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 2.1, Problem 3E
Program Plan Intro
To write the pseudo code for linear search and also discuss the three necessary properties.
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Question # 01:
Consider the searching problem:
Input: A sequence of n numbers A = (a1, a2, a3,
an ) a value V.
Output: An index i such that v = A[i] or the special value NIL if v does not appear in A.
Write pseudocode for linear search, which scans through the sequence, looking for v. Using a
loop invariant, prove that your algorithm is correct. Make sure that your loop invariant fulfills
the three necessary properties.
The intersection of two sets contains the list of elements that are common to both, without
repetition. As such, you are required to write an algorithm that finds and returns the intersection
of two sets A and B. The sets A and B and the resulting intersection set are all lists. Assume
the size of A is n and that of B is m.
(a) Write an iterative algorithm that finds the intersection.
(b) Prove the correctness of your algorithm; i.e. state its loop invariant and prove it by induc-
tion.
(c) Analyze the time complexity of your algorithm.
(d) Think of a way to enhance the complexity of your algorithm by sorting both lists A and
B before calculating their intersection. Write the enhanced version and prove that it has a
better time complexity.
2.3-1
Using Figure 2.4 as a model, illustrate the operation of merge sort on the array
A = (3, 41, 52, 26, 38, 57, 9, 49).
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sorted sequence
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merge
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Chapter 2 Solutions
Introduction to Algorithms
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- The SparseVector class should be extended to include a method sum() that accepts a SparseVector as an argument and returns a SparseVector that represents the term-by-term total of the input and output vectors. Note: To handle the situation where an entry becomes 0, you must use delete() (and pay close mind to precision).use java to codearrow_forwardLet's say you're going to invite some people to a party. You're considering n friends, but you know that they will have a good time only if each of them knows at least k others at the party. (Assume that if A knows B, then B automatically knows A.) Solve your problem by designing an algorithm for finding the largest possible subset of your friends where everyone knows at least k of the others, if such a subset exists. Please comment on code and show all work so I can better understand the problem and please use python thank you, To help you get started l've made a graph of "my friends": G={'Rachel':['Ross Monica'], 'Ross':['Rachel', 'Monica'], 'Monica':['Rachel', 'Ross'], 'Jon Snow':['Daenerys' "Sansa' "Arya'), 'Daenerys':['Jon Snow "Sansa "Arya' Khal Drogo'), 'Sansa':['Jon Snow DaenerysArya'), 'Arya':['Jon Snow "Daenerys' Sansa'), 'Khal Drogo':['Daenerys'], 'Cersei':['Jaime'], 'Jaime':['Cersei'], 'Bart':['Milhouse'], 'Milhouse':['Bart','Lisa'], 'Lisa':['Milhouse'],…arrow_forwardApply the Binary Search method to find a target using Recursive method.arrow_forward
- An n x n matrix is called a positive Markov matrix if eachelement is positive and the sum of the elements in each column is 1. Write thefollowing method to check whether a matrix is a Markov matrix:public static boolean isMarkovMatrix(double[][] m) Write a test program that prompts the user to enter a 3 x 3 matrix of doublevalues and tests whether it is a Markov matrix. Here are sample runs: Enter a 3−by−3 matrix row by row:0.15 0.875 0.375 ↵Enter0.55 0.005 0.225 ↵Enter0.30 0.12 0.4 ↵EnterIt is a Markov matrix Enter a 3−by−3 matrix row by row:0.95 −0.875 0.375 ↵Enter0.65 0.005 0.225 ↵Enter0.30 0.22 −0.4 ↵EnterIt is not a Markov matrixarrow_forwardIN JAVA, USING RECURSION PLEASE Create a method int[][] generateMatrix(int row, int col, int boundary1, int boundary2, int iteration) that generates a random matrix with random numbers between [min(boundary1, boundary2), max(boundary1, boundary2)). The sum of the diagonal and the sub-diagonal should be the same. If not, regenerate it again, until a matrix that satisfies the condition is generated (return that matrix). If you try iteration times and none of the matrixes satisfy the condition, return null.arrow_forwardWrite a program that inputs a matrix and displays the trans- pose of that matrix. A transpose of a matrix is obtained by converting all the rows of a given matrix into columns and vice versa.arrow_forward
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