and worst case.
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Consider sorting n numbers stored in array A by first finding the smallest elementof A and exchanging it with the element in A[1]. Then find the second smallestelement of A, and exchange it with A[2]. Continue in this manner for the first n-1elements of A. Write pseudocode for this algorithm, which is known as selectionsort. What loop invariant does this algorithm maintain? Why does it need to run foronly the first n - 1 elements, rather than for all n elements? Give the best-case andworst-case running times of selection sort in Θ -notation.
Q: Suppose you are given an array A of n elements. 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.
A binary search only works if the values in the list are sorted. A bubble
sort is a simple way to sort entries. The basic idea is to compare two
adjacent entries in a list-call them entry[j] and entry[j+1]. If entry[j] is
larger, then swap the entries. If this is repeated until the last two entries
are compared, the largest element in the list will now be last. The smallest
entry will ultimately get swapped, or "bubbled" up to the top. The
algorithm could be described in C as:
last = num;
while (last > 0) {
pairs = last – 1:
for (j = 0; j entry (j+1]
{
temp = entry[il:
entryli] = entrylj+1];
entrylj+1] = temp;
last = i:
}
}
Here, num is the number of entries in the list. Write an assembly
language program to implement a bubble sort algorithm, and test it using
a list of 8 elements. Each element should be a halfword in length. Please
show your code works with the Keil tools or VisUAL, by grabbing a screen
shot with your name somewhere on the screen.
Chapter 2 Solutions
Introduction to Algorithms
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- We can express insertion sort as a recursive procedure as follows. In order to sort A[1..n], we recursively sort A[1..n-1] and then insert A[n] into the sorted array A[1..n-1]. Write a recursive solution. Input: 1. User input the number of integers to be sorted, named n. User input n integers. Output: print out the sorted integers with space between numbers. Test your program on the following five inputs and print out the output for each input. Input 1: 1 2 3 4 5 6 7 8 Input 2: 8 7 6 5 4 3 2 1 We can express insertion sort as a recursive procedure as follows. In order to sort A[1..n], we recursively sort A[1..n-1] and then insert A[n] into the sorted array A[1..n-1]. Write a recursive solution. Input: 1. User input the number of integers to be sorted, named n. User input n integers. Output: print out the sorted integers with space between numbers. Test your program on the…arrow_forwardThe Binary Search algorithm works by testing a mid-point, then eliminating half of the list. In this exercise, you are going to take our binary search algorithm and add print statements so that you can track how the search executes. Inside of the recursive binary search function, add print statements to print out the starting, ending, and midpoint values each time. Then as you test a value, print out the results, either too high, too low, or a match. Sample Output Starting value: 0 Ending value: 9 Testing midpoint value: 4 Too high! Starting value: 0 Ending value: 3 Testing midpoint value: 1 Too low! Starting value: 2 Ending value: 3 Testing midpoint value: 2 Match! public class BinaryExplorer { public static void main(String[] args) {int[] testArray = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}; binaryRec(testArray, 8, 0, testArray.length - 1); } /*** Add Print statements to the binaryRec method:* * Print Starting, ending, and midpoint values.* * Print when you find a match* * Print if you are…arrow_forwardWrite an algorithm that sorts a list of n items by dividing it into three sublists of about n/3 items, sorting each sublist recursively and merging the three sorted sublists. Analyze your algorithm, and give the results under order notation.arrow_forward
- The Binary Search algorithm works by testing a mid-point, then eliminating half of the list. In this exercise, you are going to take our binary search algorithm and add print statements so that you can track how the search executes. Inside of the recursive binary search function, add print statements to print out the starting, ending, and midpoint values each time. Then as you test a value, print out the results, either too high, too low, or a match. Sample Output Starting value: 0 Ending value: 9 Testing midpoint value: 4 Too high! Starting value: 0 Ending value: 3 Testing midpoint value: 1 Too low! Starting value: 2 Ending value: 3 Testing midpoint value: 2 Match!arrow_forwardWrite an algorithm that sorts a list of n items by dividing it into three sublistsof about n/3 items, sorting each sublist recursively and merging the threesorted sublists. Analyze your algorithm, and give the results under orderarrow_forwardImagine that your best friend's birthday is just around the corner and she has a list of birthday gifts she wants. You want to buy not one but two gifts to her. There is a set of prices of gifts like following (2200, 850, 300, 5000, 1550, 480) Design and implement a way of finding the minimum two prices, sum up them, and display. While finding these numbers, make sure that you are using one searching and one sorting algorithm. You should also write your design in a few sentences as a comment.arrow_forward
- Write a program that sorts an array of random or sorted numbers using Radix sort algorithms, fill the array with a nearly ordered list. Construct your nearly ordered list by reversing elements 19 and 20 in the sorted random list. Count the number of comparisons and moves necessary to order this list. Run the program three times, once with an array of 100 items, once with an array of 500 items, and once with an array of 1000 items. For the first execution only (100 elements), Print the unsorted data followed by the sort data 10x10 matrixes (10 rows of 10 numbers each).arrow_forwardWrite 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. Analyze your algorithm and give the results using order notation.arrow_forwardGiven an unsorted array of integers, A, its size n, and two numbers x and y both elements of A,write an algorithm that returns the distance between x and y. The distance between two numbersof an array is the number of elements that lie between them in the sorted order. Achieve theasymptotically fastest time for this problem.arrow_forward
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